Optimal. Leaf size=57 \[ 8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-\frac {2}{5} (\tanh (x)+1)^{5/2}-\frac {4}{3} (\tanh (x)+1)^{3/2}-8 \sqrt {\tanh (x)+1} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ 8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-\frac {2}{5} (\tanh (x)+1)^{5/2}-\frac {4}{3} (\tanh (x)+1)^{3/2}-8 \sqrt {\tanh (x)+1} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3478
Rule 3480
Rubi steps
\begin {align*} \int (1+\tanh (x))^{7/2} \, dx &=-\frac {2}{5} (1+\tanh (x))^{5/2}+2 \int (1+\tanh (x))^{5/2} \, dx\\ &=-\frac {4}{3} (1+\tanh (x))^{3/2}-\frac {2}{5} (1+\tanh (x))^{5/2}+4 \int (1+\tanh (x))^{3/2} \, dx\\ &=-8 \sqrt {1+\tanh (x)}-\frac {4}{3} (1+\tanh (x))^{3/2}-\frac {2}{5} (1+\tanh (x))^{5/2}+8 \int \sqrt {1+\tanh (x)} \, dx\\ &=-8 \sqrt {1+\tanh (x)}-\frac {4}{3} (1+\tanh (x))^{3/2}-\frac {2}{5} (1+\tanh (x))^{5/2}+16 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right )\\ &=8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-8 \sqrt {1+\tanh (x)}-\frac {4}{3} (1+\tanh (x))^{3/2}-\frac {2}{5} (1+\tanh (x))^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 65, normalized size = 1.14 \[ \frac {\cosh ^3(x) \left (8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right ) (\tanh (x)+1)^3-\frac {2}{15} (\tanh (x)+1)^{7/2} \left (16 \tanh (x)-3 \text {sech}^2(x)+76\right )\right )}{(\sinh (x)+\cosh (x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 434, normalized size = 7.61 \[ -\frac {4 \, {\left (2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \relax (x)^{5} + 115 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{4} + 23 \, \sqrt {2} \sinh \relax (x)^{5} + 5 \, {\left (46 \, \sqrt {2} \cosh \relax (x)^{2} + 7 \, \sqrt {2}\right )} \sinh \relax (x)^{3} + 35 \, \sqrt {2} \cosh \relax (x)^{3} + 5 \, {\left (46 \, \sqrt {2} \cosh \relax (x)^{3} + 21 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 5 \, {\left (23 \, \sqrt {2} \cosh \relax (x)^{4} + 21 \, \sqrt {2} \cosh \relax (x)^{2} + 3 \, \sqrt {2}\right )} \sinh \relax (x) + 15 \, \sqrt {2} \cosh \relax (x)\right )} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 15 \, {\left (\sqrt {2} \cosh \relax (x)^{6} + 6 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{5} + \sqrt {2} \sinh \relax (x)^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x)^{4} + 3 \, \sqrt {2} \cosh \relax (x)^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{4} + 6 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x)^{2} + 3 \, \sqrt {2} \cosh \relax (x)^{2} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{5} + 2 \, \sqrt {2} \cosh \relax (x)^{3} + \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - 2 \, \cosh \relax (x)^{2} - 4 \, \cosh \relax (x) \sinh \relax (x) - 2 \, \sinh \relax (x)^{2} - 1\right )\right )}}{15 \, {\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 140, normalized size = 2.46 \[ \frac {4}{15} \, \sqrt {2} {\left (\frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} - 135 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 170 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 100 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 100 \, e^{\left (2 \, x\right )} + 23\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{5}} - 15 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 43, normalized size = 0.75 \[ 8 \arctanh \left (\frac {\sqrt {1+\tanh \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-8 \sqrt {1+\tanh \relax (x )}-\frac {4 \left (1+\tanh \relax (x )\right )^{\frac {3}{2}}}{3}-\frac {2 \left (1+\tanh \relax (x )\right )^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 83, normalized size = 1.46 \[ -4 \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}}}{\sqrt {2} + \frac {\sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}}}\right ) - \frac {8 \, \sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}} - \frac {8 \, \sqrt {2}}{3 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {8 \, \sqrt {2}}{5 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 44, normalized size = 0.77 \[ -8\,\sqrt {\mathrm {tanh}\relax (x)+1}-\frac {4\,{\left (\mathrm {tanh}\relax (x)+1\right )}^{3/2}}{3}-\frac {2\,{\left (\mathrm {tanh}\relax (x)+1\right )}^{5/2}}{5}-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\relax (x)+1}\,1{}\mathrm {i}}{2}\right )\,8{}\mathrm {i} \]
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 \[ \int \left (\tanh {\relax (x )} + 1\right )^{\frac {7}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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