Optimal. Leaf size=48 \[ \frac {x}{a}-\frac {3 \text {ArcTan}(\sinh (x))}{8 a}-\frac {(8-3 \text {sech}(x)) \tanh (x)}{8 a}-\frac {(4-3 \text {sech}(x)) \tanh ^3(x)}{12 a} \]
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Rubi [A]
time = 0.07, 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 = {3973, 3966,
3855} \begin {gather*} -\frac {3 \text {ArcTan}(\sinh (x))}{8 a}+\frac {x}{a}-\frac {\tanh ^3(x) (4-3 \text {sech}(x))}{12 a}-\frac {\tanh (x) (8-3 \text {sech}(x))}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3966
Rule 3973
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 = 0.08, size = 60, normalized size = 1.25 \begin {gather*} \frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (6 \left (4 x-3 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )\right )+\left (-32+15 \text {sech}(x)+8 \text {sech}^2(x)-6 \text {sech}^3(x)\right ) \tanh (x)\right )}{12 a (1+\text {sech}(x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 75, normalized size = 1.56
method | result | size |
default | \(\frac {\frac {2 \left (-\frac {11 \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{8}-\frac {137 \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{24}-\frac {71 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {5 \tanh \left (\frac {x}{2}\right )}{8}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\) | \(75\) |
risch | \(\frac {x}{a}+\frac {15 \,{\mathrm e}^{7 x}+48 \,{\mathrm e}^{6 x}-9 \,{\mathrm e}^{5 x}+96 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{3 x}+80 \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{x}+32}{12 \left (1+{\mathrm e}^{2 x}\right )^{4} a}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8 a}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8 a}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (42) = 84\).
time = 0.49, size = 93, normalized size = 1.94 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 686 vs.
\(2 (42) = 84\).
time = 0.39, size = 686, normalized size = 14.29 \begin {gather*} \frac {12 \, x \cosh \left (x\right )^{8} + 12 \, x \sinh \left (x\right )^{8} + 3 \, {\left (32 \, x \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{7} + 48 \, {\left (x + 1\right )} \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{7} + 3 \, {\left (112 \, x \cosh \left (x\right )^{2} + 16 \, x + 35 \, \cosh \left (x\right ) + 16\right )} \sinh \left (x\right )^{6} + 3 \, {\left (224 \, x \cosh \left (x\right )^{3} + 96 \, {\left (x + 1\right )} \cosh \left (x\right ) + 105 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{5} + 24 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{5} + 3 \, {\left (280 \, x \cosh \left (x\right )^{4} + 240 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + 175 \, \cosh \left (x\right )^{3} + 24 \, x - 15 \, \cosh \left (x\right ) + 32\right )} \sinh \left (x\right )^{4} + 3 \, {\left (224 \, x \cosh \left (x\right )^{5} + 320 \, {\left (x + 1\right )} \cosh \left (x\right )^{3} + 175 \, \cosh \left (x\right )^{4} + 32 \, {\left (3 \, x + 4\right )} \cosh \left (x\right ) - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{3} + 16 \, {\left (3 \, x + 5\right )} \cosh \left (x\right )^{2} + 9 \, \cosh \left (x\right )^{3} + {\left (336 \, x \cosh \left (x\right )^{6} + 720 \, {\left (x + 1\right )} \cosh \left (x\right )^{4} + 315 \, \cosh \left (x\right )^{5} + 144 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{2} - 90 \, \cosh \left (x\right )^{3} + 48 \, x + 27 \, \cosh \left (x\right ) + 80\right )} \sinh \left (x\right )^{2} - 9 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (96 \, x \cosh \left (x\right )^{7} + 288 \, {\left (x + 1\right )} \cosh \left (x\right )^{5} + 105 \, \cosh \left (x\right )^{6} + 96 \, {\left (3 \, x + 4\right )} \cosh \left (x\right )^{3} - 45 \, \cosh \left (x\right )^{4} + 32 \, {\left (3 \, x + 5\right )} \cosh \left (x\right ) + 27 \, \cosh \left (x\right )^{2} - 15\right )} \sinh \left (x\right ) + 12 \, x - 15 \, \cosh \left (x\right ) + 32}{12 \, {\left (a \cosh \left (x\right )^{8} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + a \sinh \left (x\right )^{8} + 4 \, a \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{6} + 8 \, {\left (7 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 6 \, a \cosh \left (x\right )^{4} + 2 \, {\left (35 \, a \cosh \left (x\right )^{4} + 30 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (7 \, a \cosh \left (x\right )^{5} + 10 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + 4 \, {\left (7 \, a \cosh \left (x\right )^{6} + 15 \, a \cosh \left (x\right )^{4} + 9 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 8 \, {\left (a \cosh \left (x\right )^{7} + 3 \, a \cosh \left (x\right )^{5} + 3 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\tanh ^{6}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 69, normalized size = 1.44 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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