Optimal. Leaf size=86 \[ \frac{2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{5 \log (\sin (c+d x))}{a^3 d}-\frac{5 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.0697399, antiderivative size = 86, 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, 77} \[ \frac{2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{5 \log (\sin (c+d x))}{a^3 d}-\frac{5 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 77
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{3}{a^2 x^2}+\frac{5}{a^3 x}-\frac{2}{a^2 (a+x)^2}-\frac{5}{a^3 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{3 \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{5 \log (\sin (c+d x))}{a^3 d}-\frac{5 \log (1+\sin (c+d x))}{a^3 d}+\frac{2}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.19094, size = 61, normalized size = 0.71 \[ \frac{\frac{4}{\sin (c+d x)+1}-\csc ^2(c+d x)+6 \csc (c+d x)+10 \log (\sin (c+d x))-10 \log (\sin (c+d x)+1)}{2 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.129, size = 84, normalized size = 1. \begin{align*} 2\,{\frac{1}{d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-5\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{2\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{1}{d{a}^{3}\sin \left ( dx+c \right ) }}+5\,{\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 = 1.95102, size = 108, normalized size = 1.26 \begin{align*} \frac{\frac{10 \, \sin \left (d x + c\right )^{2} + 5 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2}} - \frac{10 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{10 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54676, size = 393, normalized size = 4.57 \begin{align*} \frac{10 \, \cos \left (d x + c\right )^{2} + 10 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 10 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, \sin \left (d x + c\right ) - 9}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{2} - 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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.73369, size = 208, normalized size = 2.42 \begin{align*} -\frac{\frac{80 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{40 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 53 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{2} a^{3}} + \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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