Optimal. Leaf size=26 \[ x-\frac {2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac {2 \sinh (x)}{\cosh (x)+1} \]
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Rubi [A] time = 0.12, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4392, 2670, 2680, 8} \[ x-\frac {2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac {2 \sinh (x)}{\cosh (x)+1} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2670
Rule 2680
Rule 4392
Rubi steps
\begin {align*} \int (-\coth (x)+\text {csch}(x))^4 \, dx &=\int (i-i \cosh (x))^4 \text {csch}^4(x) \, 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 [A] time = 0.01, size = 30, normalized size = 1.15 \[ 2 \tanh ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2}{3} \tanh ^3\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 68, normalized size = 2.62 \[ \frac {3 \, x \cosh \relax (x)^{2} + 3 \, x \sinh \relax (x)^{2} + 4 \, {\left (3 \, x + 10\right )} \cosh \relax (x) + 2 \, {\left (3 \, x \cosh \relax (x) + 3 \, x + 4\right )} \sinh \relax (x) + 9 \, x + 24}{3 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 0.85 \[ x + \frac {8 \, {\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2\right )}}{3 \, {\left (e^{x} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 49, normalized size = 1.88 \[ x -\coth \relax (x )-\frac {\left (\coth ^{3}\relax (x )\right )}{3}+\frac {4 \left (\cosh ^{2}\relax (x )\right )}{\sinh \relax (x )^{3}}-\frac {4}{3 \sinh \relax (x )^{3}}-\frac {3 \cosh \relax (x )}{\sinh \relax (x )^{3}}-2 \left (\frac {2}{3}-\frac {\mathrm {csch}\relax (x )^{2}}{3}\right ) \coth \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 183, normalized size = 7.04 \[ -2 \, \coth \relax (x)^{3} + x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {8 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac {16 \, e^{\left (-3 \, x\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {8 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac {4}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {32}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 57, normalized size = 2.19 \[ x+\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{3}+\frac {8}{3}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {8\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}+\frac {8}{3\,\left ({\mathrm {e}}^x+1\right )} \]
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 \[ \int \left (- \coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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