Optimal. Leaf size=68 \[ \frac{1}{8 a (1-\sin (x))}+\frac{3}{4 a (\sin (x)+1)}-\frac{1}{8 a (\sin (x)+1)^2}+\frac{5 \log (1-\sin (x))}{16 a}+\frac{11 \log (\sin (x)+1)}{16 a} \]
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Rubi [A] time = 0.0703221, antiderivative size = 68, 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 = {3879, 88} \[ \frac{1}{8 a (1-\sin (x))}+\frac{3}{4 a (\sin (x)+1)}-\frac{1}{8 a (\sin (x)+1)^2}+\frac{5 \log (1-\sin (x))}{16 a}+\frac{11 \log (\sin (x)+1)}{16 a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tan ^3(x)}{a+a \csc (x)} \, dx &=a^4 \operatorname{Subst}\left (\int \frac{x^4}{(a-a x)^2 (a+a x)^3} \, dx,x,\sin (x)\right )\\ &=a^4 \operatorname{Subst}\left (\int \left (\frac{1}{8 a^5 (-1+x)^2}+\frac{5}{16 a^5 (-1+x)}+\frac{1}{4 a^5 (1+x)^3}-\frac{3}{4 a^5 (1+x)^2}+\frac{11}{16 a^5 (1+x)}\right ) \, dx,x,\sin (x)\right )\\ &=\frac{5 \log (1-\sin (x))}{16 a}+\frac{11 \log (1+\sin (x))}{16 a}+\frac{1}{8 a (1-\sin (x))}-\frac{1}{8 a (1+\sin (x))^2}+\frac{3}{4 a (1+\sin (x))}\\ \end{align*}
Mathematica [A] time = 0.131365, size = 50, normalized size = 0.74 \[ \frac{\frac{2 \left (5 \sin ^2(x)-3 \sin (x)-6\right )}{(\sin (x)-1) (\sin (x)+1)^2}+5 \log (1-\sin (x))+11 \log (\sin (x)+1)}{16 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 55, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,a \left ( \sin \left ( x \right ) +1 \right ) ^{2}}}+{\frac{3}{4\,a \left ( \sin \left ( x \right ) +1 \right ) }}+{\frac{11\,\ln \left ( \sin \left ( x \right ) +1 \right ) }{16\,a}}-{\frac{1}{8\,a \left ( \sin \left ( x \right ) -1 \right ) }}+{\frac{5\,\ln \left ( \sin \left ( x \right ) -1 \right ) }{16\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962337, size = 78, normalized size = 1.15 \begin{align*} \frac{5 \, \sin \left (x\right )^{2} - 3 \, \sin \left (x\right ) - 6}{8 \,{\left (a \sin \left (x\right )^{3} + a \sin \left (x\right )^{2} - a \sin \left (x\right ) - a\right )}} + \frac{11 \, \log \left (\sin \left (x\right ) + 1\right )}{16 \, a} + \frac{5 \, \log \left (\sin \left (x\right ) - 1\right )}{16 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.522906, size = 227, normalized size = 3.34 \begin{align*} \frac{10 \, \cos \left (x\right )^{2} + 11 \,{\left (\cos \left (x\right )^{2} \sin \left (x\right ) + \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) + 5 \,{\left (\cos \left (x\right )^{2} \sin \left (x\right ) + \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) + 6 \, \sin \left (x\right ) + 2}{16 \,{\left (a \cos \left (x\right )^{2} \sin \left (x\right ) + a \cos \left (x\right )^{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*} \frac{\int \frac{\tan ^{3}{\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 = 1.39496, size = 70, normalized size = 1.03 \begin{align*} \frac{11 \, \log \left (\sin \left (x\right ) + 1\right )}{16 \, a} + \frac{5 \, \log \left (-\sin \left (x\right ) + 1\right )}{16 \, a} + \frac{5 \, \sin \left (x\right )^{2} - 3 \, \sin \left (x\right ) - 6}{8 \, a{\left (\sin \left (x\right ) + 1\right )}^{2}{\left (\sin \left (x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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