1,1,154,119,0.7244427,"\int (a+a \sin (c+d x))^{7/2} \, dx","Integrate[(a + a*Sin[c + d*x])^(7/2),x]","-\frac{a^3 (\sin (c+d x)+1)^3 \sqrt{a (\sin (c+d x)+1)} \left(-1225 \sin \left(\frac{1}{2} (c+d x)\right)+245 \sin \left(\frac{3}{2} (c+d x)\right)+49 \sin \left(\frac{5}{2} (c+d x)\right)-5 \sin \left(\frac{7}{2} (c+d x)\right)+1225 \cos \left(\frac{1}{2} (c+d x)\right)+245 \cos \left(\frac{3}{2} (c+d x)\right)-49 \cos \left(\frac{5}{2} (c+d x)\right)-5 \cos \left(\frac{7}{2} (c+d x)\right)\right)}{140 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","-\frac{256 a^4 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}",1,"-1/140*(a^3*(1 + Sin[c + d*x])^3*Sqrt[a*(1 + Sin[c + d*x])]*(1225*Cos[(c + d*x)/2] + 245*Cos[(3*(c + d*x))/2] - 49*Cos[(5*(c + d*x))/2] - 5*Cos[(7*(c + d*x))/2] - 1225*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] + 49*Sin[(5*(c + d*x))/2] - 5*Sin[(7*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","A",1
2,1,117,89,0.3277994,"\int (a+a \sin (c+d x))^{5/2} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2),x]","-\frac{(a (\sin (c+d x)+1))^{5/2} \left(-150 \sin \left(\frac{1}{2} (c+d x)\right)+25 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+150 \cos \left(\frac{1}{2} (c+d x)\right)+25 \cos \left(\frac{3}{2} (c+d x)\right)-3 \cos \left(\frac{5}{2} (c+d x)\right)\right)}{30 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"-1/30*((a*(1 + Sin[c + d*x]))^(5/2)*(150*Cos[(c + d*x)/2] + 25*Cos[(3*(c + d*x))/2] - 3*Cos[(5*(c + d*x))/2] - 150*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
3,1,89,59,0.1480218,"\int (a+a \sin (c+d x))^{3/2} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2),x]","-\frac{(a (\sin (c+d x)+1))^{3/2} \left(-9 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+9 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}",1,"-1/3*((a*(1 + Sin[c + d*x]))^(3/2)*(9*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] - 9*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
4,1,65,26,0.0382649,"\int \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"(2*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
5,1,73,47,0.073962,"\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Sin[c + d*x]],x]","\frac{(2+2 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)}{d \sqrt{a (\sin (c+d x)+1)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2 + 2*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
6,1,108,77,0.1782381,"\int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+(1+i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{2 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2] + (1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(1 + Sin[c + d*x])))/(2*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
7,1,196,107,0.1664516,"\int \frac{1}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(8 \sin \left(\frac{1}{2} (c+d x)\right)-3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+6 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(3+3 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{16 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{3 \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(8*Sin[(c + d*x)/2] - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 6*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (3 + 3*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4))/(16*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
8,1,341,67,2.704908,"\int (a+a \sin (c+d x))^{4/3} \, dx","Integrate[(a + a*Sin[c + d*x])^(4/3),x]","\frac{(a (\sin (c+d x)+1))^{4/3} \left(-\frac{3}{2} (\cos (c+d x)-5) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{3 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{4 \sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"(((-3*(-5 + Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/2 + (3*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(4*Sqrt[2]*E^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]))*(a*(1 + Sin[c + d*x]))^(4/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
9,1,124,66,0.2223041,"\int (a+a \sin (c+d x))^{2/3} \, dx","Integrate[(a + a*Sin[c + d*x])^(2/3),x]","-\frac{3 (a (\sin (c+d x)+1))^{2/3} \left(\sqrt{2-2 \sin (c+d x)}-2 \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \sqrt{2-2 \sin (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 \sqrt[6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{7/6}}",1,"(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + Sqrt[2 - 2*Sin[c + d*x]])*(a*(1 + Sin[c + d*x]))^(2/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Sqrt[2 - 2*Sin[c + d*x]])","A",1
10,1,270,66,2.6150465,"\int \sqrt[3]{a+a \sin (c+d x)} \, dx","Integrate[(a + a*Sin[c + d*x])^(1/3),x]","\frac{\sqrt[3]{a (\sin (c+d x)+1)} \left(3+\frac{\left(\frac{3}{10}+\frac{3 i}{10}\right) (-1)^{3/4} e^{-i (c+d x)} \left(-2 \left(1+i e^{-i (c+d x)}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+5 i \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right) \sqrt{\cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)}\right)}{\sqrt{2} \left(1+i e^{-i (c+d x)}\right)^{2/3} \sqrt{i e^{-i (c+d x)} \left(e^{i (c+d x)}-i\right)^2}}\right)}{d}","-\frac{2^{5/6} \cos (c+d x) \sqrt[3]{a \sin (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (\sin (c+d x)+1)^{5/6}}",1,"((3 + ((3/10 + (3*I)/10)*(-1)^(3/4)*(20*E^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(Sqrt[2]*E^(I*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]))*(a*(1 + Sin[c + d*x]))^(1/3))/d","C",0
11,1,70,66,0.1127879,"\int \frac{1}{\sqrt[3]{a+a \sin (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-1/3),x]","\frac{3 \sqrt{2} \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)}{d \sqrt{1-\sin (c+d x)} \sqrt[3]{a (\sin (c+d x)+1)}}","-\frac{\sqrt[6]{2} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(3*Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2])/(d*Sqrt[1 - Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(1/3))","A",1
