Optimal. Leaf size=119 \[ -\frac{\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac{a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d} \]
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Rubi [A] time = 0.172154, antiderivative size = 119, 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 = {2837, 12, 894} \[ -\frac{\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac{a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (b^2-x^2\right )}{b^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (b^2-x^2\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 \left (1-\frac{b^2}{a^2}\right )-\left (a^2-b^2\right ) x+a x^2-x^3+\frac{a^2 \left (-a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac{\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.347514, size = 104, normalized size = 0.87 \[ \frac{6 b^2 \left (b^2-a^2\right ) \sin ^2(c+d x)+12 a b \left (a^2-b^2\right ) \sin (c+d x)+12 a^2 \left (b^2-a^2\right ) \log (a+b \sin (c+d x))+4 a b^3 \sin ^3(c+d x)-3 b^4 \sin ^4(c+d x)}{12 b^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 144, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,bd}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d{b}^{3}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,bd}}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d{b}^{4}}}-{\frac{a\sin \left ( dx+c \right ) }{{b}^{2}d}}-{\frac{{a}^{4}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{5}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980881, size = 142, normalized size = 1.19 \begin{align*} -\frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \,{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \,{\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - a^{2} b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58068, size = 230, normalized size = 1.93 \begin{align*} -\frac{3 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 12 \,{\left (a^{4} - a^{2} b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 2 \, a b^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} 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 = 1.22803, size = 158, normalized size = 1.33 \begin{align*} -\frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b \sin \left (d x + c\right )^{2} - 6 \, b^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) + 12 \, a b^{2} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - a^{2} b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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