Optimal. Leaf size=78 \[ \frac {a^2}{24 (a \cosh (x)+a)^3}-\frac {a}{32 (a-a \cosh (x))^2}+\frac {3 a}{32 (a \cosh (x)+a)^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {3}{16 (a \cosh (x)+a)}-\frac {5 \tanh ^{-1}(\cosh (x))}{16 a} \]
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Rubi [A] time = 0.11, antiderivative size = 78, 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} \[ \frac {a^2}{24 (a \cosh (x)+a)^3}-\frac {a}{32 (a-a \cosh (x))^2}+\frac {3 a}{32 (a \cosh (x)+a)^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {3}{16 (a \cosh (x)+a)}-\frac {5 \tanh ^{-1}(\cosh (x))}{16 a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx &=-\left (a^5 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^4} \, dx,x,a \cosh (x)\right )\right )\\ &=-\left (a^5 \operatorname {Subst}\left (\int \left (\frac {1}{16 a^4 (a-x)^3}+\frac {1}{8 a^5 (a-x)^2}+\frac {1}{8 a^3 (a+x)^4}+\frac {3}{16 a^4 (a+x)^3}+\frac {3}{16 a^5 (a+x)^2}+\frac {5}{16 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \cosh (x)\right )\right )\\ &=-\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))}-\frac {5}{16} \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right )\\ &=-\frac {5 \tanh ^{-1}(\cosh (x))}{16 a}-\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 89, normalized size = 1.14 \[ \frac {\cosh ^2\left (\frac {x}{2}\right ) \left (-3 \text {csch}^4\left (\frac {x}{2}\right )+24 \text {csch}^2\left (\frac {x}{2}\right )+2 \text {sech}^6\left (\frac {x}{2}\right )+9 \text {sech}^4\left (\frac {x}{2}\right )+36 \text {sech}^2\left (\frac {x}{2}\right )+120 \log \left (\sinh \left (\frac {x}{2}\right )\right )-120 \log \left (\cosh \left (\frac {x}{2}\right )\right )\right )}{192 (a \cosh (x)+a)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 1551, normalized size = 19.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 116, normalized size = 1.49 \[ -\frac {5 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{32 \, a} + \frac {5 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{32 \, a} - \frac {15 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 76 \, e^{\left (-x\right )} - 76 \, e^{x} + 100}{64 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}^{2}} + \frac {55 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 402 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 1020 \, e^{\left (-x\right )} + 1020 \, e^{x} + 936}{192 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 67, normalized size = 0.86 \[ -\frac {\tanh ^{6}\left (\frac {x}{2}\right )}{192 a}+\frac {5 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{128 a}-\frac {5 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{32 a}-\frac {1}{128 a \tanh \left (\frac {x}{2}\right )^{4}}+\frac {5}{64 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 155, normalized size = 1.99 \[ \frac {15 \, e^{\left (-x\right )} + 30 \, e^{\left (-2 \, x\right )} - 40 \, e^{\left (-3 \, x\right )} - 110 \, e^{\left (-4 \, x\right )} + 18 \, e^{\left (-5 \, x\right )} - 110 \, e^{\left (-6 \, x\right )} - 40 \, e^{\left (-7 \, x\right )} + 30 \, e^{\left (-8 \, x\right )} + 15 \, e^{\left (-9 \, x\right )}}{24 \, {\left (2 \, a e^{\left (-x\right )} - 3 \, a e^{\left (-2 \, x\right )} - 8 \, a e^{\left (-3 \, x\right )} + 2 \, a e^{\left (-4 \, x\right )} + 12 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 8 \, a e^{\left (-7 \, x\right )} - 3 \, a e^{\left (-8 \, x\right )} + 2 \, a e^{\left (-9 \, x\right )} + a e^{\left (-10 \, x\right )} + a\right )}} - \frac {5 \, \log \left (e^{\left (-x\right )} + 1\right )}{16 \, a} + \frac {5 \, \log \left (e^{\left (-x\right )} - 1\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 244, normalized size = 3.13 \[ \frac {1}{a\,\left (10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1\right )}+\frac {1}{4\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}+\frac {1}{8\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}-\frac {1}{8\,a\,\left (6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1\right )}-\frac {5}{8\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}+\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {3}{8\,a\,\left ({\mathrm {e}}^x+1\right )}-\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{8\,\sqrt {-a^2}}-\frac {1}{3\,a\,\left (15\,{\mathrm {e}}^{2\,x}+20\,{\mathrm {e}}^{3\,x}+15\,{\mathrm {e}}^{4\,x}+6\,{\mathrm {e}}^{5\,x}+{\mathrm {e}}^{6\,x}+6\,{\mathrm {e}}^x+1\right )}-\frac {5}{12\,a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \]
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 \[ \frac {\int \frac {\operatorname {csch}^{5}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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