Optimal. Leaf size=36 \[ -\frac {\text {sech}^3(x)}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {\text {sech}(x)}{a}+\frac {\log (\cosh (x))}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 36, 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 = {3879, 75} \[ -\frac {\text {sech}^3(x)}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {\text {sech}(x)}{a}+\frac {\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 75
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{a+a \text {sech}(x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^2 (a+a x)}{x^4} \, dx,x,\cosh (x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^3}{x^3}-\frac {a^3}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\cosh (x)\right )}{a^4}\\ &=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 1.06 \[ \frac {\text {sech}^3(x) (6 \cosh (2 x)+3 \cosh (3 x) \log (\cosh (x))+\cosh (x) (9 \log (\cosh (x))+6)+2)}{12 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 437, normalized size = 12.14 \[ -\frac {3 \, x \cosh \relax (x)^{6} + 3 \, x \sinh \relax (x)^{6} + 6 \, {\left (3 \, x \cosh \relax (x) - 1\right )} \sinh \relax (x)^{5} + 3 \, {\left (3 \, x - 2\right )} \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{5} + 3 \, {\left (15 \, x \cosh \relax (x)^{2} + 3 \, x - 10 \, \cosh \relax (x) - 2\right )} \sinh \relax (x)^{4} + 4 \, {\left (15 \, x \cosh \relax (x)^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \relax (x) - 15 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{3} + 3 \, {\left (3 \, x - 2\right )} \cosh \relax (x)^{2} - 4 \, \cosh \relax (x)^{3} + 3 \, {\left (15 \, x \cosh \relax (x)^{4} + 6 \, {\left (3 \, x - 2\right )} \cosh \relax (x)^{2} - 20 \, \cosh \relax (x)^{3} + 3 \, x - 4 \, \cosh \relax (x) - 2\right )} \sinh \relax (x)^{2} - 3 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 6 \, {\left (3 \, x \cosh \relax (x)^{5} + 2 \, {\left (3 \, x - 2\right )} \cosh \relax (x)^{3} - 5 \, \cosh \relax (x)^{4} + {\left (3 \, x - 2\right )} \cosh \relax (x) - 2 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) + 3 \, x - 6 \, \cosh \relax (x)}{3 \, {\left (a \cosh \relax (x)^{6} + 6 \, a \cosh \relax (x) \sinh \relax (x)^{5} + a \sinh \relax (x)^{6} + 3 \, a \cosh \relax (x)^{4} + 3 \, {\left (5 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{4} + 4 \, {\left (5 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, a \cosh \relax (x)^{2} + 3 \, {\left (5 \, a \cosh \relax (x)^{4} + 6 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{2} + 6 \, {\left (a \cosh \relax (x)^{5} + 2 \, a \cosh \relax (x)^{3} + a \cosh \relax (x)\right )} \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 61, normalized size = 1.69 \[ \frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {11 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 12 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 12 \, e^{\left (-x\right )} - 12 \, e^{x} + 16}{6 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 34, normalized size = 0.94 \[ -\frac {\mathrm {sech}\relax (x )^{3}}{3 a}+\frac {\mathrm {sech}\relax (x )^{2}}{2 a}+\frac {\mathrm {sech}\relax (x )}{a}-\frac {\ln \left (\mathrm {sech}\relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 74, normalized size = 2.06 \[ \frac {x}{a} + \frac {2 \, {\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 3 \, e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, a e^{\left (-2 \, x\right )} + 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 96, normalized size = 2.67 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{a}-\frac {\frac {2}{a}+\frac {8\,{\mathrm {e}}^x}{3\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {\frac {2}{a}+\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}+\frac {8\,{\mathrm {e}}^x}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )} \]
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 \[ \frac {\int \frac {\tanh ^{5}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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