Optimal. Leaf size=30 \[ \frac {2}{(1-\cosh (x))^2}-\frac {4}{1-\cosh (x)}-\log (1-\cosh (x)) \]
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Rubi [A]
time = 0.05, antiderivative size = 30, 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 {4}{1-\cosh (x)}+\frac {2}{(1-\cosh (x))^2}-\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))^5} \, dx &=i \int \frac {\sinh ^5(x)}{(i-i \cosh (x))^5} \, dx\\ &=-\text {Subst}\left (\int \frac {(i-x)^2}{(i+x)^3} \, dx,x,-i \cosh (x)\right )\\ &=-\text {Subst}\left (\int \left (-\frac {4}{(i+x)^3}-\frac {4 i}{(i+x)^2}+\frac {1}{i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac {2}{(i-i \cosh (x))^2}-\frac {4 i}{i-i \cosh (x)}-\log (1-\cosh (x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.07 \begin {gather*} 2 \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{2} \text {csch}^4\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 = 1.05, size = 37, normalized size = 1.23
method | result | size |
risch | \(x +\frac {8 \,{\mathrm e}^{x} \left (1-{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{x}-1\right )^{4}}-2 \ln \left ({\mathrm e}^{x}-1\right )\) | \(30\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{\tanh \left (\frac {x}{2}\right )^{2}}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(37\) |
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. 58 vs.
\(2 (26) = 52\).
time = 0.27, size = 58, normalized size = 1.93 \begin {gather*} -x - \frac {8 \, {\left (e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, 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. 269 vs.
\(2 (26) = 52\).
time = 0.38, size = 269, normalized size = 8.97 \begin {gather*} \frac {x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} - 4 \, {\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \, {\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} - 6 \, {\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} - 4 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} - 3 \, {\left (x - 2\right )} \cosh \left (x\right )^{2} + {\left (3 \, x - 4\right )} \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \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 ^{5}{\left (x \right )} - 5 \coth ^{4}{\left (x \right )} \operatorname {csch}{\left (x \right )} + 10 \coth ^{3}{\left (x \right )} \operatorname {csch}^{2}{\left (x \right )} - 10 \coth ^{2}{\left (x \right )} \operatorname {csch}^{3}{\left (x \right )} + 5 \coth {\left (x \right )} \operatorname {csch}^{4}{\left (x \right )} - \operatorname {csch}^{5}{\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 = 31, normalized size = 1.03 \begin {gather*} x + \frac {8 \, {\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} - 1\right )}^{4}} - 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 = 1.52, size = 79, normalized size = 2.63 \begin {gather*} x-2\,\ln \left ({\mathrm {e}}^x-1\right )-\frac {16}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}+\frac {16}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}+\frac {8}{6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1}+\frac {8}{{\mathrm {e}}^x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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