Optimal. Leaf size=66 \[ \frac{b^2 \log (a+b \csc (x))}{a \left (a^2-b^2\right )}-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (\csc (x)+1)}{2 (a-b)}-\frac{\log (\sin (x))}{a} \]
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Rubi [A] time = 0.0906753, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3885, 894} \[ \frac{b^2 \log (a+b \csc (x))}{a \left (a^2-b^2\right )}-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (\csc (x)+1)}{2 (a-b)}-\frac{\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\tan (x)}{a+b \csc (x)} \, dx &=b^2 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (b^2-x^2\right )} \, dx,x,b \csc (x)\right )\\ &=b^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 b^2 (a+b) (b-x)}+\frac{1}{a b^2 x}+\frac{1}{a (a-b) (a+b) (a+x)}-\frac{1}{2 (a-b) b^2 (b+x)}\right ) \, dx,x,b \csc (x)\right )\\ &=-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (1+\csc (x))}{2 (a-b)}+\frac{b^2 \log (a+b \csc (x))}{a \left (a^2-b^2\right )}-\frac{\log (\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0611355, size = 56, normalized size = 0.85 \[ -\frac{-2 b^2 \log (a \sin (x)+b)+a (a-b) \log (1-\sin (x))+a (a+b) \log (\sin (x)+1)}{2 a (a-b) (a+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 60, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}\ln \left ( b+a\sin \left ( x \right ) \right ) }{ \left ( a+b \right ) \left ( a-b \right ) a}}-{\frac{\ln \left ( \sin \left ( x \right ) +1 \right ) }{2\,a-2\,b}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{2\,a+2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976723, size = 68, normalized size = 1.03 \begin{align*} \frac{b^{2} \log \left (a \sin \left (x\right ) + b\right )}{a^{3} - a b^{2}} - \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.551987, size = 143, normalized size = 2.17 \begin{align*} \frac{2 \, b^{2} \log \left (a \sin \left (x\right ) + b\right ) -{\left (a^{2} + a b\right )} \log \left (\sin \left (x\right ) + 1\right ) -{\left (a^{2} - a b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a^{3} - a b^{2}\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{\tan{\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 = 1.40585, size = 72, normalized size = 1.09 \begin{align*} \frac{b^{2} \log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a^{3} - a b^{2}} - \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \,{\left (a - b\right )}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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