12,1,604,66,6.1068384,"\int \frac{1}{(a+a \sin (c+d x))^{2/3}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-2/3),x]","\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\frac{3 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-3\right)}{d (a (\sin (c+d x)+1))^{2/3}}-\frac{2 \sqrt{2} \sqrt[6]{\sin (c+d x)+1} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\frac{3 \sin \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right) \cos ^2\left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2\left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)\right)}{5 \sqrt{\sin ^2\left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)} \sqrt[6]{\cos \left(2 \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)\right)+1}}-\frac{i \left(-\frac{3 i \left(e^{-i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}+e^{i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}\right)^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}\right)}{2^{2/3} \left(1+e^{2 i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}\right)^{2/3}}-\frac{3 i e^{i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)} \sqrt[3]{1+e^{2 i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}\right)}{2\ 2^{2/3} \sqrt[3]{e^{-i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}+e^{i \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}}}\right) \sqrt[3]{\cos \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)}}{2 \sqrt[6]{\cos \left(2 \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)\right)+1}}\right)}{d (a (\sin (c+d x)+1))^{2/3}}","-\frac{\sqrt[6]{\sin (c+d x)+1} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{\sqrt[6]{2} d (a \sin (c+d x)+a)^{2/3}}",1,"(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-3 + (3*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(d*(a*(1 + Sin[c + d*x]))^(2/3)) - (2*Sqrt[2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x])^(1/6)*(((-1/2*I)*Cos[Pi/4 + (-c - d*x)/2]^(1/3)*(((-3*I)*(E^((-I)*(Pi/4 + (-c - d*x)/2)) + E^(I*(Pi/4 + (-c - d*x)/2)))^(2/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(Pi/4 + (-c - d*x)/2))])/(2^(2/3)*(1 + E^((2*I)*(Pi/4 + (-c - d*x)/2)))^(2/3)) - (((3*I)/2)*E^(I*(Pi/4 + (-c - d*x)/2))*(1 + E^((2*I)*(Pi/4 + (-c - d*x)/2)))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(Pi/4 + (-c - d*x)/2))])/(2^(2/3)*(E^((-I)*(Pi/4 + (-c - d*x)/2)) + E^(I*(Pi/4 + (-c - d*x)/2)))^(1/3))))/(1 + Cos[2*(Pi/4 + (-c - d*x)/2)])^(1/6) + (3*Cos[Pi/4 + (-c - d*x)/2]^2*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[Pi/4 + (-c - d*x)/2]^2]*Sin[Pi/4 + (-c - d*x)/2])/(5*(1 + Cos[2*(Pi/4 + (-c - d*x)/2)])^(1/6)*Sqrt[Sin[Pi/4 + (-c - d*x)/2]^2])))/(d*(a*(1 + Sin[c + d*x]))^(2/3))","C",1
13,1,130,69,0.2942494,"\int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx","Integrate[(a + a*Sin[c + d*x])^(-4/3),x]","-\frac{3 \left(\sqrt{2-2 \sin (c+d x)}-2 (\sin (c+d x)+1) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{5 d \sqrt{2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}","-\frac{\cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\sin (c+d x)+1} \sqrt[3]{a \sin (c+d x)+a}}",1,"(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sqrt[2 - 2*Sin[c + d*x]] - 2*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x])))/(5*d*Sqrt[2 - 2*Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^(4/3))","A",1
14,1,90,74,0.1926347,"\int (a+a \sin (c+d x))^n \, dx","Integrate[(a + a*Sin[c + d*x])^n,x]","\frac{\sqrt{2} \cos (c+d x) (a (\sin (c+d x)+1))^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{(2 d n+d) \sqrt{1-\sin (c+d x)}}","-\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac{1}{2}} (a \sin (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"(Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(a*(1 + Sin[c + d*x]))^n)/((d + 2*d*n)*Sqrt[1 - Sin[c + d*x]])","A",1
15,1,91,74,0.1510734,"\int (a-a \sin (c+d x))^n \, dx","Integrate[(a - a*Sin[c + d*x])^n,x]","\frac{\cos (c+d x) \cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)^{-n-\frac{1}{2}} (a-a \sin (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{d}","\frac{2^{n+\frac{1}{2}} \cos (c+d x) (1-\sin (c+d x))^{-n-\frac{1}{2}} (a-a \sin (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d}",1,"(Cos[c + d*x]*(Cos[(2*c + Pi + 2*d*x)/4]^2)^(-1/2 - n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(a - a*Sin[c + d*x])^n)/d","A",0
16,1,90,60,0.112618,"\int (2+2 \sin (c+d x))^n \, dx","Integrate[(2 + 2*Sin[c + d*x])^n,x]","\frac{2^{n+\frac{1}{2}} \cos (c+d x) (\sin (c+d x)+1)^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{(2 d n+d) \sqrt{1-\sin (c+d x)}}","-\frac{2^{2 n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(1 + Sin[c + d*x])^n)/((d + 2*d*n)*Sqrt[1 - Sin[c + d*x]])","A",1
17,1,90,59,0.1354002,"\int (2-2 \sin (c+d x))^n \, dx","Integrate[(2 - 2*Sin[c + d*x])^n,x]","\frac{\cos (c+d x) (2-2 \sin (c+d x))^n \cos ^2\left(\frac{1}{4} (2 c+2 d x+\pi )\right)^{-n-\frac{1}{2}} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)}{d}","\frac{2^{2 n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"(Cos[c + d*x]*(Cos[(2*c + Pi + 2*d*x)/4]^2)^(-1/2 - n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (Cos[c + d*x]^2*Csc[(2*c - Pi + 2*d*x)/4]^2)/4]*(2 - 2*Sin[c + d*x])^n)/d","A",0
18,1,56,31,0.0270848,"\int \frac{1}{5+3 \sin (c+d x)} \, dx","Integrate[(5 + 3*Sin[c + d*x])^(-1),x]","\frac{\tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{2 d}","\frac{\tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{2 d}+\frac{x}{4}",1,"ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])]/(2*d)","A",1
19,1,91,56,0.1325221,"\int \frac{1}{(5+3 \sin (c+d x))^2} \, dx","Integrate[(5 + 3*Sin[c + d*x])^(-2),x]","\frac{\frac{6 (-3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)+5}+25 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{160 d}","\frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{32 d}+\frac{5 x}{64}",1,"(25*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (6*(-5 + 5*Cos[c + d*x] - 3*Sin[c + d*x]))/(5 + 3*Sin[c + d*x]))/(160*d)","A",1
