3.587 \(\int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx\)
Optimal. Leaf size=317 \[ -\frac{16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}} \]
[Out]
(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + d*Sin[e + f*x])^(9/2)) + (2*a^3*(c -
d)*(3*c + 19*d)*Cos[e + f*x])/(63*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a
^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5
/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x])^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^5*f*Sqrt[a + a*Sin[e + f*x]]*
Sqrt[c + d*Sin[e + f*x]])
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Rubi [A] time = 0.775709, antiderivative size = 317, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used =
{2762, 2980, 2772, 2771} \[ -\frac{16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac{2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]
[Out]
(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + d*Sin[e + f*x])^(9/2)) + (2*a^3*(c -
d)*(3*c + 19*d)*Cos[e + f*x])/(63*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a
^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5
/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x])^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^5*f*Sqrt[a + a*Sin[e + f*x]]*
Sqrt[c + d*Sin[e + f*x]])
Rule 2762
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
Rule 2980
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Rule 2772
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Rule 2771
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx &=\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}-\frac{(2 a) \int \frac{\sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a (c-19 d)-\frac{3}{2} a (c+5 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{9/2}} \, dx}{9 d (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}+\frac{\left (a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx}{21 d^2 (c+d)^2}\\ &=\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac{\left (4 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{105 d^2 (c+d)^3}\\ &=\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac{8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{\left (8 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{315 d^2 (c+d)^4}\\ &=\frac{2 a^2 (c-d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac{2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac{2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac{8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac{16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.53461, size = 616, normalized size = 1.94 \[ \frac{(a (\sin (e+f x)+1))^{5/2} \sqrt{c+d \sin (e+f x)} \left (-\frac{16 \left (-c^2 \sin \left (\frac{1}{2} (e+f x)\right )+c^2 \cos \left (\frac{1}{2} (e+f x)\right )-10 c d \sin \left (\frac{1}{2} (e+f x)\right )+10 c d \cos \left (\frac{1}{2} (e+f x)\right )-73 d^2 \sin \left (\frac{1}{2} (e+f x)\right )+73 d^2 \cos \left (\frac{1}{2} (e+f x)\right )\right )}{315 d^2 (c+d)^5 (c+d \sin (e+f x))}-\frac{8 \left (-c^2 \sin \left (\frac{1}{2} (e+f x)\right )+c^2 \cos \left (\frac{1}{2} (e+f x)\right )-10 c d \sin \left (\frac{1}{2} (e+f x)\right )+10 c d \cos \left (\frac{1}{2} (e+f x)\right )-73 d^2 \sin \left (\frac{1}{2} (e+f x)\right )+73 d^2 \cos \left (\frac{1}{2} (e+f x)\right )\right )}{315 d^2 (c+d)^4 (c+d \sin (e+f x))^2}-\frac{2 \left (-c^2 \sin \left (\frac{1}{2} (e+f x)\right )+c^2 \cos \left (\frac{1}{2} (e+f x)\right )-10 c d \sin \left (\frac{1}{2} (e+f x)\right )+10 c d \cos \left (\frac{1}{2} (e+f x)\right )-73 d^2 \sin \left (\frac{1}{2} (e+f x)\right )+73 d^2 \cos \left (\frac{1}{2} (e+f x)\right )\right )}{105 d^2 (c+d)^3 (c+d \sin (e+f x))^3}-\frac{4 \left (5 c^2 \sin \left (\frac{1}{2} (e+f x)\right )-5 c^2 \cos \left (\frac{1}{2} (e+f x)\right )+8 c d \sin \left (\frac{1}{2} (e+f x)\right )-8 c d \cos \left (\frac{1}{2} (e+f x)\right )-13 d^2 \sin \left (\frac{1}{2} (e+f x)\right )+13 d^2 \cos \left (\frac{1}{2} (e+f x)\right )\right )}{63 d^2 (c+d)^2 (c+d \sin (e+f x))^4}-\frac{2 \left (-c^2 \sin \left (\frac{1}{2} (e+f x)\right )+c^2 \cos \left (\frac{1}{2} (e+f x)\right )+2 c d \sin \left (\frac{1}{2} (e+f x)\right )-2 c d \cos \left (\frac{1}{2} (e+f x)\right )-d^2 \sin \left (\frac{1}{2} (e+f x)\right )+d^2 \cos \left (\frac{1}{2} (e+f x)\right )\right )}{9 d^2 (c+d) (c+d \sin (e+f x))^5}\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]
[Out]
((a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]]*((-2*(c^2*Cos[(e + f*x)/2] - 2*c*d*Cos[(e + f*x)/2] + d
^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] + 2*c*d*Sin[(e + f*x)/2] - d^2*Sin[(e + f*x)/2]))/(9*d^2*(c + d)*(c
+ d*Sin[e + f*x])^5) - (4*(-5*c^2*Cos[(e + f*x)/2] - 8*c*d*Cos[(e + f*x)/2] + 13*d^2*Cos[(e + f*x)/2] + 5*c^2
*Sin[(e + f*x)/2] + 8*c*d*Sin[(e + f*x)/2] - 13*d^2*Sin[(e + f*x)/2]))/(63*d^2*(c + d)^2*(c + d*Sin[e + f*x])^
4) - (2*(c^2*Cos[(e + f*x)/2] + 10*c*d*Cos[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*
c*d*Sin[(e + f*x)/2] - 73*d^2*Sin[(e + f*x)/2]))/(105*d^2*(c + d)^3*(c + d*Sin[e + f*x])^3) - (8*(c^2*Cos[(e +
f*x)/2] + 10*c*d*Cos[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*c*d*Sin[(e + f*x)/2]
- 73*d^2*Sin[(e + f*x)/2]))/(315*d^2*(c + d)^4*(c + d*Sin[e + f*x])^2) - (16*(c^2*Cos[(e + f*x)/2] + 10*c*d*Co
s[(e + f*x)/2] + 73*d^2*Cos[(e + f*x)/2] - c^2*Sin[(e + f*x)/2] - 10*c*d*Sin[(e + f*x)/2] - 73*d^2*Sin[(e + f*
x)/2]))/(315*d^2*(c + d)^5*(c + d*Sin[e + f*x]))))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)
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Maple [B] time = 0.395, size = 1730, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x)
[Out]
-2/315/f/(c+d)^5*(a*(1+sin(f*x+e)))^(5/2)*(c+d*sin(f*x+e))^(1/2)*(11711*cos(f*x+e)^4*c*d^8-51540*cos(f*x+e)^4*
c^2*d^7+336*sin(f*x+e)*cos(f*x+e)^2*c^9+8112*sin(f*x+e)*cos(f*x+e)^2*d^9-3312*cos(f*x+e)^2*c^8*d+16192*cos(f*x
+e)^2*c^7*d^2+8*cos(f*x+e)^12*c^2*d^7+80*cos(f*x+e)^12*c*d^8+1460*sin(f*x+e)*cos(f*x+e)^10*d^9+37*cos(f*x+e)^1
0*c^4*d^5+360*cos(f*x+e)^10*c^3*d^6+2538*cos(f*x+e)^10*c^2*d^7-1360*cos(f*x+e)^10*c*d^8-7535*sin(f*x+e)*cos(f*
x+e)^8*d^9+310*cos(f*x+e)^8*c^6*d^3+1875*cos(f*x+e)^8*c^5*d^4+6805*cos(f*x+e)^8*c^4*d^5-2930*cos(f*x+e)^8*c^3*
d^6-16320*cos(f*x+e)^8*c^2*d^7+6095*cos(f*x+e)^8*c*d^8+15474*sin(f*x+e)*cos(f*x+e)^6*d^9+28864*cos(f*x+e)^2*c^
6*d^3-23904*cos(f*x+e)^2*c^5*d^4-47520*cos(f*x+e)^2*c^4*d^5+15168*cos(f*x+e)^2*c^3*d^6+30912*cos(f*x+e)^2*c^2*
d^7-5776*cos(f*x+e)^2*c*d^8-4224*sin(f*x+e)*c^8*d+7168*sin(f*x+e)*c^7*d^2+12288*sin(f*x+e)*c^6*d^3-7424*sin(f*
x+e)*c^5*d^4+4224*c^8*d-12288*c^6*d^3-13568*sin(f*x+e)*c^4*d^5+4096*sin(f*x+e)*c^3*d^6+7168*sin(f*x+e)*c^2*d^7
-1152*sin(f*x+e)*c*d^8-1482*cos(f*x+e)^6*c^6*d^3-17010*cos(f*x+e)^6*c^5*d^4-35980*cos(f*x+e)^6*c^4*d^5+11406*c
