Optimal. Leaf size=47 \[ -\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.105745, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x)^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{3}{x}+\frac{4}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (1+\sin (c+d x))}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.050403, size = 35, normalized size = 0.74 \[ -\frac{\csc (c+d x)+3 \log (\sin (c+d x))-4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.139, size = 50, normalized size = 1.1 \begin{align*} 4\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{d{a}^{3}\sin \left ( dx+c \right ) }}-3\,{\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 = 1.10166, size = 59, normalized size = 1.26 \begin{align*} \frac{\frac{4 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{3 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac{1}{a^{3} \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.14028, size = 142, normalized size = 3.02 \begin{align*} -\frac{3 \, \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 4 \, \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 1}{a^{3} d \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 [B] time = 1.27088, size = 136, normalized size = 2.89 \begin{align*} -\frac{\frac{2 \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac{16 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac{6 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{3}} - \frac{6 \, \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 )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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