20,1,113,81,0.3289183,"\int \frac{1}{(5+3 \sin (c+d x))^3} \, dx","Integrate[(5 + 3*Sin[c + d*x])^(-3),x]","\frac{\frac{546 \cos (c+d x)+9 (-60 \sin (c+d x)+15 \sin (2 (c+d x))+9 \cos (2 (c+d x))-59)}{(3 \sin (c+d x)+5)^2}+59 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{1024 d}","\frac{45 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)}+\frac{3 \cos (c+d x)}{32 d (3 \sin (c+d x)+5)^2}+\frac{59 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{1024 d}+\frac{59 x}{2048}",1,"(59*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (546*Cos[c + d*x] + 9*(-59 + 9*Cos[2*(c + d*x)] - 60*Sin[c + d*x] + 15*Sin[2*(c + d*x)]))/(5 + 3*Sin[c + d*x])^2)/(1024*d)","A",1
21,1,133,106,0.4586221,"\int \frac{1}{(5+3 \sin (c+d x))^4} \, dx","Integrate[(5 + 3*Sin[c + d*x])^(-4),x]","\frac{\frac{-305091 \sin (c+d x)+105300 \sin (2 (c+d x))+8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)+5)^3}+1925 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{81920 d}","\frac{311 \cos (c+d x)}{8192 d (3 \sin (c+d x)+5)}+\frac{25 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)^2}+\frac{\cos (c+d x)}{16 d (3 \sin (c+d x)+5)^3}+\frac{385 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{16384 d}+\frac{385 x}{32768}",1,"(1925*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (-239470 + 219735*Cos[c + d*x] + 83970*Cos[2*(c + d*x)] - 13995*Cos[3*(c + d*x)] - 305091*Sin[c + d*x] + 105300*Sin[2*(c + d*x)] + 8397*Sin[3*(c + d*x)])/(2*(5 + 3*Sin[c + d*x])^3))/(81920*d)","A",1
22,1,56,33,0.0275613,"\int \frac{1}{5-3 \sin (c+d x)} \, dx","Integrate[(5 - 3*Sin[c + d*x])^(-1),x]","-\frac{\tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{2 d}","\frac{x}{4}-\frac{\tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{2 d}",1,"-1/2*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])]/d","A",1
23,1,91,58,0.1444831,"\int \frac{1}{(5-3 \sin (c+d x))^2} \, dx","Integrate[(5 - 3*Sin[c + d*x])^(-2),x]","\frac{\frac{6 (3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)-5}-25 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{160 d}","-\frac{3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}-\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{32 d}+\frac{5 x}{64}",1,"(-25*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (6*(-5 + 5*Cos[c + d*x] + 3*Sin[c + d*x]))/(-5 + 3*Sin[c + d*x]))/(160*d)","A",1
24,1,113,83,0.4453535,"\int \frac{1}{(5-3 \sin (c+d x))^3} \, dx","Integrate[(5 - 3*Sin[c + d*x])^(-3),x]","-\frac{\frac{546 \cos (c+d x)+9 (60 \sin (c+d x)-15 \sin (2 (c+d x))+9 \cos (2 (c+d x))-59)}{(5-3 \sin (c+d x))^2}+59 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{1024 d}","-\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}-\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}-\frac{59 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{1024 d}+\frac{59 x}{2048}",1,"-1/1024*(59*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (546*Cos[c + d*x] + 9*(-59 + 9*Cos[2*(c + d*x)] + 60*Sin[c + d*x] - 15*Sin[2*(c + d*x)]))/(5 - 3*Sin[c + d*x])^2)/d","A",1
25,1,133,108,0.4803719,"\int \frac{1}{(5-3 \sin (c+d x))^4} \, dx","Integrate[(5 - 3*Sin[c + d*x])^(-4),x]","\frac{\frac{305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)-5)^3}-1925 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{81920 d}","-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{16384 d}+\frac{385 x}{32768}",1,"(-1925*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (-239470 + 219735*Cos[c + d*x] + 83970*Cos[2*(c + d*x)] - 13995*Cos[3*(c + d*x)] + 305091*Sin[c + d*x] - 105300*Sin[2*(c + d*x)] - 8397*Sin[3*(c + d*x)])/(2*(-5 + 3*Sin[c + d*x])^3))/(81920*d)","A",1
26,1,56,33,0.0242104,"\int \frac{1}{-5+3 \sin (c+d x)} \, dx","Integrate[(-5 + 3*Sin[c + d*x])^(-1),x]","\frac{\tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{2 d}","\frac{\tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{2 d}-\frac{x}{4}",1,"ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])]/(2*d)","A",1
27,1,91,58,0.0242651,"\int \frac{1}{(-5+3 \sin (c+d x))^2} \, dx","Integrate[(-5 + 3*Sin[c + d*x])^(-2),x]","\frac{\frac{6 (3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)-5}-25 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{160 d}","-\frac{3 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}-\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{32 d}+\frac{5 x}{64}",1,"(-25*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (6*(-5 + 5*Cos[c + d*x] + 3*Sin[c + d*x]))/(-5 + 3*Sin[c + d*x]))/(160*d)","A",1
28,1,113,83,0.2875575,"\int \frac{1}{(-5+3 \sin (c+d x))^3} \, dx","Integrate[(-5 + 3*Sin[c + d*x])^(-3),x]","\frac{\frac{546 \cos (c+d x)+9 (60 \sin (c+d x)-15 \sin (2 (c+d x))+9 \cos (2 (c+d x))-59)}{(5-3 \sin (c+d x))^2}+59 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{1024 d}","\frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{59 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{1024 d}-\frac{59 x}{2048}",1,"(59*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (546*Cos[c + d*x] + 9*(-59 + 9*Cos[2*(c + d*x)] + 60*Sin[c + d*x] - 15*Sin[2*(c + d*x)]))/(5 - 3*Sin[c + d*x])^2)/(1024*d)","A",1
29,1,133,108,0.0982422,"\int \frac{1}{(-5+3 \sin (c+d x))^4} \, dx","Integrate[(-5 + 3*Sin[c + d*x])^(-4),x]","\frac{\frac{305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)-5)^3}-1925 \tan ^{-1}\left(\frac{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{81920 d}","-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left(\frac{\cos (c+d x)}{3-\sin (c+d x)}\right)}{16384 d}+\frac{385 x}{32768}",1,"(-1925*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (-239470 + 219735*Cos[c + d*x] + 83970*Cos[2*(c + d*x)] - 13995*Cos[3*(c + d*x)] + 305091*Sin[c + d*x] - 105300*Sin[2*(c + d*x)] - 8397*Sin[3*(c + d*x)])/(2*(-5 + 3*Sin[c + d*x])^3))/(81920*d)","A",1
