Optimal. Leaf size=46 \[ -\frac {\tanh ^5(x)}{5 a}+\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {\tanh ^3(x) \text {sech}(x)}{4 a}-\frac {3 \tanh (x) \text {sech}(x)}{8 a} \]
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Rubi [A] time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {\tanh ^5(x)}{5 a}+\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {\tanh ^3(x) \text {sech}(x)}{4 a}-\frac {3 \tanh (x) \text {sech}(x)}{8 a} \]
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 ^6(x)}{a+a \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \tanh ^4(x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh ^4(x) \, dx}{a}\\ &=-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}+\frac {i \operatorname {Subst}\left (\int x^4 \, dx,x,i \tanh (x)\right )}{a}+\frac {3 \int \text {sech}(x) \tanh ^2(x) \, dx}{4 a}\\ &=-\frac {3 \text {sech}(x) \tanh (x)}{8 a}-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}-\frac {\tanh ^5(x)}{5 a}+\frac {3 \int \text {sech}(x) \, dx}{8 a}\\ &=\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {3 \text {sech}(x) \tanh (x)}{8 a}-\frac {\text {sech}(x) \tanh ^3(x)}{4 a}-\frac {\tanh ^5(x)}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 58, normalized size = 1.26 \[ \frac {\cosh ^2\left (\frac {x}{2}\right ) \left (30 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+\tanh (x) \left (-8 \text {sech}^4(x)+10 \text {sech}^3(x)+16 \text {sech}^2(x)-25 \text {sech}(x)-8\right )\right )}{20 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 750, normalized size = 16.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 58, normalized size = 1.26 \[ \frac {3 \, \arctan \left (e^{x}\right )}{4 \, a} - \frac {25 \, e^{\left (9 \, x\right )} - 40 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (7 \, x\right )} - 80 \, e^{\left (4 \, x\right )} - 10 \, e^{\left (3 \, x\right )} - 25 \, e^{x} - 8}{20 \, a {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 115, normalized size = 2.50 \[ \frac {3 \left (\tanh ^{9}\left (\frac {x}{2}\right )\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}+\frac {7 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{2 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}-\frac {32 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}-\frac {7 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}-\frac {3 \tanh \left (\frac {x}{2}\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{5}}+\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 89, normalized size = 1.93 \[ -\frac {25 \, e^{\left (-x\right )} + 10 \, e^{\left (-3 \, x\right )} + 80 \, e^{\left (-4 \, x\right )} - 10 \, e^{\left (-7 \, x\right )} + 40 \, e^{\left (-8 \, x\right )} - 25 \, e^{\left (-9 \, x\right )} + 8}{20 \, {\left (5 \, a e^{\left (-2 \, x\right )} + 10 \, a e^{\left (-4 \, x\right )} + 10 \, a e^{\left (-6 \, x\right )} + 5 \, a e^{\left (-8 \, x\right )} + a e^{\left (-10 \, x\right )} + a\right )}} - \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 183, normalized size = 3.98 \[ \frac {\frac {16}{a}-\frac {6\,{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {8}{a}-\frac {9\,{\mathrm {e}}^x}{2\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {32}{5\,a\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {\frac {16}{a}-\frac {4\,{\mathrm {e}}^x}{a}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {2}{a}-\frac {5\,{\mathrm {e}}^x}{4\,a}}{{\mathrm {e}}^{2\,x}+1}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{4\,\sqrt {a^2}} \]
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 \[ \frac {\int \frac {\tanh ^{6}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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