Optimal. Leaf size=14 \[ \tanh (x) \sqrt {\coth ^2(x)} \log (\sinh (x)) \]
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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 = {4121, 3658, 3475} \[ \tanh (x) \sqrt {\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \sqrt {1+\text {csch}^2(x)} \, dx &=\int \sqrt {\coth ^2(x)} \, dx\\ &=\left (\sqrt {\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx\\ &=\sqrt {\coth ^2(x)} \log (\sinh (x)) \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \tanh (x) \sqrt {\coth ^2(x)} \log (\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 18, normalized size = 1.29 \[ -x + \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 27, normalized size = 1.93 \[ -x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 79, normalized size = 5.64 \[ -\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 x}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 22, normalized size = 1.57 \[ -x - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]
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 \[ \int \sqrt {\frac {1}{{\mathrm {sinh}\relax (x)}^2}+1} \,d x \]
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 \[ \int \sqrt {\operatorname {csch}^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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