30,1,56,31,0.0263286,"\int \frac{1}{-5-3 \sin (c+d x)} \, dx","Integrate[(-5 - 3*Sin[c + d*x])^(-1),x]","-\frac{\tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{2 d}","-\frac{\tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{2 d}-\frac{x}{4}",1,"-1/2*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])]/d","A",1
31,1,91,56,0.1157817,"\int \frac{1}{(-5-3 \sin (c+d x))^2} \, dx","Integrate[(-5 - 3*Sin[c + d*x])^(-2),x]","\frac{\frac{6 (-3 \sin (c+d x)+5 \cos (c+d x)-5)}{3 \sin (c+d x)+5}+25 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{160 d}","\frac{3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{32 d}+\frac{5 x}{64}",1,"(25*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (6*(-5 + 5*Cos[c + d*x] - 3*Sin[c + d*x]))/(5 + 3*Sin[c + d*x]))/(160*d)","A",1
32,1,114,81,0.3588022,"\int \frac{1}{(-5-3 \sin (c+d x))^3} \, dx","Integrate[(-5 - 3*Sin[c + d*x])^(-3),x]","\frac{\frac{3 (3 (60 \sin (c+d x)-15 \sin (2 (c+d x))-9 \cos (2 (c+d x))+59)-182 \cos (c+d x))}{(3 \sin (c+d x)+5)^2}-59 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{1024 d}","-\frac{45 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)}-\frac{3 \cos (c+d x)}{32 d (3 \sin (c+d x)+5)^2}-\frac{59 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{1024 d}-\frac{59 x}{2048}",1,"(-59*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (3*(-182*Cos[c + d*x] + 3*(59 - 9*Cos[2*(c + d*x)] + 60*Sin[c + d*x] - 15*Sin[2*(c + d*x)])))/(5 + 3*Sin[c + d*x])^2)/(1024*d)","A",1
33,1,133,106,0.3261662,"\int \frac{1}{(-5-3 \sin (c+d x))^4} \, dx","Integrate[(-5 - 3*Sin[c + d*x])^(-4),x]","\frac{\frac{-305091 \sin (c+d x)+105300 \sin (2 (c+d x))+8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)+5)^3}+1925 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{81920 d}","\frac{311 \cos (c+d x)}{8192 d (3 \sin (c+d x)+5)}+\frac{25 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)^2}+\frac{\cos (c+d x)}{16 d (3 \sin (c+d x)+5)^3}+\frac{385 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{16384 d}+\frac{385 x}{32768}",1,"(1925*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])] + (-239470 + 219735*Cos[c + d*x] + 83970*Cos[2*(c + d*x)] - 13995*Cos[3*(c + d*x)] - 305091*Sin[c + d*x] + 105300*Sin[2*(c + d*x)] + 8397*Sin[3*(c + d*x)])/(2*(5 + 3*Sin[c + d*x])^3))/(81920*d)","A",1
34,1,63,63,0.0256625,"\int \frac{1}{3+5 \sin (c+d x)} \, dx","Integrate[(3 + 5*Sin[c + d*x])^(-1),x]","\frac{\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"-1/4*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/d + Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]]/(4*d)","A",1
35,1,126,88,0.2065167,"\int \frac{1}{(3+5 \sin (c+d x))^2} \, dx","Integrate[(3 + 5*Sin[c + d*x])^(-2),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{3}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{1}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}\right)+9 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","-\frac{5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*(Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]]) + 20*Sin[(c + d*x)/2]*((3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-1) + 3/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(192*d)","A",1
36,1,180,113,0.502348,"\int \frac{1}{(3+5 \sin (c+d x))^3} \, dx","Integrate[(3 + 5*Sin[c + d*x])^(-3),x]","\frac{\sin \left(\frac{1}{2} (c+d x)\right) \left(-\frac{180}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{60}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}\right)+\frac{40}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{40}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)+43 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","\frac{45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}-\frac{5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(-43*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 43*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]] + 40/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 40/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^2 + Sin[(c + d*x)/2]*(-60/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 180/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(2048*d)","A",1
37,1,235,138,0.8911893,"\int \frac{1}{(3+5 \sin (c+d x))^4} \, dx","Integrate[(3 + 5*Sin[c + d*x])^(-4),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{597}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{240}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{199}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}+\frac{80}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)-\frac{2320}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{720}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2511 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-2511 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{294912 d}","-\frac{995 \cos (c+d x)}{24576 d (5 \sin (c+d x)+3)}+\frac{25 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)^2}-\frac{5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}+\frac{279 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(2511*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2511*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]] - 2320/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 720/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^2 + 20*Sin[(c + d*x)/2]*(80/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 199/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 240/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^3 + 597/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(294912*d)","A",1
38,1,65,65,0.0250675,"\int \frac{1}{3-5 \sin (c+d x)} \, dx","Integrate[(3 - 5*Sin[c + d*x])^(-1),x]","\frac{\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"-1/4*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]]/d + Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d)","A",1
