Optimal. Leaf size=14 \[ \frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \]
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Rubi [A] time = 0.06, antiderivative size = 14, 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 = {4392, 2667, 43} \[ \frac {2}{\cosh (x)+1}+\log (\cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
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 )\\ &=-\operatorname {Subst}\left (\int \frac {i-x}{(i+x)^2} \, dx,x,i \cosh (x)\right )\\ &=-\operatorname {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 = 1.29 \[ \text {sech}^2\left (\frac {x}{2}\right )+2 \log \left (\cosh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 89, normalized size = 6.36 \[ -\frac {x \cosh \relax (x)^{2} + x \sinh \relax (x)^{2} + 2 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (x \cosh \relax (x) + x - 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 21, normalized size = 1.50 \[ -x + \frac {4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 28, normalized size = 2.00 \[ -\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 31, normalized size = 2.21 \[ x + \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 33, normalized size = 2.36 \[ 2\,\ln \left ({\mathrm {e}}^x+1\right )-x-\frac {4}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {4}{{\mathrm {e}}^x+1} \]
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 \[ \int \frac {1}{\left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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