Optimal. Leaf size=48 \[ \frac {x}{a}-\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {\tanh ^3(x) (4-3 \text {sech}(x))}{12 a}-\frac {\tanh (x) (8-3 \text {sech}(x))}{8 a} \]
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Rubi [A] time = 0.10, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ \frac {x}{a}-\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {\tanh ^3(x) (4-3 \text {sech}(x))}{12 a}-\frac {\tanh (x) (8-3 \text {sech}(x))}{8 a} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3881
Rule 3888
Rubi steps
\begin {align*} \int \frac {\tanh ^6(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\int (-a+a \text {sech}(x)) \tanh ^4(x) \, dx}{a^2}\\ &=-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {\int (-4 a+3 a \text {sech}(x)) \tanh ^2(x) \, dx}{4 a^2}\\ &=-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {\int (-8 a+3 a \text {sech}(x)) \, dx}{8 a^2}\\ &=\frac {x}{a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}-\frac {3 \int \text {sech}(x) \, dx}{8 a}\\ &=\frac {x}{a}-\frac {3 \tan ^{-1}(\sinh (x))}{8 a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 60, normalized size = 1.25 \[ \frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (6 \left (4 x-3 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )\right )+\tanh (x) \left (-6 \text {sech}^3(x)+8 \text {sech}^2(x)+15 \text {sech}(x)-32\right )\right )}{12 a (\text {sech}(x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 686, normalized size = 14.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 69, normalized size = 1.44 \[ \frac {x}{a} - \frac {3 \, \arctan \left (e^{x}\right )}{4 \, a} + \frac {15 \, e^{\left (7 \, x\right )} + 48 \, e^{\left (6 \, x\right )} - 9 \, e^{\left (5 \, x\right )} + 96 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (3 \, x\right )} + 80 \, e^{\left (2 \, x\right )} - 15 \, e^{x} + 32}{12 \, a {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 117, normalized size = 2.44 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {11 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {137 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{12 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {71 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{12 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {5 \tanh \left (\frac {x}{2}\right )}{4 a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 93, normalized size = 1.94 \[ \frac {x}{a} + \frac {15 \, e^{\left (-x\right )} - 80 \, e^{\left (-2 \, x\right )} - 9 \, e^{\left (-3 \, x\right )} - 96 \, e^{\left (-4 \, x\right )} + 9 \, e^{\left (-5 \, x\right )} - 48 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} - 32}{12 \, {\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} + \frac {3 \, \arctan \left (e^{\left (-x\right )}\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 143, normalized size = 2.98 \[ \frac {\frac {8}{3\,a}+\frac {6\,{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4}{a}+\frac {9\,{\mathrm {e}}^x}{2\,a}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {x}{a}+\frac {\frac {4}{a}+\frac {5\,{\mathrm {e}}^x}{4\,a}}{{\mathrm {e}}^{2\,x}+1}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{4\,\sqrt {a^2}}-\frac {4\,{\mathrm {e}}^x}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )} \]
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 \[ \frac {\int \frac {\tanh ^{6}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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