39,1,130,90,0.2152893,"\int \frac{1}{(3-5 \sin (c+d x))^2} \, dx","Integrate[(3 - 5*Sin[c + d*x])^(-2),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{3}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}\right)+9 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*(Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + 20*(3/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) + (3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-1))*Sin[(c + d*x)/2])/(192*d)","A",1
40,1,184,115,0.5528562,"\int \frac{1}{(3-5 \sin (c+d x))^3} \, dx","Integrate[(3 - 5*Sin[c + d*x])^(-3),x]","\frac{\sin \left(\frac{1}{2} (c+d x)\right) \left(-\frac{60}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{180}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}\right)+\frac{40}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{40}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)+43 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","-\frac{45 \cos (c+d x)}{512 d (3-5 \sin (c+d x))}+\frac{5 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}-\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(-43*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] + 43*Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 40/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^2 + (-180/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) - 60/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sin[(c + d*x)/2] - 40/(-3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(2048*d)","A",1
41,1,241,140,0.9430278,"\int \frac{1}{(3-5 \sin (c+d x))^4} \, dx","Integrate[(3 - 5*Sin[c + d*x])^(-4),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{199}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{80}{\left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{597}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}+\frac{240}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)-\frac{720}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2320}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2511 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)-2511 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{294912 d}","\frac{995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac{5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}+\frac{279 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(2511*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - 2511*Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 720/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^2 + 20*(240/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^3 + 597/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) + 80/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + 199/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sin[(c + d*x)/2] + 2320/(-3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(294912*d)","A",1
42,1,65,65,0.0210244,"\int \frac{1}{-3+5 \sin (c+d x)} \, dx","Integrate[(-3 + 5*Sin[c + d*x])^(-1),x]","\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]]/(4*d) - Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d)","A",1
43,1,130,90,0.019881,"\int \frac{1}{(-3+5 \sin (c+d x))^2} \, dx","Integrate[(-3 + 5*Sin[c + d*x])^(-2),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{3}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}\right)+9 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*(Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + 20*(3/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) + (3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-1))*Sin[(c + d*x)/2])/(192*d)","A",1
44,1,183,115,0.5638595,"\int \frac{1}{(-3+5 \sin (c+d x))^3} \, dx","Integrate[(-3 + 5*Sin[c + d*x])^(-3),x]","\frac{60 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{3}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}\right)-\frac{40}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{40}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)-43 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","\frac{45 \cos (c+d x)}{512 d (3-5 \sin (c+d x))}-\frac{5 \cos (c+d x)}{32 d (3-5 \sin (c+d x))^2}+\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(43*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - 43*Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 40/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^2 + 60*(3/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) + (3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-1))*Sin[(c + d*x)/2] + 40/(-3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(2048*d)","A",1
45,1,241,140,0.0260324,"\int \frac{1}{(-3+5 \sin (c+d x))^4} \, dx","Integrate[(-3 + 5*Sin[c + d*x])^(-4),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{199}{3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{80}{\left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{597}{\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)}+\frac{240}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)-\frac{720}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2320}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2511 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)-2511 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{294912 d}","\frac{995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac{5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}+\frac{279 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(3 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(2511*Log[Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]] - 2511*Log[3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 720/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^2 + 20*(240/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2])^3 + 597/(Cos[(c + d*x)/2] - 3*Sin[(c + d*x)/2]) + 80/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + 199/(3*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sin[(c + d*x)/2] + 2320/(-3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(294912*d)","A",1
