Optimal. Leaf size=33 \[ -\frac {\tanh ^3(x)}{3 a}+\frac {\tan ^{-1}(\sinh (x))}{2 a}-\frac {\tanh (x) \text {sech}(x)}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2611, 3770} \[ -\frac {\tanh ^3(x)}{3 a}+\frac {\tan ^{-1}(\sinh (x))}{2 a}-\frac {\tanh (x) \text {sech}(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \tanh ^2(x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh ^2(x) \, dx}{a}\\ &=-\frac {\text {sech}(x) \tanh (x)}{2 a}-\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,i \tanh (x)\right )}{a}+\frac {\int \text {sech}(x) \, dx}{2 a}\\ &=\frac {\tan ^{-1}(\sinh (x))}{2 a}-\frac {\text {sech}(x) \tanh (x)}{2 a}-\frac {\tanh ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 46, normalized size = 1.39 \[ \frac {\cosh ^2\left (\frac {x}{2}\right ) \left (6 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\tanh (x) \left (2 \text {sech}^2(x)-3 \text {sech}(x)-2\right )\right )}{3 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 315, normalized size = 9.55 \[ -\frac {3 \, \cosh \relax (x)^{5} + 3 \, {\left (5 \, \cosh \relax (x) - 2\right )} \sinh \relax (x)^{4} + 3 \, \sinh \relax (x)^{5} - 6 \, \cosh \relax (x)^{4} + 6 \, {\left (5 \, \cosh \relax (x)^{2} - 4 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 6 \, {\left (5 \, \cosh \relax (x)^{3} - 6 \, \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{2} - 3 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 8 \, \cosh \relax (x)^{3} - 1\right )} \sinh \relax (x) - 3 \, \cosh \relax (x) - 2}{3 \, {\left (a \cosh \relax (x)^{6} + 6 \, a \cosh \relax (x) \sinh \relax (x)^{5} + a \sinh \relax (x)^{6} + 3 \, a \cosh \relax (x)^{4} + 3 \, {\left (5 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{4} + 4 \, {\left (5 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, a \cosh \relax (x)^{2} + 3 \, {\left (5 \, a \cosh \relax (x)^{4} + 6 \, a \cosh \relax (x)^{2} + a\right )} \sinh \relax (x)^{2} + 6 \, {\left (a \cosh \relax (x)^{5} + 2 \, a \cosh \relax (x)^{3} + a \cosh \relax (x)\right )} \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 39, normalized size = 1.18 \[ \frac {\arctan \left (e^{x}\right )}{a} - \frac {3 \, e^{\left (5 \, x\right )} - 6 \, e^{\left (4 \, x\right )} - 3 \, e^{x} - 2}{3 \, a {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 71, normalized size = 2.15 \[ \frac {\tanh ^{5}\left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {\tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 57, normalized size = 1.73 \[ -\frac {3 \, e^{\left (-x\right )} + 6 \, e^{\left (-4 \, x\right )} - 3 \, e^{\left (-5 \, x\right )} + 2}{3 \, {\left (3 \, a e^{\left (-2 \, x\right )} + 3 \, a e^{\left (-4 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} - \frac {\arctan \left (e^{\left (-x\right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 95, normalized size = 2.88 \[ \frac {8}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {\frac {4}{a}-\frac {2\,{\mathrm {e}}^x}{a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {2}{a}-\frac {{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}} \]
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 \[ \frac {\int \frac {\tanh ^{4}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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