Optimal. Leaf size=16 \[ \frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3556, 30}
\begin {gather*} \frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3556
Rule 3801
Rubi steps
\begin {align*} \int x \coth ^2(x) \, dx &=-x \coth (x)+\int x \, dx+\int \coth (x) \, dx\\ &=\frac {x^2}{2}-x \coth (x)+\log (\sinh (x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}-x \coth (x)+\log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 28, normalized size = 1.75
method | result | size |
risch | \(\frac {x^{2}}{2}-2 x -\frac {2 x}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (14) = 28\).
time = 1.71, size = 53, normalized size = 3.31 \begin {gather*} -\frac {x e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} - 1} - \frac {x^{2} - {\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 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. 95 vs.
\(2 (14) = 28\).
time = 0.86, size = 95, normalized size = 5.94 \begin {gather*} \frac {{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (x^{2} - 4 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (x^{2} - 4 \, x\right )} \sinh \left (x\right )^{2} - x^{2} + 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 22, normalized size = 1.38 \begin {gather*} \frac {x^{2}}{2} + x - \frac {x}{\tanh {\left (x \right )}} - \log {\left (\tanh {\left (x \right )} + 1 \right )} + \log {\left (\tanh {\left (x \right )} \right )} \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. 53 vs.
\(2 (14) = 28\).
time = 0.44, size = 53, normalized size = 3.31 \begin {gather*} \frac {x^{2} e^{\left (2 \, x\right )} - x^{2} - 4 \, x e^{\left (2 \, x\right )} + 2 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 2 \, \log \left (e^{\left (2 \, x\right )} - 1\right )}{2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 27, normalized size = 1.69 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x-\frac {2\,x}{{\mathrm {e}}^{2\,x}-1}+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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