46,1,63,63,0.0256971,"\int \frac{1}{-3-5 \sin (c+d x)} \, dx","Integrate[(-3 - 5*Sin[c + d*x])^(-1),x]","\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d) - Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]]/(4*d)","A",1
47,1,126,88,0.1350774,"\int \frac{1}{(-3-5 \sin (c+d x))^2} \, dx","Integrate[(-3 - 5*Sin[c + d*x])^(-2),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{3}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{1}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}\right)+9 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","-\frac{5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*(Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]]) + 20*Sin[(c + d*x)/2]*((3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-1) + 3/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(192*d)","A",1
48,1,179,113,0.5038714,"\int \frac{1}{(-3-5 \sin (c+d x))^3} \, dx","Integrate[(-3 - 5*Sin[c + d*x])^(-3),x]","\frac{60 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{3}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{1}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}\right)-\frac{40}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{40}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-43 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","-\frac{45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}+\frac{5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}+\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(43*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 43*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]] - 40/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 40/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^2 + 60*Sin[(c + d*x)/2]*((3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-1) + 3/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(2048*d)","A",1
49,1,235,138,0.0259493,"\int \frac{1}{(-3-5 \sin (c+d x))^4} \, dx","Integrate[(-3 - 5*Sin[c + d*x])^(-4),x]","\frac{20 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{597}{3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{240}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{199}{\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)}+\frac{80}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)-\frac{2320}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{720}{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2511 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-2511 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{294912 d}","-\frac{995 \cos (c+d x)}{24576 d (5 \sin (c+d x)+3)}+\frac{25 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)^2}-\frac{5 \cos (c+d x)}{48 d (5 \sin (c+d x)+3)^3}+\frac{279 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(2511*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2511*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]] - 2320/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 720/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^2 + 20*Sin[(c + d*x)/2]*(80/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 199/(3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 240/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])^3 + 597/(Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2])))/(294912*d)","A",1
50,1,220,256,1.0627418,"\int (a+b \sin (c+d x))^{7/2} \, dx","Integrate[(a + b*Sin[c + d*x])^(7/2),x]","\frac{4 \left(71 a^4-46 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-b \cos (c+d x) \left(488 a^3+b \left(752 a^2+145 b^2\right) \sin (c+d x)-162 a b^2 \cos (2 (c+d x))+262 a b^2-15 b^3 \sin (3 (c+d x))\right)-64 a \left(11 a^3+11 a^2 b+13 a b^2+13 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{210 d \sqrt{a+b \sin (c+d x)}}","-\frac{2 b \left(71 a^2+25 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}+\frac{32 a \left(11 a^2+13 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \left(71 a^4-46 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \sin (c+d x)}}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}",1,"(-64*a*(11*a^3 + 11*a^2*b + 13*a*b^2 + 13*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 4*(71*a^4 - 46*a^2*b^2 - 25*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(488*a^3 + 262*a*b^2 - 162*a*b^2*Cos[2*(c + d*x)] + b*(752*a^2 + 145*b^2)*Sin[c + d*x] - 15*b^3*Sin[3*(c + d*x)]))/(210*d*Sqrt[a + b*Sin[c + d*x]])","A",1
51,1,185,207,0.9857363,"\int (a+b \sin (c+d x))^{5/2} \, dx","Integrate[(a + b*Sin[c + d*x])^(5/2),x]","\frac{b \cos (c+d x) \left(-22 a^2-28 a b \sin (c+d x)+3 b^2 \cos (2 (c+d x))-3 b^2\right)+16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-2 \left(23 a^3+23 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \sin (c+d x)}}","-\frac{16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}-\frac{16 a b \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 d}",1,"(-2*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 16*a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-22*a^2 - 3*b^2 + 3*b^2*Cos[2*(c + d*x)] - 28*a*b*Sin[c + d*x]))/(15*d*Sqrt[a + b*Sin[c + d*x]])","A",1
52,1,142,167,0.8438034,"\int (a+b \sin (c+d x))^{3/2} \, dx","Integrate[(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-2 b \cos (c+d x) (a+b \sin (c+d x))-8 a (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 b \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 d}+\frac{8 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(-2*b*Cos[c + d*x]*(a + b*Sin[c + d*x]) - 8*a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 2*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]])","A",1
