Optimal. Leaf size=96 \[ -\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.0677812, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^5 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^5}-\frac{3}{x^4}+\frac{4}{a x^3}-\frac{4}{a^2 x^2}+\frac{4}{a^3 x}-\frac{4}{a^3 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{4 \csc (c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (1+\sin (c+d x))}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.318515, size = 69, normalized size = 0.72 \[ \frac{-\csc ^4(c+d x)+4 \csc ^3(c+d x)-8 \csc ^2(c+d x)+16 \csc (c+d x)+16 \log (\sin (c+d x))-16 \log (\sin (c+d x)+1)}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 97, normalized size = 1. \begin{align*} -4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{4\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{1}{d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{1}{d{a}^{3}\sin \left ( dx+c \right ) }}+4\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19002, size = 101, normalized size = 1.05 \begin{align*} -\frac{\frac{16 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{16 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac{16 \, \sin \left (d x + c\right )^{3} - 8 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{4}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13131, size = 348, normalized size = 3.62 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{2} + 16 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 16 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \,{\left (4 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 9}{4 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\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 [A] time = 1.25489, size = 235, normalized size = 2.45 \begin{align*} -\frac{\frac{1536 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{768 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{1600 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 456 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 108 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}} + \frac{3 \,{\left (a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 152 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{12}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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