Optimal. Leaf size=49 \[ -\frac {a}{8 (a \cosh (x)+a)^2}+\frac {1}{8 (a-a \cosh (x))}-\frac {1}{4 (a \cosh (x)+a)}+\frac {3 \tanh ^{-1}(\cosh (x))}{8 a} \]
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Rubi [A] time = 0.08, 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 = {2667, 44, 206} \[ -\frac {a}{8 (a \cosh (x)+a)^2}+\frac {1}{8 (a-a \cosh (x))}-\frac {1}{4 (a \cosh (x)+a)}+\frac {3 \tanh ^{-1}(\cosh (x))}{8 a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{a+a \cosh (x)} \, dx &=a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \cosh (x)\right )\\ &=a^3 \operatorname {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} \operatorname {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 = 0.17, size = 60, normalized size = 1.22 \[ -\frac {2 \coth ^2\left (\frac {x}{2}\right )+\text {sech}^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+4}{16 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 631, normalized size = 12.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 94, normalized size = 1.92 \[ \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 10}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 52 \, e^{\left (-x\right )} + 52 \, e^{x} + 84}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 45, normalized size = 0.92 \[ -\frac {\tanh ^{4}\left (\frac {x}{2}\right )}{32 a}+\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{16 a}-\frac {1}{16 a \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 103, normalized size = 2.10 \[ -\frac {3 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \]
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 \[ \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \]
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 \[ \frac {\int \frac {\operatorname {csch}^{3}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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