53,1,61,62,2.2208807,"\int \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{2 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(-2*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])","A",1
54,1,61,62,0.0705311,"\int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"(-2*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",1
55,1,87,111,0.2994505,"\int \frac{1}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-3/2),x]","\frac{2 b \cos (c+d x)-2 (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{d (a-b) (a+b) \sqrt{a+b \sin (c+d x)}}","\frac{2 b \cos (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{2 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*b*Cos[c + d*x] - 2*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/((a - b)*(a + b)*d*Sqrt[a + b*Sin[c + d*x]])","A",1
56,1,166,231,0.9349504,"\int \frac{1}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-5/2),x]","\frac{2 \left(b \cos (c+d x) \left(5 a^2+4 a b \sin (c+d x)-b^2\right)+(a-b) (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-4 a (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{3 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^{3/2}}","\frac{8 a b \cos (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \cos (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{8 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*(-4*a*(a + b)^2*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (a - b)*(a + b)^2*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + b*Cos[c + d*x]*(5*a^2 - b^2 + 4*a*b*Sin[c + d*x])))/(3*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^(3/2))","A",1
57,1,198,292,1.4179077,"\int \frac{1}{(a+b \sin (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-7/2),x]","\frac{2 \left(\frac{b \cos (c+d x) \left(34 a^4+b^2 \left(23 a^2+9 b^2\right) \sin ^2(c+d x)+2 a b \left(27 a^2+5 b^2\right) \sin (c+d x)-5 a^2 b^2+3 b^4\right)}{\left(a^2-b^2\right)^3}-\frac{\left(\frac{a+b \sin (c+d x)}{a+b}\right)^{5/2} \left(\left(23 a^2+9 b^2\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+8 a (b-a) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{(a-b)^3}\right)}{15 d (a+b \sin (c+d x))^{5/2}}","\frac{2 b \left(23 a^2+9 b^2\right) \cos (c+d x)}{15 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}+\frac{16 a b \cos (c+d x)}{15 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}+\frac{2 b \cos (c+d x)}{5 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}-\frac{16 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*(-((((23*a^2 + 9*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + 8*a*(-a + b)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(5/2))/(a - b)^3) + (b*Cos[c + d*x]*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*Sin[c + d*x] + b^2*(23*a^2 + 9*b^2)*Sin[c + d*x]^2))/(a^2 - b^2)^3))/(15*d*(a + b*Sin[c + d*x])^(5/2))","A",1
58,1,244,109,1.6843635,"\int (a+b \sin (c+d x))^{4/3} \, dx","Integrate[(a + b*Sin[c + d*x])^(4/3),x]","-\frac{3 \sec (c+d x) \sqrt[3]{a+b \sin (c+d x)} \left(4 \left(a^2-b^2\right) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)-5 a \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)+4 b^2 \cos ^2(c+d x)\right)}{16 b d}","-\frac{\sqrt{2} (a+b) \cos (c+d x) \sqrt[3]{a+b \sin (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1} \sqrt[3]{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(-3*Sec[c + d*x]*(a + b*Sin[c + d*x])^(1/3)*(4*b^2*Cos[c + d*x]^2 + 4*(a^2 - b^2)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)] - 5*a*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])))/(16*b*d)","B",0
59,1,118,106,0.2100245,"\int (a+b \sin (c+d x))^{2/3} \, dx","Integrate[(a + b*Sin[c + d*x])^(2/3),x]","\frac{3 \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{5/3} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{5 b d}","-\frac{\sqrt{2} \cos (c+d x) (a+b \sin (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1} \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{2/3}}",1,"(3*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(5/3))/(5*b*d)","A",0
60,1,118,106,0.2075012,"\int \sqrt[3]{a+b \sin (c+d x)} \, dx","Integrate[(a + b*Sin[c + d*x])^(1/3),x]","\frac{3 \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{4/3} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{4 b d}","-\frac{\sqrt{2} \cos (c+d x) \sqrt[3]{a+b \sin (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1} \sqrt[3]{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(3*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(4/3))/(4*b*d)","A",0
61,1,118,106,0.1952483,"\int \frac{1}{\sqrt[3]{a+b \sin (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-1/3),x]","\frac{3 \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{2/3} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{2 b d}","-\frac{\sqrt{2} \cos (c+d x) \sqrt[3]{\frac{a+b \sin (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1} \sqrt[3]{a+b \sin (c+d x)}}",1,"(3*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(2/3))/(2*b*d)","A",0
62,1,116,106,0.1858938,"\int \frac{1}{(a+b \sin (c+d x))^{2/3}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-2/3),x]","\frac{3 \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \sqrt[3]{a+b \sin (c+d x)} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d}","-\frac{\sqrt{2} \cos (c+d x) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1} (a+b \sin (c+d x))^{2/3}}",1,"(3*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(1/3))/(b*d)","A",0
63,1,262,111,1.9271001,"\int \frac{1}{(a+b \sin (c+d x))^{4/3}} \, dx","Integrate[(a + b*Sin[c + d*x])^(-4/3),x]","-\frac{3 \sec (c+d x) \left(5 a \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} (a+b \sin (c+d x)) F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)-2 \left(2 \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^2 F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)+5 b^2 \cos ^2(c+d x)\right)\right)}{10 b d \left(a^2-b^2\right) \sqrt[3]{a+b \sin (c+d x)}}","-\frac{\sqrt{2} \cos (c+d x) \sqrt[3]{\frac{a+b \sin (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sin (c+d x)+1} \sqrt[3]{a+b \sin (c+d x)}}",1,"(-3*Sec[c + d*x]*(5*a*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))]*(a + b*Sin[c + d*x]) - 2*(5*b^2*Cos[c + d*x]^2 + 2*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^2)))/(10*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(1/3))","B",0
