Optimal. Leaf size=61 \[ \frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0691863, antiderivative size = 61, 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 = {2833, 12, 43} \[ \frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{(a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (1+\frac{a^3}{x^3}+\frac{3 a^2}{x^2}+\frac{3 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{3 a^3 \log (\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.020793, size = 53, normalized size = 0.87 \[ a^3 \left (\frac{\sin (c+d x)}{d}-\frac{\csc ^2(c+d x)}{2 d}-\frac{3 \csc (c+d x)}{d}+\frac{3 \log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 62, normalized size = 1. \begin{align*}{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}-3\,{\frac{{a}^{3}}{d\sin \left ( dx+c \right ) }}+3\,{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06912, size = 73, normalized size = 1.2 \begin{align*} \frac{6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 2 \, a^{3} \sin \left (d x + c\right ) - \frac{6 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73561, size = 180, normalized size = 2.95 \begin{align*} \frac{a^{3} + 6 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right )^{2} - 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 = 1.21023, size = 74, normalized size = 1.21 \begin{align*} \frac{6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a^{3} \sin \left (d x + c\right ) - \frac{6 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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