Optimal. Leaf size=46 \[ -\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a} \]
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Rubi [A]
time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2785, 2687, 30,
2691, 3855} \begin {gather*} \frac {\coth ^4(x)}{4 a}-\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rule 2691
Rule 2785
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx &=\frac {\int \coth ^4(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth ^3(x) \text {csch}^2(x) \, dx}{a}\\ &=-\frac {\coth ^3(x) \text {csch}(x)}{4 a}+\frac {3 \int \coth ^2(x) \text {csch}(x) \, dx}{4 a}+\frac {\text {Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )}{a}\\ &=\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a}+\frac {3 \int \text {csch}(x) \, dx}{8 a}\\ &=-\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 60, normalized size = 1.30 \begin {gather*} \frac {-8-2 \coth ^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )}{16 a (1+\cosh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 38, normalized size = 0.83
method | result | size |
default | \(\frac {\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}+3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}\) | \(38\) |
risch | \(-\frac {{\mathrm e}^{x} \left (5 \,{\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+5\right )}{4 \left ({\mathrm e}^{x}-1\right )^{2} a \left ({\mathrm e}^{x}+1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}\) | \(65\) |
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. 103 vs.
\(2 (38) = 76\).
time = 0.26, size = 103, normalized size = 2.24 \begin {gather*} -\frac {5 \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} - \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \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. 631 vs.
\(2 (38) = 76\).
time = 0.48, size = 631, normalized size = 13.72 \begin {gather*} -\frac {10 \, \cosh \left (x\right )^{5} + 2 \, {\left (25 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{4} + 10 \, \sinh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{4} + 4 \, {\left (25 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 4 \, \cosh \left (x\right )^{3} + 4 \, {\left (25 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 3 \, {\left (\cosh \left (x\right )^{6} + 2 \, {\left (3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 2 \, \cosh \left (x\right )^{5} + {\left (15 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 3 \, {\left (\cosh \left (x\right )^{6} + 2 \, {\left (3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 2 \, \cosh \left (x\right )^{5} + {\left (15 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 20 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, {\left (25 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )}{8 \, {\left (a \cosh \left (x\right )^{6} + a \sinh \left (x\right )^{6} + 2 \, a \cosh \left (x\right )^{5} + 2 \, {\left (3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{5} - a \cosh \left (x\right )^{4} + {\left (15 \, a \cosh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{4} - 4 \, a \cosh \left (x\right )^{3} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 5 \, a \cosh \left (x\right )^{2} - a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{3} - a \cosh \left (x\right )^{2} + {\left (15 \, a \cosh \left (x\right )^{4} + 20 \, a \cosh \left (x\right )^{3} - 6 \, a \cosh \left (x\right )^{2} - 12 \, a \cosh \left (x\right ) - a\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{4} - 2 \, a \cosh \left (x\right )^{3} - 6 \, a \cosh \left (x\right )^{2} - a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + 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 ^{3}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (38) = 76\).
time = 0.43, size = 94, normalized size = 2.04 \begin {gather*} -\frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} - \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 2}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} + \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} + 4 \, e^{x} - 12}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 132, normalized size = 2.87 \begin {gather*} \frac {3}{2\,a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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