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 = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 894
Rule 3885
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 1.25, size = 58, normalized size = 0.88 \[ \frac {2 \, b^{2} \log \left (a \sin \relax (x) + b\right ) - {\left (a^{2} + a b\right )} \log \left (\sin \relax (x) + 1\right ) - {\left (a^{2} - a b\right )} \log \left (-\sin \relax (x) + 1\right )}{2 \, {\left (a^{3} - a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 53, normalized size = 0.80 \[ \frac {b^{2} \log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a^{3} - a b^{2}} - \frac {\log \left (\sin \relax (x) + 1\right )}{2 \, {\left (a - b\right )}} - \frac {\log \left (-\sin \relax (x) + 1\right )}{2 \, {\left (a + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 60, normalized size = 0.91 \[ \frac {b^{2} \ln \left (b +a \sin \relax (x )\right )}{\left (a +b \right ) \left (a -b \right ) a}-\frac {\ln \left (-1+\sin \relax (x )\right )}{2 a +2 b}-\frac {\ln \left (1+\sin \relax (x )\right )}{2 a -2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 50, normalized size = 0.76 \[ \frac {b^{2} \log \left (a \sin \relax (x) + b\right )}{a^{3} - a b^{2}} - \frac {\log \left (\sin \relax (x) + 1\right )}{2 \, {\left (a - b\right )}} - \frac {\log \left (\sin \relax (x) - 1\right )}{2 \, {\left (a + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 80, normalized size = 1.21 \[ \frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{a-b}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}{a+b}+\frac {b^2\,\ln \left (b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )}{a\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan {\relax (x )}}{a + b \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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