Optimal. Leaf size=14 \[ \sqrt {\coth ^2(x)} \log (\sinh (x)) \tanh (x) \]
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Rubi [A]
time = 0.01, 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 = {4206, 3739,
3556} \begin {gather*} \tanh (x) \sqrt {\coth ^2(x)} \log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3739
Rule 4206
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 \begin {gather*} \sqrt {\coth ^2(x)} \log (\sinh (x)) \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs.
\(2(12)=24\).
time = 1.51, size = 79, normalized size = 5.64
method | result | size |
risch | \(-\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 x}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 22, normalized size = 1.57 \begin {gather*} -x - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 1.29 \begin {gather*} -x + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {\operatorname {csch}^{2}{\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (12) = 24\).
time = 0.39, size = 27, normalized size = 1.93 \begin {gather*} -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 ) \end {gather*}
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 \begin {gather*} \int \sqrt {\frac {1}{{\mathrm {sinh}\left (x\right )}^2}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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