Optimal. Leaf size=50 \[ \frac{a^2-b^2}{2 a^3 (a \cos (x)+b)^2}+\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3} \]
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Rubi [A] time = 0.0787822, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4392, 2668, 697} \[ \frac{a^2-b^2}{2 a^3 (a \cos (x)+b)^2}+\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{1}{(a \cot (x)+b \csc (x))^3} \, dx &=\int \frac{\sin ^3(x)}{(b+a \cos (x))^3} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a^2-x^2}{(b+x)^3} \, dx,x,a \cos (x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{-b-x}+\frac{a^2-b^2}{(b+x)^3}+\frac{2 b}{(b+x)^2}\right ) \, dx,x,a \cos (x)\right )}{a^3}\\ &=\frac{a^2-b^2}{2 a^3 (b+a \cos (x))^2}+\frac{2 b}{a^3 (b+a \cos (x))}+\frac{\log (b+a \cos (x))}{a^3}\\ \end{align*}
Mathematica [A] time = 0.111372, size = 77, normalized size = 1.54 \[ \frac{a^2 \cos (2 x) \log (a \cos (x)+b)+a^2 \log (a \cos (x)+b)+a^2+2 b^2 \log (a \cos (x)+b)+4 a b \cos (x) (\log (a \cos (x)+b)+1)+3 b^2}{2 a^3 (a \cos (x)+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 56, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( b+a\cos \left ( x \right ) \right ) }{{a}^{3}}}+2\,{\frac{b}{{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) }}+{\frac{1}{2\,a \left ( b+a\cos \left ( x \right ) \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( b+a\cos \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51703, size = 239, normalized size = 4.78 \begin{align*} \frac{2 \,{\left (a b + b^{2} + \frac{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{a^{5} + a^{4} b - a^{3} b^{2} - a^{2} b^{3} - \frac{2 \,{\left (a^{5} - a^{4} b - a^{3} b^{2} + a^{2} b^{3}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} + \frac{\log \left (a + b - \frac{{\left (a - b\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{3}} - \frac{\log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31057, size = 181, normalized size = 3.62 \begin{align*} \frac{4 \, a b \cos \left (x\right ) + a^{2} + 3 \, b^{2} + 2 \,{\left (a^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + b^{2}\right )} \log \left (a \cos \left (x\right ) + b\right )}{2 \,{\left (a^{5} \cos \left (x\right )^{2} + 2 \, a^{4} b \cos \left (x\right ) + a^{3} b^{2}\right )}} \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 = 1.11597, size = 61, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left | a \cos \left (x\right ) + b \right |}\right )}{a^{3}} + \frac{4 \, b \cos \left (x\right ) + \frac{a^{2} + 3 \, b^{2}}{a}}{2 \,{\left (a \cos \left (x\right ) + b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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