Optimal. Leaf size=61 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{5 (1+\tanh (x))^{5/2}}-\frac {1}{6 (1+\tanh (x))^{3/2}}-\frac {1}{4 \sqrt {1+\tanh (x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, 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 = {3560, 3561, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{4 \sqrt {\tanh (x)+1}}-\frac {1}{6 (\tanh (x)+1)^{3/2}}-\frac {1}{5 (\tanh (x)+1)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3560
Rule 3561
Rubi steps
\begin {align*} \int \frac {1}{(1+\tanh (x))^{5/2}} \, dx &=-\frac {1}{5 (1+\tanh (x))^{5/2}}+\frac {1}{2} \int \frac {1}{(1+\tanh (x))^{3/2}} \, dx\\ &=-\frac {1}{5 (1+\tanh (x))^{5/2}}-\frac {1}{6 (1+\tanh (x))^{3/2}}+\frac {1}{4} \int \frac {1}{\sqrt {1+\tanh (x)}} \, dx\\ &=-\frac {1}{5 (1+\tanh (x))^{5/2}}-\frac {1}{6 (1+\tanh (x))^{3/2}}-\frac {1}{4 \sqrt {1+\tanh (x)}}+\frac {1}{8} \int \sqrt {1+\tanh (x)} \, dx\\ &=-\frac {1}{5 (1+\tanh (x))^{5/2}}-\frac {1}{6 (1+\tanh (x))^{3/2}}-\frac {1}{4 \sqrt {1+\tanh (x)}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{5 (1+\tanh (x))^{5/2}}-\frac {1}{6 (1+\tanh (x))^{3/2}}-\frac {1}{4 \sqrt {1+\tanh (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 62, normalized size = 1.02 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {(-\cosh (2 x)+\sinh (2 x)) (11+26 \cosh (2 x)+20 \sinh (2 x))}{60 \sqrt {1+\tanh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 43, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {1}{4 \sqrt {1+\tanh \left (x \right )}}-\frac {1}{5 \left (1+\tanh \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}\) | \(43\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {1}{4 \sqrt {1+\tanh \left (x \right )}}-\frac {1}{5 \left (1+\tanh \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 79, normalized size = 1.30 \begin {gather*} -\frac {1}{120} \, \sqrt {2} {\left (\frac {5}{e^{\left (-2 \, x\right )} + 1} + \frac {15}{{\left (e^{\left (-2 \, x\right )} + 1\right )}^{2}} + 3\right )} {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}} - \frac {1}{16} \, \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 ) \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. 266 vs.
\(2 (42) = 84\).
time = 0.47, size = 266, normalized size = 4.36 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \left (x\right )^{4} + 92 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 23 \, \sqrt {2} \sinh \left (x\right )^{4} + {\left (138 \, \sqrt {2} \cosh \left (x\right )^{2} + 11 \, \sqrt {2}\right )} \sinh \left (x\right )^{2} + 11 \, \sqrt {2} \cosh \left (x\right )^{2} + 2 \, {\left (46 \, \sqrt {2} \cosh \left (x\right )^{3} + 11 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3 \, \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 15 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 5 \, \sqrt {2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sqrt {2} \sinh \left (x\right )^{5}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{240 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\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*} \int \frac {1}{\left (\tanh {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs.
\(2 (42) = 84\).
time = 0.42, size = 139, normalized size = 2.28 \begin {gather*} -\frac {1}{240} \, \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} - 45 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 35 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 15 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 15 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 40, normalized size = 0.66 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{8}-\frac {\frac {\mathrm {tanh}\left (x\right )}{6}+\frac {{\left (\mathrm {tanh}\left (x\right )+1\right )}^2}{4}+\frac {11}{30}}{{\left (\mathrm {tanh}\left (x\right )+1\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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