Optimal. Leaf size=61 \[ \frac{\left (a^2-b^2\right ) \sin (x)}{a^3}-\frac{b \left (a^2-b^2\right ) \log (a \sin (x)+b)}{a^4}+\frac{b \sin ^2(x)}{2 a^2}-\frac{\sin ^3(x)}{3 a} \]
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Rubi [A] time = 0.131112, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3872, 2837, 12, 772} \[ \frac{\left (a^2-b^2\right ) \sin (x)}{a^3}-\frac{b \left (a^2-b^2\right ) \log (a \sin (x)+b)}{a^4}+\frac{b \sin ^2(x)}{2 a^2}-\frac{\sin ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 772
Rubi steps
\begin{align*} \int \frac{\cos ^3(x)}{a+b \csc (x)} \, dx &=\int \frac{\cos ^3(x) \sin (x)}{b+a \sin (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{a (b+x)} \, dx,x,a \sin (x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a^2-x^2\right )}{b+x} \, dx,x,a \sin (x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 \left (1-\frac{b^2}{a^2}\right )+b x-x^2+\frac{-a^2 b+b^3}{b+x}\right ) \, dx,x,a \sin (x)\right )}{a^4}\\ &=-\frac{b \left (a^2-b^2\right ) \log (b+a \sin (x))}{a^4}+\frac{\left (a^2-b^2\right ) \sin (x)}{a^3}+\frac{b \sin ^2(x)}{2 a^2}-\frac{\sin ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.117915, size = 60, normalized size = 0.98 \[ \frac{6 a \left (a^2-b^2\right ) \sin (x)+6 b \left (b^2-a^2\right ) \log (a \sin (x)+b)+3 a^2 b \sin ^2(x)-2 a^3 \sin ^3(x)}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 64, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3\,a}}+{\frac{b \left ( \sin \left ( x \right ) \right ) ^{2}}{2\,{a}^{2}}}+{\frac{\sin \left ( x \right ) }{a}}-{\frac{{b}^{2}\sin \left ( x \right ) }{{a}^{3}}}-{\frac{b\ln \left ( b+a\sin \left ( x \right ) \right ) }{{a}^{2}}}+{\frac{{b}^{3}\ln \left ( b+a\sin \left ( x \right ) \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964254, size = 81, normalized size = 1.33 \begin{align*} -\frac{2 \, a^{2} \sin \left (x\right )^{3} - 3 \, a b \sin \left (x\right )^{2} - 6 \,{\left (a^{2} - b^{2}\right )} \sin \left (x\right )}{6 \, a^{3}} - \frac{{\left (a^{2} b - b^{3}\right )} \log \left (a \sin \left (x\right ) + b\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.510558, size = 147, normalized size = 2.41 \begin{align*} -\frac{3 \, a^{2} b \cos \left (x\right )^{2} + 6 \,{\left (a^{2} b - b^{3}\right )} \log \left (a \sin \left (x\right ) + b\right ) - 2 \,{\left (a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (x\right )}{6 \, a^{4}} \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 = 1.21786, size = 84, normalized size = 1.38 \begin{align*} -\frac{2 \, a^{2} \sin \left (x\right )^{3} - 3 \, a b \sin \left (x\right )^{2} - 6 \, a^{2} \sin \left (x\right ) + 6 \, b^{2} \sin \left (x\right )}{6 \, a^{3}} - \frac{{\left (a^{2} b - b^{3}\right )} \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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