Optimal. Leaf size=67 \[ \frac{3 a^2 b \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0449595, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2721, 43} \[ \frac{3 a^2 b \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2721
Rule 43
Rubi steps
\begin{align*} \int \cot (c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^3}{x} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a^2+\frac{a^3}{x}+3 a x+x^2\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0252867, size = 67, normalized size = 1. \[ \frac{3 a^2 b \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 64, normalized size = 1. \begin{align*}{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{2}b\sin \left ( dx+c \right ) }{d}}+{\frac{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}a{b}^{2}}{2\,d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}{b}^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56722, size = 77, normalized size = 1.15 \begin{align*} \frac{2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 18 \, a^{2} b \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49174, size = 158, normalized size = 2.36 \begin{align*} -\frac{9 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{3} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) + 2 \,{\left (b^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \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*} \int \left (a + b \sin{\left (c + d x \right )}\right )^{3} \cot{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.99966, size = 78, normalized size = 1.16 \begin{align*} \frac{2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 18 \, a^{2} b \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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