Optimal. Leaf size=29 \[ \frac {3 x}{2}-\frac {3 \coth (x)}{2}-\log (\sinh (x))+\frac {\coth (x)}{2 (1+\tanh (x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3633, 3610,
3612, 3556} \begin {gather*} \frac {3 x}{2}-\frac {3 \coth (x)}{2}-\log (\sinh (x))+\frac {\coth (x)}{2 (\tanh (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3633
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{1+\tanh (x)} \, dx &=\frac {\coth (x)}{2 (1+\tanh (x))}-\frac {1}{2} \int \coth ^2(x) (-3+2 \tanh (x)) \, dx\\ &=-\frac {3 \coth (x)}{2}+\frac {\coth (x)}{2 (1+\tanh (x))}-\frac {1}{2} i \int \coth (x) (-2 i+3 i \tanh (x)) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth (x)}{2 (1+\tanh (x))}-\int \coth (x) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}-\log (\sinh (x))+\frac {\coth (x)}{2 (1+\tanh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 0.93 \begin {gather*} \frac {1}{4} (6 x-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))+\sinh (2 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. \(58\) vs.
\(2(23)=46\).
time = 0.43, size = 59, normalized size = 2.03
method | result | size |
risch | \(\frac {5 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}-\frac {2}{{\mathrm e}^{2 x}-1}-\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(30\) |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 38, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, x + \frac {2}{e^{\left (-2 \, x\right )} - 1} - \frac {1}{4} \, e^{\left (-2 \, x\right )} - \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs.
\(2 (23) = 46\).
time = 0.60, size = 196, normalized size = 6.76 \begin {gather*} \frac {10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} - {\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} + {\left (60 \, x \cosh \left (x\right )^{2} - 10 \, x - 9\right )} \sinh \left (x\right )^{2} - 4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (20 \, x \cosh \left (x\right )^{3} - {\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\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 \frac {\coth ^{2}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 36, normalized size = 1.24 \begin {gather*} \frac {5}{2} \, x - \frac {{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 29, normalized size = 1.00 \begin {gather*} \frac {5\,x}{2}-\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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