Optimal. Leaf size=40 \[ \frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac{a \cos (x)}{b^2}+\frac{\cos ^2(x)}{2 b} \]
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Rubi [A] time = 0.06125, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac{a \cos (x)}{b^2}+\frac{\cos ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+b \cos (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{a+x} \, dx,x,b \cos (x)\right )}{b^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a-x+\frac{-a^2+b^2}{a+x}\right ) \, dx,x,b \cos (x)\right )}{b^3}\\ &=-\frac{a \cos (x)}{b^2}+\frac{\cos ^2(x)}{2 b}+\frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0573246, size = 40, normalized size = 1. \[ \frac{\left (a^2-b^2\right ) \log (a+b \cos (x))}{b^3}-\frac{a \cos (x)}{b^2}+\frac{\cos (2 x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 45, normalized size = 1.1 \begin{align*}{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{2\,b}}-{\frac{a\cos \left ( x \right ) }{{b}^{2}}}+{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ){a}^{2}}{{b}^{3}}}-{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52093, size = 51, normalized size = 1.27 \begin{align*} \frac{b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (b \cos \left (x\right ) + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69147, size = 100, normalized size = 2.5 \begin{align*} \frac{b^{2} \cos \left (x\right )^{2} - 2 \, a b \cos \left (x\right ) + 2 \,{\left (a^{2} - b^{2}\right )} \log \left (-b \cos \left (x\right ) - a\right )}{2 \, b^{3}} \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 = 1.17596, size = 53, normalized size = 1.32 \begin{align*} \frac{b \cos \left (x\right )^{2} - 2 \, a \cos \left (x\right )}{2 \, b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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