Optimal. Leaf size=89 \[ -\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.114042, antiderivative size = 89, 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 = {2837, 12, 772} \[ -\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 772
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (b^2-x^2\right )}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (b^2-x^2\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^2 \left (1-\frac{b^2}{a^2}\right )+a x-x^2+\frac{a^3-a b^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac{a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^4 d}-\frac{\left (a^2-b^2\right ) \sin (c+d x)}{b^3 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.200605, size = 79, normalized size = 0.89 \[ \frac{6 b \left (b^2-a^2\right ) \sin (c+d x)+6 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))+3 a b^2 \sin ^2(c+d x)-2 b^3 \sin ^3(c+d x)}{6 b^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 106, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,bd}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}-{\frac{{a}^{2}\sin \left ( dx+c \right ) }{{b}^{3}d}}+{\frac{\sin \left ( dx+c \right ) }{bd}}+{\frac{{a}^{3}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{4}d}}-{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98536, size = 107, normalized size = 1.2 \begin{align*} -\frac{\frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \,{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49913, size = 185, normalized size = 2.08 \begin{align*} -\frac{3 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \,{\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \,{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, b^{4} d} \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 = 1.17481, size = 115, normalized size = 1.29 \begin{align*} -\frac{\frac{2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right ) - 6 \, b^{2} \sin \left (d x + c\right )}{b^{3}} - \frac{6 \,{\left (a^{3} - a b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{4}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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