Optimal. Leaf size=11 \[ \frac {\tanh (x)}{\sqrt {\text {sech}^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3657, 4122, 191} \[ \frac {\tanh (x)}{\sqrt {\text {sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\tanh ^2(x)}} \, dx &=\int \frac {1}{\sqrt {\text {sech}^2(x)}} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{\sqrt {\text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\tanh (x)}{\sqrt {\text {sech}^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 2, normalized size = 0.18 \[ \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 14, normalized size = 1.27 \[ \frac {\tanh \relax (x )}{\sqrt {1-\left (\tanh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 2, normalized size = 0.18 \[ \mathrm {sinh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 12, normalized size = 1.09 \[ \frac {\tanh {\relax (x )}}{\sqrt {1 - \tanh ^{2}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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