Optimal. Leaf size=11 \[ \sqrt {\cosh ^2(x)} \tanh (x) \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3176, 3207, 2637} \[ \sqrt {\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {1+\sinh ^2(x)} \, dx &=\int \sqrt {\cosh ^2(x)} \, dx\\ &=\left (\sqrt {\cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\sqrt {\cosh ^2(x)} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \sqrt {\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 3.47, 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 = 0.11, 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 = 0.08, size = 14, normalized size = 1.27 \[ \frac {\sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \sinh \relax (x )}{\cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, 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 = 0.07, 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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sinh ^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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