Optimal. Leaf size=30 \[ -\frac {\tanh ^4(x)}{4 a}+\frac {\text {sech}^3(x)}{3 a}-\frac {\text {sech}(x)}{a} \]
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Rubi [A] time = 0.09, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ -\frac {\tanh ^4(x)}{4 a}+\frac {\text {sech}^3(x)}{3 a}-\frac {\text {sech}(x)}{a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2706
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \tanh ^3(x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh ^3(x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int x^3 \, dx,x,i \tanh (x)\right )}{a}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\text {sech}(x)\right )}{a}\\ &=-\frac {\text {sech}(x)}{a}+\frac {\text {sech}^3(x)}{3 a}-\frac {\tanh ^4(x)}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.83 \[ \frac {2 \sinh ^6\left (\frac {x}{2}\right ) (5 \cosh (x)+3) \text {sech}^4(x)}{3 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 174, normalized size = 5.80 \[ -\frac {2 \, {\left (3 \, \cosh \relax (x)^{4} + 3 \, {\left (4 \, \cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + 3 \, \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{3} + {\left (18 \, \cosh \relax (x)^{2} - 9 \, \cosh \relax (x) + 8\right )} \sinh \relax (x)^{2} + 8 \, \cosh \relax (x)^{2} + {\left (12 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) + 3\right )} \sinh \relax (x) - 3 \, \cosh \relax (x) + 5\right )}}{3 \, {\left (a \cosh \relax (x)^{5} + 5 \, a \cosh \relax (x) \sinh \relax (x)^{4} + a \sinh \relax (x)^{5} + 5 \, a \cosh \relax (x)^{3} + {\left (10 \, a \cosh \relax (x)^{2} + 3 \, a\right )} \sinh \relax (x)^{3} + 5 \, {\left (2 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 10 \, a \cosh \relax (x) + {\left (5 \, a \cosh \relax (x)^{4} + 9 \, a \cosh \relax (x)^{2} + 2 \, a\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.60 \[ -\frac {2 \, {\left (3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4 \, e^{\left (-x\right )} - 4 \, e^{x} + 6\right )}}{3 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 30, normalized size = 1.00 \[ \frac {\frac {1}{3 \cosh \relax (x )^{3}}-\frac {1}{4 \cosh \relax (x )^{4}}-\frac {1}{\cosh \relax (x )}+\frac {1}{2 \cosh \relax (x )^{2}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 223, normalized size = 7.43 \[ -\frac {2 \, e^{\left (-x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} + \frac {2 \, e^{\left (-2 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac {10 \, e^{\left (-3 \, x\right )}}{3 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} - \frac {10 \, e^{\left (-5 \, x\right )}}{3 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} + \frac {2 \, e^{\left (-6 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac {2 \, e^{\left (-7 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 117, normalized size = 3.90 \[ \frac {\frac {8}{a}-\frac {8\,{\mathrm {e}}^x}{3\,a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {6}{a}-\frac {8\,{\mathrm {e}}^x}{3\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {2}{a}-\frac {2\,{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}-\frac {4}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )} \]
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 \[ \frac {\int \frac {\tanh ^{5}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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