64,1,120,104,0.249913,"\int (a+b \sin (c+d x))^n \, dx","Integrate[(a + b*Sin[c + d*x])^n,x]","\frac{\sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} (a+b \sin (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (n+1)}","-\frac{\sqrt{2} \cos (c+d x) (a+b \sin (c+d x))^n \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{b (1-\sin (c+d x))}{a+b}\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(1 + n))","A",0
65,1,83,72,0.1561711,"\int (3+4 \sin (c+d x))^n \, dx","Integrate[(3 + 4*Sin[c + d*x])^n,x]","\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) (4 \sin (c+d x)+3)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;-4 \sin (c+d x)-3,\frac{1}{7} (4 \sin (c+d x)+3)\right)}{\sqrt{7} d (n+1)}","-\frac{\sqrt{2} 7^n \cos (c+d x) F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),\frac{4}{7} (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, -3 - 4*Sin[c + d*x], (3 + 4*Sin[c + d*x])/7]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*(3 + 4*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n))","A",0
66,1,84,69,0.1688519,"\int (3-4 \sin (c+d x))^n \, dx","Integrate[(3 - 4*Sin[c + d*x])^n,x]","-\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) (3-4 \sin (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (3-4 \sin (c+d x)),4 \sin (c+d x)-3\right)}{\sqrt{7} d (n+1)}","\frac{\sqrt{2} 7^n \cos (c+d x) F_1\left(\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{4}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"-((AppellF1[1 + n, 1/2, 1/2, 2 + n, (3 - 4*Sin[c + d*x])/7, -3 + 4*Sin[c + d*x]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*(3 - 4*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n)))","A",0
67,1,99,64,0.1310162,"\int (4+3 \sin (c+d x))^n \, dx","Integrate[(4 + 3*Sin[c + d*x])^n,x]","\frac{\sqrt{-\sin (c+d x)-1} \sqrt{1-\sin (c+d x)} \sec (c+d x) (3 \sin (c+d x)+4)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right)}{\sqrt{7} d (n+1)}","\frac{\sqrt{2} \cos (c+d x) F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1),-3 (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, 4 + 3*Sin[c + d*x], (4 + 3*Sin[c + d*x])/7]*Sec[c + d*x]*Sqrt[-1 - Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]*(4 + 3*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n))","A",0
68,1,96,69,0.1349089,"\int (4-3 \sin (c+d x))^n \, dx","Integrate[(4 - 3*Sin[c + d*x])^n,x]","-\frac{\sqrt{\sin (c+d x)-1} \sqrt{\sin (c+d x)+1} \sec (c+d x) (4-3 \sin (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right)}{\sqrt{7} d (n+1)}","\frac{\sqrt{2} 7^n \cos (c+d x) F_1\left(\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"-((AppellF1[1 + n, 1/2, 1/2, 2 + n, (4 - 3*Sin[c + d*x])/7, 4 - 3*Sin[c + d*x]]*Sec[c + d*x]*(4 - 3*Sin[c + d*x])^(1 + n)*Sqrt[-1 + Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/(Sqrt[7]*d*(1 + n)))","A",0
69,1,83,67,0.150223,"\int (-3+4 \sin (c+d x))^n \, dx","Integrate[(-3 + 4*Sin[c + d*x])^n,x]","\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) (4 \sin (c+d x)-3)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (3-4 \sin (c+d x)),4 \sin (c+d x)-3\right)}{\sqrt{7} d (n+1)}","-\frac{\sqrt{2} \cos (c+d x) F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x)),4 (1-\sin (c+d x))\right)}{d \sqrt{\sin (c+d x)+1}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, (3 - 4*Sin[c + d*x])/7, -3 + 4*Sin[c + d*x]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*(-3 + 4*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n))","A",0
70,1,84,64,0.178641,"\int (-3-4 \sin (c+d x))^n \, dx","Integrate[(-3 - 4*Sin[c + d*x])^n,x]","-\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) (-4 \sin (c+d x)-3)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;-4 \sin (c+d x)-3,\frac{1}{7} (4 \sin (c+d x)+3)\right)}{\sqrt{7} d (n+1)}","\frac{\sqrt{2} \cos (c+d x) F_1\left(\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};4 (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"-((AppellF1[1 + n, 1/2, 1/2, 2 + n, -3 - 4*Sin[c + d*x], (3 + 4*Sin[c + d*x])/7]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*(-3 - 4*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n)))","A",0
71,1,95,95,0.1177286,"\int (-4+3 \sin (c+d x))^n \, dx","Integrate[(-4 + 3*Sin[c + d*x])^n,x]","\frac{\sqrt{\sin (c+d x)-1} \sqrt{\sin (c+d x)+1} \sec (c+d x) (3 \sin (c+d x)-4)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right)}{\sqrt{7} d (n+1)}","\frac{\sqrt{2} 7^n \cos (c+d x) (4-3 \sin (c+d x))^{-n} (3 \sin (c+d x)-4)^n F_1\left(\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{1-\sin (c+d x)}}",1,"(AppellF1[1 + n, 1/2, 1/2, 2 + n, (4 - 3*Sin[c + d*x])/7, 4 - 3*Sin[c + d*x]]*Sec[c + d*x]*Sqrt[-1 + Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]*(-4 + 3*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n))","A",0
72,1,100,110,0.1393654,"\int (-4-3 \sin (c+d x))^n \, dx","Integrate[(-4 - 3*Sin[c + d*x])^n,x]","-\frac{\sqrt{-\sin (c+d x)-1} \sqrt{1-\sin (c+d x)} \sec (c+d x) (-3 \sin (c+d x)-4)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right)}{\sqrt{7} d (n+1)}","-\frac{\sqrt{-\sin (c+d x)-1} \cos (c+d x) (-3 \sin (c+d x)-4)^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right)}{\sqrt{7} d (n+1) \sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)}",1,"-((AppellF1[1 + n, 1/2, 1/2, 2 + n, 4 + 3*Sin[c + d*x], (4 + 3*Sin[c + d*x])/7]*Sec[c + d*x]*(-4 - 3*Sin[c + d*x])^(1 + n)*Sqrt[-1 - Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[7]*d*(1 + n)))","A",1