Optimal. Leaf size=30 \[ x+\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3} \]
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Rubi [A]
time = 0.06, 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, 2759, 8}
\begin {gather*} x+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac {2 \sinh (x)}{1-\cosh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2759
Rule 4477
Rubi steps
\begin {align*} \int \frac {1}{(-\coth (x)+\text {csch}(x))^4} \, dx &=\int \frac {\sinh ^4(x)}{(i-i \cosh (x))^4} \, dx\\ &=\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}-\int \frac {\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\int 1 \, dx\\ &=x+\frac {2 \sinh (x)}{1-\cosh (x)}+\frac {2 \sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 28, normalized size = 0.93 \begin {gather*} -\frac {2}{3} \coth ^3\left (\frac {x}{2}\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2\left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.07, size = 34, normalized size = 1.13
method | result | size |
risch | \(x -\frac {8 \left (3 \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{x}+2\right )}{3 \left ({\mathrm e}^{x}-1\right )^{3}}\) | \(23\) |
default | \(-\frac {2}{3 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {2}{\tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 1.27 \begin {gather*} x - \frac {8 \, {\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, 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. 68 vs.
\(2 (24) = 48\).
time = 0.37, size = 68, normalized size = 2.27 \begin {gather*} \frac {3 \, x \cosh \left (x\right )^{2} + 3 \, x \sinh \left (x\right )^{2} - 4 \, {\left (3 \, x + 10\right )} \cosh \left (x\right ) + 2 \, {\left (3 \, x \cosh \left (x\right ) - 3 \, x - 4\right )} \sinh \left (x\right ) + 9 \, x + 24}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\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 {1}{\left (- \coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{4}}\, 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 = 0.73 \begin {gather*} x - \frac {8 \, {\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 2\right )}}{3 \, {\left (e^{x} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.53, size = 59, normalized size = 1.97 \begin {gather*} x-\frac {8\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {8}{3}}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}-\frac {8}{3\,\left ({\mathrm {e}}^x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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