Optimal. Leaf size=49 \[ \frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))} \]
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Rubi [A]
time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 212}
\begin {gather*} -\frac {a}{8 (a \cosh (x)+a)^2}+\frac {1}{8 (a-a \cosh (x))}-\frac {1}{4 (a \cosh (x)+a)}+\frac {3 \tanh ^{-1}(\cosh (x))}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx &=a^3 \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \cosh (x)\right )\\ &=a^3 \text {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \cosh (x)\right )\\ &=\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))}+\frac {3}{8} \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right )\\ &=\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac {1}{8 (a-a \cosh (x))}-\frac {a}{8 (a+a \cosh (x))^2}-\frac {1}{4 (a+a \cosh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 60, normalized size = 1.22 \begin {gather*} -\frac {4+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 = 0.60, size = 38, normalized size = 0.78
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 | \(-\frac {{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}+3\right )}{4 \left ({\mathrm e}^{x}-1\right )^{2} a \left ({\mathrm e}^{x}+1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}\) | \(65\) |
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.
\(2 (42) = 84\).
time = 0.26, size = 103, normalized size = 2.10 \begin {gather*} -\frac {3 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 3 \, 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.
\(2 (42) = 84\).
time = 0.37, size = 631, normalized size = 12.88 \begin {gather*} -\frac {6 \, \cosh \left (x\right )^{5} + 6 \, {\left (5 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{4} + 6 \, \sinh \left (x\right )^{5} + 12 \, \cosh \left (x\right )^{4} + 4 \, {\left (15 \, \cosh \left (x\right )^{2} + 12 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} + 12 \, {\left (5 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 12 \, \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 ) + 6 \, {\left (5 \, \cosh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 6 \, \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 {\operatorname {csch}^{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.
\(2 (42) = 84\).
time = 0.41, size = 94, normalized size = 1.92 \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} - 10}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 52 \, e^{\left (-x\right )} + 52 \, e^{x} + 84}{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 = 0.93, size = 114, normalized size = 2.33 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\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}{2\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\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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