Optimal. Leaf size=18 \[ \frac {2}{1-\cosh (x)}+\log (1-\cosh (x)) \]
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Rubi [A]
time = 0.05, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, 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 (\coth (x)+\text {csch}(x))^3 \, dx &=i \int (i+i \cosh (x))^3 \text {csch}^3(x) \, dx\\ &=\text {Subst}\left (\int \frac {i+x}{(i-x)^2} \, dx,x,i \cosh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {2 i}{(-i+x)^2}+\frac {1}{-i+x}\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 [B] Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).
time = 0.04, size = 41, normalized size = 2.28 \begin {gather*} -\text {csch}^2\left (\frac {x}{2}\right )+2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\log (\sinh (x))+3 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 22, normalized size = 1.22
method | result | size |
risch | \(-x -\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}-1\right )^{2}}+2 \ln \left ({\mathrm e}^{x}-1\right )\) | \(22\) |
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. 66 vs.
\(2 (16) = 32\).
time = 0.26, size = 66, normalized size = 3.67 \begin {gather*} -\frac {3}{2} \, \coth \left (x\right )^{2} + x + \frac {4 \, {\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, 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. 91 vs.
\(2 (16) = 32\).
time = 0.35, size = 91, normalized size = 5.06 \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 \left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 22, normalized size = 1.22 \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 = 0.05, size = 33, normalized size = 1.83 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^x-1\right )-x-\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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