Optimal. Leaf size=30 \[ \frac {\coth ^3(x)}{3 a}-\frac {\text {csch}(x)}{a}-\frac {\text {csch}^3(x)}{3 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2687, 30,
2686} \begin {gather*} \frac {\coth ^3(x)}{3 a}-\frac {\text {csch}^3(x)}{3 a}-\frac {\text {csch}(x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2687
Rule 2785
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{a+a \cosh (x)} \, dx &=\frac {\int \coth ^3(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth ^2(x) \text {csch}^2(x) \, dx}{a}\\ &=\frac {i \text {Subst}\left (\int x^2 \, dx,x,i \coth (x)\right )}{a}+\frac {i \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,-i \text {csch}(x)\right )}{a}\\ &=\frac {\coth ^3(x)}{3 a}-\frac {\text {csch}(x)}{a}-\frac {\text {csch}^3(x)}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 0.83 \begin {gather*} \frac {(-3-4 \cosh (x)+\cosh (2 x)) \text {csch}(x)}{6 a (1+\cosh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 29, normalized size = 0.97
method | result | size |
default | \(\frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+2 \tanh \left (\frac {x}{2}\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a}\) | \(29\) |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{3 x}+3 \,{\mathrm e}^{2 x}+{\mathrm e}^{x}-1\right )}{3 \left ({\mathrm e}^{x}+1\right )^{3} a \left ({\mathrm e}^{x}-1\right )}\) | \(34\) |
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. 121 vs.
\(2 (26) = 52\).
time = 0.28, size = 121, normalized size = 4.03 \begin {gather*} -\frac {2 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a} + \frac {2}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\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.
\(2 (26) = 52\).
time = 0.37, size = 91, normalized size = 3.03 \begin {gather*} -\frac {2 \, {\left (3 \, \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )}}{3 \, {\left (a \cosh \left (x\right )^{3} + a \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{2} - a \cosh \left (x\right ) + {\left (3 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - 2 \, a\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*} \frac {\int \frac {\coth ^{2}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 35, normalized size = 1.17 \begin {gather*} -\frac {1}{2 \, a {\left (e^{x} - 1\right )}} - \frac {9 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 7}{6 \, a {\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 = 0.93, size = 92, normalized size = 3.07 \begin {gather*} -\frac {\frac {{\mathrm {e}}^{2\,x}}{2\,a}+\frac {1}{2\,a}+\frac {{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{6\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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