Optimal. Leaf size=83 \[ -\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.10044, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a-x)^2}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^4}-\frac{3}{x^3}+\frac{4}{a x^2}-\frac{4}{a^2 x}+\frac{4}{a^2 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{4 \csc (c+d x)}{a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{\csc ^3(c+d x)}{3 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 = 0.115998, size = 59, normalized size = 0.71 \[ -\frac{2 \csc ^3(c+d x)-9 \csc ^2(c+d x)+24 \csc (c+d x)+24 \log (\sin (c+d x))-24 \log (\sin (c+d x)+1)}{6 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 82, normalized size = 1. \begin{align*} 4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{3\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3}{2\,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 = 1.04906, size = 88, normalized size = 1.06 \begin{align*} \frac{\frac{24 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{24 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac{24 \, \sin \left (d x + c\right )^{2} - 9 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12039, size = 292, normalized size = 3.52 \begin{align*} -\frac{24 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 24 \,{\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 24 \, \cos \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) - 26}{6 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\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 = 1.26211, size = 196, normalized size = 2.36 \begin{align*} \frac{\frac{192 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{96 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{176 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 51 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} - \frac{a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 51 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{9}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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