Optimal. Leaf size=18 \[ \frac {2}{1-\sin (x)}+\log (1-\sin (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 43
Rule 2667
Rule 4391
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.91, size = 21, normalized size = 1.17 \[ \frac {{\left (\sin \relax (x) - 1\right )} \log \left (-\sin \relax (x) + 1\right ) - 2}{\sin \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 48, normalized size = 2.67 \[ -\frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, x\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{2}} - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 45, normalized size = 2.50 \[ \frac {\sec \relax (x ) \tan \relax (x )}{2}-\ln \left (\sec \relax (x )+\tan \relax (x )\right )+\frac {3}{2 \cos \relax (x )^{2}}+\frac {3 \left (\sin ^{3}\relax (x )\right )}{2 \cos \relax (x )^{2}}+\frac {3 \sin \relax (x )}{2}+\frac {\left (\tan ^{2}\relax (x )\right )}{2}+\ln \left (\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 52, normalized size = 2.89 \[ \frac {3}{2} \, \tan \relax (x)^{2} - \frac {2 \, \sin \relax (x)}{\sin \relax (x)^{2} - 1} - \frac {1}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2} - 1\right ) - \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\sin \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 43, normalized size = 2.39 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.81, size = 44, normalized size = 2.44 \[ \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} - \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} - \frac {\log {\left (\sec ^{2}{\relax (x )} \right )}}{2} + 2 \sec ^{2}{\relax (x )} - \frac {4 \sin {\relax (x )}}{2 \sin ^{2}{\relax (x )} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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