Optimal. Leaf size=3 \[ \tan ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3675, 203} \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 3675
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{1+\tanh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tanh (x)\right )\\ &=\tan ^{-1}(\tanh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 19, normalized size = 6.33 \[ -\arctan \left (-\frac {\cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 5, normalized size = 1.67 \[ \arctan \left (e^{\left (2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 116, normalized size = 38.67 \[ \frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {2 \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 35, normalized size = 11.67 \[ \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 5, normalized size = 1.67 \[ \mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right ) \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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