Optimal. Leaf size=78 \[ -\frac {5 \tanh ^{-1}(\cosh (x))}{16 a}-\frac {a}{32 (a-a \cosh (x))^2}-\frac {1}{8 (a-a \cosh (x))}+\frac {a^2}{24 (a+a \cosh (x))^3}+\frac {3 a}{32 (a+a \cosh (x))^2}+\frac {3}{16 (a+a \cosh (x))} \]
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Rubi [A]
time = 0.08, 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 = {2746, 46, 212}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \frac {\text {csch}^5(x)}{a+a \cosh (x)} \, dx &=-\left (a^5 \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^4} \, dx,x,a \cosh (x)\right )\right )\\ &=-\left (a^5 \text {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} \text {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 = 0.20, size = 89, normalized size = 1.14 \begin {gather*} \frac {\cosh ^2\left (\frac {x}{2}\right ) \left (24 \text {csch}^2\left (\frac {x}{2}\right )-3 \text {csch}^4\left (\frac {x}{2}\right )-120 \log \left (\cosh \left (\frac {x}{2}\right )\right )+120 \log \left (\sinh \left (\frac {x}{2}\right )\right )+36 \text {sech}^2\left (\frac {x}{2}\right )+9 \text {sech}^4\left (\frac {x}{2}\right )+2 \text {sech}^6\left (\frac {x}{2}\right )\right )}{192 (a+a \cosh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 54, normalized size = 0.69
method | result | size |
default | \(\frac {-\frac {\left (\tanh ^{6}\left (\frac {x}{2}\right )\right )}{6}+\frac {5 \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{4}-5 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+\frac {5}{2 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {1}{4 \tanh \left (\frac {x}{2}\right )^{4}}+10 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{32 a}\) | \(54\) |
risch | \(\frac {{\mathrm e}^{x} \left (15 \,{\mathrm e}^{8 x}+30 \,{\mathrm e}^{7 x}-40 \,{\mathrm e}^{6 x}-110 \,{\mathrm e}^{5 x}+18 \,{\mathrm e}^{4 x}-110 \,{\mathrm e}^{3 x}-40 \,{\mathrm e}^{2 x}+30 \,{\mathrm e}^{x}+15\right )}{24 \left ({\mathrm e}^{x}+1\right )^{6} a \left ({\mathrm e}^{x}-1\right )^{4}}-\frac {5 \ln \left ({\mathrm e}^{x}+1\right )}{16 a}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{16 a}\) | \(89\) |
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. 155 vs.
\(2 (68) = 136\).
time = 0.28, size = 155, normalized size = 1.99 \begin {gather*} \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} \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. 1551 vs.
\(2 (68) = 136\).
time = 0.34, size = 1551, normalized size = 19.88 \begin {gather*} \text {Too large to display} \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}^{5}{\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 = 116, normalized size = 1.49 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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