os(f*x+e)^6*c^3*d^6+41570*cos(f*x+e)^6*c^2*d^7-11902*cos(f*x+e)^6*c*d^8-63*sin(f*x+e)*cos(f*x+e)^4*c^9-15847*s
in(f*x+e)*cos(f*x+e)^4*d^9+87*cos(f*x+e)^4*c^8*d-7220*cos(f*x+e)^4*c^7*d^2-15204*cos(f*x+e)^4*c^6*d^3+31650*co
s(f*x+e)^4*c^5*d^4+63090*cos(f*x+e)^4*c^4*d^5-19908*cos(f*x+e)^4*c^3*d^6-279*cos(f*x+e)^6*c^8*d-1310*cos(f*x+e
)^6*c^7*d^2+13568*c^4*d^5+2688*c^9-4096*d^6*c^3-7168*d^7*c^2+1152*d^8*c-7168*c^7*d^2+7424*c^5*d^4-15*sin(f*x+e
)*cos(f*x+e)^4*c^8*d+1812*sin(f*x+e)*cos(f*x+e)^4*c^7*d^2+5380*sin(f*x+e)*cos(f*x+e)^4*c^6*d^3-22482*sin(f*x+e
)*cos(f*x+e)^4*c^5*d^4-44418*sin(f*x+e)*cos(f*x+e)^4*c^4*d^5+13860*sin(f*x+e)*cos(f*x+e)^4*c^3*d^6+38772*sin(f
*x+e)*cos(f*x+e)^4*c^2*d^7-9255*sin(f*x+e)*cos(f*x+e)^4*c*d^8+4*sin(f*x+e)*cos(f*x+e)^10*c^3*d^6+60*sin(f*x+e)
*cos(f*x+e)^10*c^2*d^7+492*sin(f*x+e)*cos(f*x+e)^10*c*d^8-35*sin(f*x+e)*cos(f*x+e)^8*c^5*d^4+5*sin(f*x+e)*cos(
f*x+e)^8*c^4*d^5+1650*sin(f*x+e)*cos(f*x+e)^8*c^3*d^6+5850*sin(f*x+e)*cos(f*x+e)^8*c^2*d^7-3295*sin(f*x+e)*cos
(f*x+e)^8*c*d^8+1200*sin(f*x+e)*cos(f*x+e)^2*c^8*d-12608*sin(f*x+e)*cos(f*x+e)^2*c^7*d^2-22720*sin(f*x+e)*cos(
f*x+e)^2*c^6*d^3+20192*sin(f*x+e)*cos(f*x+e)^2*c^5*d^4+40736*sin(f*x+e)*cos(f*x+e)^2*c^4*d^5-13120*sin(f*x+e)*
cos(f*x+e)^2*c^3*d^6-27328*sin(f*x+e)*cos(f*x+e)^2*c^2*d^7+5200*sin(f*x+e)*cos(f*x+e)^2*c*d^8+458*sin(f*x+e)*c
os(f*x+e)^6*c^7*d^2+2730*sin(f*x+e)*cos(f*x+e)^6*c^6*d^3+8774*sin(f*x+e)*cos(f*x+e)^6*c^5*d^4+17070*sin(f*x+e)
*cos(f*x+e)^6*c^4*d^5-6490*sin(f*x+e)*cos(f*x+e)^6*c^3*d^6-24522*sin(f*x+e)*cos(f*x+e)^6*c^2*d^7+8010*sin(f*x+
e)*cos(f*x+e)^6*c*d^8+1664*d^9+584*cos(f*x+e)^12*d^9-4599*cos(f*x+e)^10*d^9+14245*cos(f*x+e)^8*d^9-22645*cos(f
*x+e)^6*d^9-105*cos(f*x+e)^4*c^9+19695*cos(f*x+e)^4*d^9-1680*cos(f*x+e)^2*c^9-8944*cos(f*x+e)^2*d^9-2688*sin(f
*x+e)*c^9-1664*sin(f*x+e)*d^9)/cos(f*x+e)^5/(cos(f*x+e)^2*d^2+c^2-d^2)^5
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Maxima [B] time = 2.36932, size = 1328, normalized size = 4.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="maxima")
[Out]
-2/315*((903*c^5 + 720*c^4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/2) - (315*c^5 - 8358*c^4*d - 4770*c^
3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 70*d^5)*a^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + (4179*c^5 - 1710*c^4*d +
30878*c^3*d^2 + 11540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3*(805*c^5
- 9912*c^4*d + 2330*c^3*d^2 - 18504*c^2*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*c*d^4 + 700*d^5)*a^(5/2)*sin(f*x + e)^4/(c
os(f*x + e) + 1)^4 - 42*(149*c^5 - 894*c^4*d + 1402*c^3*d^2 - 2052*c^2*d^3 + 745*c*d^4 - 390*d^5)*a^(5/2)*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 42*(149*c^5 - 894*c^4*d + 1402*c^3*d^2 - 2052*c^2*d^3 + 745*c*d^4 - 390*d^5)
*a^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*
c*d^4 + 700*d^5)*a^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(805*c^5 - 9912*c^4*d + 2330*c^3*d^2 - 18504*
c^2*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - (4179*c^5 - 1710*c^4*d + 30878*c
^3*d^2 + 11540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + (315*c^5 - 8358*c
^4*d - 4770*c^3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 70*d^5)*a^(5/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - (903*
c^5 + 720*c^4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11)*(sin(f*
