Optimal. Leaf size=20 \[ -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4477, 2746, 45}
\begin {gather*} -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rule 4477
Rubi steps
\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^3} \, dx &=-\left (i \int \frac {\sinh ^3(x)}{(i-i \cosh (x))^3} \, dx\right )\\ &=\text {Subst}\left (\int \frac {i-x}{(i+x)^2} \, dx,x,-i \cosh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{-i-x}+\frac {2 i}{(i+x)^2}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac {2 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.90 \begin {gather*} \text {csch}^2\left (\frac {x}{2}\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 29, normalized size = 1.45
method | result | size |
risch | \(x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}-1\right )^{2}}-2 \ln \left ({\mathrm e}^{x}-1\right )\) | \(20\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )^{2}}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 1.75 \begin {gather*} -x - \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - 2 \, \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. 90 vs.
\(2 (18) = 36\).
time = 0.39, size = 90, normalized size = 4.50 \begin {gather*} \frac {x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} - 2 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, {\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1} \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 {1}{\coth ^{3}{\left (x \right )} - 3 \coth ^{2}{\left (x \right )} \operatorname {csch}{\left (x \right )} + 3 \coth {\left (x \right )} \operatorname {csch}^{2}{\left (x \right )} - \operatorname {csch}^{3}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 20, normalized size = 1.00 \begin {gather*} x + \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 31, normalized size = 1.55 \begin {gather*} x-2\,\ln \left ({\mathrm {e}}^x-1\right )+\frac {4}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}+\frac {4}{{\mathrm {e}}^x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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