Optimal. Leaf size=10 \[ \cot (x)+\log (1-\cot (x)) \]
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Rubi [A] time = 0.052689, antiderivative size = 15, normalized size of antiderivative = 1.5, 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 4342
Rule 44
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*}
Mathematica [A] time = 0.0372257, size = 16, normalized size = 1.6 \[ \cot (x)-\log (\sin (x))+\log (\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 16, normalized size = 1.6 \begin{align*} \left ( \tan \left ( x \right ) \right ) ^{-1}-\ln \left ( \tan \left ( x \right ) \right ) +\ln \left ( \tan \left ( x \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963174, size = 20, normalized size = 2. \begin{align*} \frac{1}{\tan \left (x\right )} + \log \left (\tan \left (x\right ) - 1\right ) - \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17118, size = 126, normalized size = 12.6 \begin{align*} -\frac{\log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) \sin \left (x\right ) - \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 2 \, \cos \left (x\right )}{2 \, \sin \left (x\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*} \int \frac{\sec ^{2}{\left (x \right )}}{\left (\tan{\left (x \right )} - 1\right ) \tan ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11066, size = 23, normalized size = 2.3 \begin{align*} \frac{1}{\tan \left (x\right )} + \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) - \log \left ({\left | \tan \left (x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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