Optimal. Leaf size=38 \[ -\frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac{\log (\sin (x))}{a}-\frac{\csc (x)}{b} \]
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Rubi [A] time = 0.0629755, antiderivative size = 38, 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 = {3885, 894} \[ -\frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac{\log (\sin (x))}{a}-\frac{\csc (x)}{b} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\cot ^3(x)}{a+b \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{x (a+x)} \, dx,x,b \csc (x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{b^2}{a x}+\frac{a^2-b^2}{a (a+x)}\right ) \, dx,x,b \csc (x)\right )}{b^2}\\ &=-\frac{\csc (x)}{b}-\frac{\left (1-\frac{a^2}{b^2}\right ) \log (a+b \csc (x))}{a}-\frac{\log (\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.048905, size = 39, normalized size = 1.03 \[ \frac{\left (a^2-b^2\right ) \log (a \sin (x)+b)+a^2 (-\log (\sin (x)))-a b \csc (x)}{a b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 44, normalized size = 1.2 \begin{align*}{\frac{a\ln \left ( b+a\sin \left ( x \right ) \right ) }{{b}^{2}}}-{\frac{\ln \left ( b+a\sin \left ( x \right ) \right ) }{a}}-{\frac{1}{b\sin \left ( x \right ) }}-{\frac{a\ln \left ( \sin \left ( x \right ) \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959338, size = 57, normalized size = 1.5 \begin{align*} -\frac{a \log \left (\sin \left (x\right )\right )}{b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (a \sin \left (x\right ) + b\right )}{a b^{2}} - \frac{1}{b \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.549081, size = 124, normalized size = 3.26 \begin{align*} -\frac{a^{2} \log \left (-\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \log \left (a \sin \left (x\right ) + b\right ) \sin \left (x\right ) + a b}{a b^{2} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\left (x \right )}}{a + b \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39481, size = 59, normalized size = 1.55 \begin{align*} -\frac{a \log \left ({\left | \sin \left (x\right ) \right |}\right )}{b^{2}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a b^{2}} - \frac{1}{b \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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