x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^5 + 5*c^4*d + 10*c^3*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5 + 3*(c^5 + 5*c^
4*d + 10*c^3*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(c^5 + 5*c^4*d + 10*c^3
*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^5 + 5*c^4*d + 10*c^3*d^2 + 10*c^2*
d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*(c + 2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x
+ e)^2/(cos(f*x + e) + 1)^2)^(11/2)*f)
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Fricas [B] time = 2.94489, size = 3441, normalized size = 10.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="fricas")
[Out]
2/315*(672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^
2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^5 - 4*(9*a^2*c^3*d + 89*a^2*c^2*d^2 + 647*a^2*c*d^3 - 73*a^2*d^4)*cos(f*x +
e)^4 - (63*a^2*c^4 + 648*a^2*c^3*d + 4798*a^2*c^2*d^2 + 1504*a^2*c*d^3 + 1387*a^2*d^4)*cos(f*x + e)^3 + (231*
a^2*c^4 + 3060*a^2*c^3*d - 2158*a^2*c^2*d^2 + 4580*a^2*c*d^3 - 673*a^2*d^4)*cos(f*x + e)^2 + 2*(483*a^2*c^4 +
684*a^2*c^3*d + 2642*a^2*c^2*d^2 + 812*a^2*c*d^3 + 419*a^2*d^4)*cos(f*x + e) - (672*a^2*c^4 - 2304*a^2*c^3*d +
3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^4
+ 4*(9*a^2*c^3*d + 91*a^2*c^2*d^2 + 667*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^3 - 3*(21*a^2*c^4 + 204*a^2*c^3*d
+ 1478*a^2*c^2*d^2 - 388*a^2*c*d^3 + 365*a^2*d^4)*cos(f*x + e)^2 - 2*(147*a^2*c^4 + 1836*a^2*c^3*d + 1138*a^2
*c^2*d^2 + 1708*a^2*c*d^3 + 211*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x +
e) + c)/((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^6 - 5*(c^6*d^4 + 5*c
^5*d^5 + 10*c^4*d^6 + 10*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^5 - (10*c^7*d^3 + 55*c^6*d^4 + 128*c^5*d^
5 + 165*c^4*d^6 + 130*c^3*d^7 + 65*c^2*d^8 + 20*c*d^9 + 3*d^10)*f*cos(f*x + e)^4 + 10*(c^8*d^2 + 5*c^7*d^3 + 1
1*c^6*d^4 + 15*c^5*d^5 + 15*c^4*d^6 + 11*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^3 + (5*c^9*d + 35*c^8*d^2
+ 120*c^7*d^3 + 260*c^6*d^4 + 378*c^5*d^5 + 370*c^4*d^6 + 240*c^3*d^7 + 100*c^2*d^8 + 25*c*d^9 + 3*d^10)*f*co
s(f*x + e)^2 - (c^10 + 5*c^9*d + 20*c^8*d^2 + 60*c^7*d^3 + 110*c^6*d^4 + 126*c^5*d^5 + 100*c^4*d^6 + 60*c^3*d^
7 + 25*c^2*d^8 + 5*c*d^9)*f*cos(f*x + e) - (c^10 + 10*c^9*d + 45*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5
*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)*f - ((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c
^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^5 + (5*c^6*d^4 + 26*c^5*d^5 + 55*c^4*d^6 + 60*c^3*d^7 + 35*c^2*d^8 + 1
0*c*d^9 + d^10)*f*cos(f*x + e)^4 - 2*(5*c^7*d^3 + 25*c^6*d^4 + 51*c^5*d^5 + 55*c^4*d^6 + 35*c^3*d^7 + 15*c^2*d
^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^3 - 2*(5*c^8*d^2 + 30*c^7*d^3 + 80*c^6*d^4 + 126*c^5*d^5 + 130*c^4*d^6 + 9
0*c^3*d^7 + 40*c^2*d^8 + 10*c*d^9 + d^10)*f*cos(f*x + e)^2 + (5*c^9*d + 25*c^8*d^2 + 60*c^7*d^3 + 100*c^6*d^4
+ 126*c^5*d^5 + 110*c^4*d^6 + 60*c^3*d^7 + 20*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e) + (c^10 + 10*c^9*d + 45
*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)
*f)*sin(f*x + e))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(11/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="giac")
[Out]
integrate((a*sin(f*x + e) + a)^(5/2)/(d*sin(f*x + e) + c)^(11/2), x)