Optimal. Leaf size=14 \[ \frac {\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt {\cosh ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3176, 3207, 3770} \[ \frac {\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt {\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\sinh ^2(x)}} \, dx &=\int \frac {1}{\sqrt {\cosh ^2(x)}} \, dx\\ &=\frac {\cosh (x) \int \text {sech}(x) \, dx}{\sqrt {\cosh ^2(x)}}\\ &=\frac {\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt {\cosh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.36 \[ \frac {2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {\cosh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 8, normalized size = 0.57 \[ 2 \, \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 5, normalized size = 0.36 \[ 2 \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 15, normalized size = 1.07 \[ \frac {\sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \arctan \left (\sinh \relax (x )\right )}{\cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 5, normalized size = 0.36 \[ 2 \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {{\mathrm {sinh}\relax (x)}^2+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\sinh ^{2}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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