Optimal. Leaf size=11 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {206} \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rubi steps
\begin {align*} \int \frac {1+\tan ^2(x)}{1-\tan ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \tanh ^{-1}(2 \cos (x) \sin (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 23, normalized size = 2.09 \[ \frac {1}{2} \log (\sin (x)+\cos (x))-\frac {1}{2} \log (\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1+\tan ^2(x)}{1-\tan ^2(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.91, size = 45, normalized size = 4.09 \[ \frac {1}{4} \, \log \left (\frac {\tan \relax (x)^{2} + 2 \, \tan \relax (x) + 1}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{4} \, \log \left (\frac {\tan \relax (x)^{2} - 2 \, \tan \relax (x) + 1}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 17, normalized size = 1.55 \[ \frac {1}{2} \, \log \left ({\left | \tan \relax (x) + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | \tan \relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 4, normalized size = 0.36
method | result | size |
derivativedivides | \(\arctanh \left (\tan \relax (x )\right )\) | \(4\) |
default | \(\arctanh \left (\tan \relax (x )\right )\) | \(4\) |
norman | \(-\frac {\ln \left (\tan \relax (x )-1\right )}{2}+\frac {\ln \left (1+\tan \relax (x )\right )}{2}\) | \(16\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i x}+i\right )}{2}-\frac {\ln \left ({\mathrm e}^{2 i x}-i\right )}{2}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 15, normalized size = 1.36 \[ \frac {1}{2} \, \log \left (\tan \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\tan \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 3, normalized size = 0.27 \[ \mathrm {atanh}\left (\mathrm {tan}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 15, normalized size = 1.36 \[ - \frac {\log {\left (\tan {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\tan {\relax (x )} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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