Optimal. Leaf size=68 \[ \frac {1}{8 a (1-\cosh (x))}-\frac {1}{8 a (1+\cosh (x))^2}+\frac {3}{4 a (1+\cosh (x))}+\frac {5 \log (1-\cosh (x))}{16 a}+\frac {11 \log (1+\cosh (x))}{16 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3964, 90}
\begin {gather*} \frac {1}{8 a (1-\cosh (x))}+\frac {3}{4 a (\cosh (x)+1)}-\frac {1}{8 a (\cosh (x)+1)^2}+\frac {5 \log (1-\cosh (x))}{16 a}+\frac {11 \log (\cosh (x)+1)}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 3964
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{a+a \text {sech}(x)} \, dx &=a^4 \text {Subst}\left (\int \frac {x^4}{(a-a x)^2 (a+a x)^3} \, dx,x,\cosh (x)\right )\\ &=a^4 \text {Subst}\left (\int \left (\frac {1}{8 a^5 (-1+x)^2}+\frac {5}{16 a^5 (-1+x)}+\frac {1}{4 a^5 (1+x)^3}-\frac {3}{4 a^5 (1+x)^2}+\frac {11}{16 a^5 (1+x)}\right ) \, dx,x,\cosh (x)\right )\\ &=\frac {1}{8 a (1-\cosh (x))}-\frac {1}{8 a (1+\cosh (x))^2}+\frac {3}{4 a (1+\cosh (x))}+\frac {5 \log (1-\cosh (x))}{16 a}+\frac {11 \log (1+\cosh (x))}{16 a}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 66, normalized size = 0.97 \begin {gather*} \frac {\left (12-2 \coth ^2\left (\frac {x}{2}\right )+4 \cosh ^2\left (\frac {x}{2}\right ) \left (11 \log \left (\cosh \left (\frac {x}{2}\right )\right )+5 \log \left (\sinh \left (\frac {x}{2}\right )\right )\right )-\text {sech}^2\left (\frac {x}{2}\right )\right ) \text {sech}(x)}{16 a (1+\text {sech}(x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 56, normalized size = 0.82
method | result | size |
default | \(\frac {-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{4}-\frac {5 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}-8 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-8 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}+5 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}\) | \(56\) |
risch | \(-\frac {x}{a}+\frac {{\mathrm e}^{x} \left (5 \,{\mathrm e}^{4 x}-6 \,{\mathrm e}^{3 x}-14 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+5\right )}{4 a \left ({\mathrm e}^{x}+1\right )^{4} \left ({\mathrm e}^{x}-1\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}+\frac {11 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 108, normalized size = 1.59 \begin {gather*} \frac {x}{a} + \frac {5 \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} - 14 \, e^{\left (-3 \, x\right )} - 6 \, 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 {11 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} + \frac {5 \, \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. 773 vs.
\(2 (56) = 112\).
time = 0.38, size = 773, normalized size = 11.37 \begin {gather*} -\frac {8 \, x \cosh \left (x\right )^{6} + 8 \, x \sinh \left (x\right )^{6} + 2 \, {\left (8 \, x - 5\right )} \cosh \left (x\right )^{5} + 2 \, {\left (24 \, x \cosh \left (x\right ) + 8 \, x - 5\right )} \sinh \left (x\right )^{5} - 4 \, {\left (2 \, x - 3\right )} \cosh \left (x\right )^{4} + 2 \, {\left (60 \, x \cosh \left (x\right )^{2} + 5 \, {\left (8 \, x - 5\right )} \cosh \left (x\right ) - 4 \, x + 6\right )} \sinh \left (x\right )^{4} - 4 \, {\left (8 \, x - 7\right )} \cosh \left (x\right )^{3} + 4 \, {\left (40 \, x \cosh \left (x\right )^{3} + 5 \, {\left (8 \, x - 5\right )} \cosh \left (x\right )^{2} - 4 \, {\left (2 \, x - 3\right )} \cosh \left (x\right ) - 8 \, x + 7\right )} \sinh \left (x\right )^{3} - 4 \, {\left (2 \, x - 3\right )} \cosh \left (x\right )^{2} + 4 \, {\left (30 \, x \cosh \left (x\right )^{4} + 5 \, {\left (8 \, x - 5\right )} \cosh \left (x\right )^{3} - 6 \, {\left (2 \, x - 3\right )} \cosh \left (x\right )^{2} - 3 \, {\left (8 \, x - 7\right )} \cosh \left (x\right ) - 2 \, x + 3\right )} \sinh \left (x\right )^{2} + 2 \, {\left (8 \, x - 5\right )} \cosh \left (x\right ) - 11 \, {\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 ) - 5 \, {\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 (24 \, x \cosh \left (x\right )^{5} + 5 \, {\left (8 \, x - 5\right )} \cosh \left (x\right )^{4} - 8 \, {\left (2 \, x - 3\right )} \cosh \left (x\right )^{3} - 6 \, {\left (8 \, x - 7\right )} \cosh \left (x\right )^{2} - 4 \, {\left (2 \, x - 3\right )} \cosh \left (x\right ) + 8 \, x - 5\right )} \sinh \left (x\right ) + 8 \, x}{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 )}}{\operatorname {sech}{\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 = 0.39, size = 94, normalized size = 1.38 \begin {gather*} \frac {11 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} + \frac {5 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} - \frac {5 \, e^{\left (-x\right )} + 5 \, e^{x} - 6}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {33 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 84 \, e^{\left (-x\right )} + 84 \, e^{x} + 52}{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 = 1.43, size = 160, normalized size = 2.35 \begin {gather*} \frac {\ln \left (9\,{\mathrm {e}}^{2\,x}-9\right )}{a}-\frac {x}{a}-\frac {1}{2\,\left (a+4\,a\,{\mathrm {e}}^x+6\,a\,{\mathrm {e}}^{2\,x}+4\,a\,{\mathrm {e}}^{3\,x}+a\,{\mathrm {e}}^{4\,x}\right )}+\frac {1}{a+3\,a\,{\mathrm {e}}^x+3\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{3\,x}}-\frac {1}{4\,\left (a-2\,a\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )}-\frac {2}{a+2\,a\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}+\frac {3}{2\,\left (a+a\,{\mathrm {e}}^x\right )}+\frac {1}{4\,\left (a-a\,{\mathrm {e}}^x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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