Optimal. Leaf size=18 \[ \frac{2}{1-\sin (x)}+\log (1-\sin (x)) \]
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Rubi [A] time = 0.0438103, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4391, 2667, 43} \[ \frac{2}{1-\sin (x)}+\log (1-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int (\sec (x)+\tan (x))^3 \, dx &=\int \sec ^3(x) (1+\sin (x))^3 \, dx\\ &=\operatorname{Subst}\left (\int \frac{1+x}{(1-x)^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{2}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,\sin (x)\right )\\ &=\log (1-\sin (x))+\frac{2}{1-\sin (x)}\\ \end{align*}
Mathematica [A] time = 0.0259234, size = 31, normalized size = 1.72 \[ \frac{\tan ^2(x)}{2}+\frac{3 \sec ^2(x)}{2}-\tanh ^{-1}(\sin (x))+\log (\cos (x))+2 \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 45, normalized size = 2.5 \begin{align*}{\frac{\sec \left ( x \right ) \tan \left ( x \right ) }{2}}-\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) +{\frac{3}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}}+{\frac{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}}+{\frac{3\,\sin \left ( x \right ) }{2}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( \cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.982467, size = 70, normalized size = 3.89 \begin{align*} \frac{3}{2} \, \tan \left (x\right )^{2} - \frac{2 \, \sin \left (x\right )}{\sin \left (x\right )^{2} - 1} - \frac{1}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) - \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\sin \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03005, size = 68, normalized size = 3.78 \begin{align*} \frac{{\left (\sin \left (x\right ) - 1\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2}{\sin \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.04155, size = 44, normalized size = 2.44 \begin{align*} \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{2} - \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{2} - \frac{\log{\left (\sec ^{2}{\left (x \right )} \right )}}{2} + 2 \sec ^{2}{\left (x \right )} - \frac{4 \sin{\left (x \right )}}{2 \sin ^{2}{\left (x \right )} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18844, size = 65, normalized size = 3.61 \begin{align*} -\frac{3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 10 \, \tan \left (\frac{1}{2} \, x\right ) + 3}{{\left (\tan \left (\frac{1}{2} \, x\right ) - 1\right )}^{2}} - \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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