Optimal. Leaf size=40 \[ -\frac{1}{2 a (\sin (x)+1)}-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (\sin (x)+1)}{4 a} \]
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Rubi [A] time = 0.0486917, antiderivative size = 40, 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 = {3879, 88} \[ -\frac{1}{2 a (\sin (x)+1)}-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (\sin (x)+1)}{4 a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tan (x)}{a+a \csc (x)} \, dx &=a^2 \operatorname{Subst}\left (\int \frac{x^2}{(a-a x) (a+a x)^2} \, dx,x,\sin (x)\right )\\ &=a^2 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^3 (-1+x)}+\frac{1}{2 a^3 (1+x)^2}-\frac{3}{4 a^3 (1+x)}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (1+\sin (x))}{4 a}-\frac{1}{2 a (1+\sin (x))}\\ \end{align*}
Mathematica [A] time = 0.0539248, size = 30, normalized size = 0.75 \[ -\frac{\frac{2}{\sin (x)+1}+\log (1-\sin (x))+3 \log (\sin (x)+1)}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 33, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,a \left ( \sin \left ( x \right ) +1 \right ) }}-{\frac{3\,\ln \left ( \sin \left ( x \right ) +1 \right ) }{4\,a}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957275, size = 42, normalized size = 1.05 \begin{align*} -\frac{3 \, \log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{4 \, a} - \frac{1}{2 \,{\left (a \sin \left (x\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.50747, size = 122, normalized size = 3.05 \begin{align*} -\frac{3 \,{\left (\sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right ) + 1\right ) +{\left (\sin \left (x\right ) + 1\right )} \log \left (-\sin \left (x\right ) + 1\right ) + 2}{4 \,{\left (a \sin \left (x\right ) + a\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*} \frac{\int \frac{\tan{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35769, size = 46, normalized size = 1.15 \begin{align*} -\frac{3 \, \log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{1}{2 \, a{\left (\sin \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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