Optimal. Leaf size=10 \[ \cot (x)+\log (1-\cot (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 15, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4342, 44} \[ \cot (x)+\log (1-\tan (x))-\log (\tan (x)) \]
Antiderivative was successfully verified.
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Rule 44
Rule 4342
Rubi steps
\begin {align*} \int \frac {\sec ^2(x)}{-\tan ^2(x)+\tan ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{(-1+x) x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{-1+x}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx,x,\tan (x)\right )\\ &=\cot (x)+\log (1-\tan (x))-\log (\tan (x))\\ \end {align*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.60 \[ \cot (x)-\log (\sin (x))+\log (\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 36, normalized size = 3.60 \[ -\frac {\log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right ) \sin \relax (x) - \log \left (-2 \, \cos \relax (x) \sin \relax (x) + 1\right ) \sin \relax (x) - 2 \, \cos \relax (x)}{2 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 17, normalized size = 1.70 \[ \frac {1}{\tan \relax (x)} + \log \left ({\left | \tan \relax (x) - 1 \right |}\right ) - \log \left ({\left | \tan \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 16, normalized size = 1.60 \[ \frac {1}{\tan \relax (x )}-\ln \left (\tan \relax (x )\right )+\ln \left (\tan \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 15, normalized size = 1.50 \[ \frac {1}{\tan \relax (x)} + \log \left (\tan \relax (x) - 1\right ) - \log \left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 14, normalized size = 1.40 \[ \frac {1}{\mathrm {tan}\relax (x)}-2\,\mathrm {atanh}\left (2\,\mathrm {tan}\relax (x)-1\right ) \]
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 \[ \int \frac {\sec ^{2}{\relax (x )}}{\left (\tan {\relax (x )} - 1\right ) \tan ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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