Optimal. Leaf size=221 \[ -\frac{157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{787 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{240 a^3 d}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{283 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.521103, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2765, 2977, 2983, 2968, 3023, 2751, 2649, 206} \[ -\frac{157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{787 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{240 a^3 d}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{283 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac{21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2977
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac{\int \frac{\sin ^3(c+d x) \left (4 a-\frac{13}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{\int \frac{\sin ^2(c+d x) \left (\frac{63 a^2}{2}-\frac{157}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{8 a^4}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{\sin (c+d x) \left (-\frac{157 a^3}{2}+\frac{787}{8} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{20 a^5}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{-\frac{157}{2} a^3 \sin (c+d x)+\frac{787}{8} a^3 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{20 a^5}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{787 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{240 a^3 d}-\frac{\int \frac{\frac{787 a^4}{16}-\frac{1729}{8} a^4 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{30 a^6}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{787 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{240 a^3 d}-\frac{283 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{32 a^2}\\ &=\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{787 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{240 a^3 d}+\frac{283 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{16 a^2 d}\\ &=\frac{283 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac{21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{1729 \cos (c+d x)}{120 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{787 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{240 a^3 d}\\ \end{align*}
Mathematica [C] time = 0.566465, size = 221, normalized size = 1. \[ -\frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (-2547 \sin \left (\frac{1}{2} (c+d x)\right )+3603 \sin \left (\frac{3}{2} (c+d x)\right )+872 \sin \left (\frac{5}{2} (c+d x)\right )+52 \sin \left (\frac{7}{2} (c+d x)\right )-12 \sin \left (\frac{9}{2} (c+d x)\right )+2547 \cos \left (\frac{1}{2} (c+d x)\right )+3603 \cos \left (\frac{3}{2} (c+d x)\right )-872 \cos \left (\frac{5}{2} (c+d x)\right )+52 \cos \left (\frac{7}{2} (c+d x)\right )+12 \cos \left (\frac{9}{2} (c+d x)\right )+(8490+8490 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{480 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.783, size = 323, normalized size = 1.5 \begin{align*} -{\frac{1}{ \left ( 480+480\,\sin \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) d} \left ( \sin \left ( dx+c \right ) \left ( 384\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{a}+640\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}{a}^{3/2}+7680\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{5/2}-8490\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) + \left ( -192\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{a}-320\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}{a}^{3/2}-3840\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{5/2}+4245\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+384\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{a}-470\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}{a}^{3/2}+9780\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{5/2}-8490\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )^{5}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87799, size = 1041, normalized size = 4.71 \begin{align*} \frac{4245 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \,{\left (96 \, \cos \left (d x + c\right )^{5} + 256 \, \cos \left (d x + c\right )^{4} - 1760 \, \cos \left (d x + c\right )^{3} + 2475 \, \cos \left (d x + c\right )^{2} -{\left (96 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 1920 \, \cos \left (d x + c\right )^{2} - 4395 \, \cos \left (d x + c\right ) - 60\right )} \sin \left (d x + c\right ) + 4335 \, \cos \left (d x + c\right ) - 60\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{960 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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