Optimal. Leaf size=49 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\tanh (x))^{3/2}}-\frac {1}{2 \sqrt {1+\tanh (x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 )}{2 \sqrt {2}}-\frac {1}{2 \sqrt {\tanh (x)+1}}-\frac {1}{3 (\tanh (x)+1)^{3/2}} \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))^{3/2}} \, dx &=-\frac {1}{3 (1+\tanh (x))^{3/2}}+\frac {1}{2} \int \frac {1}{\sqrt {1+\tanh (x)}} \, dx\\ &=-\frac {1}{3 (1+\tanh (x))^{3/2}}-\frac {1}{2 \sqrt {1+\tanh (x)}}+\frac {1}{4} \int \sqrt {1+\tanh (x)} \, dx\\ &=-\frac {1}{3 (1+\tanh (x))^{3/2}}-\frac {1}{2 \sqrt {1+\tanh (x)}}+\frac {1}{2} \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 )}{2 \sqrt {2}}-\frac {1}{3 (1+\tanh (x))^{3/2}}-\frac {1}{2 \sqrt {1+\tanh (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 53, normalized size = 1.08 \begin {gather*} \frac {1}{12} \left (3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2 (\cosh (x)-\sinh (x)) (5 \cosh (x)+3 \sinh (x))}{\sqrt {1+\tanh (x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 35, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\tanh \left (x \right )}}-\frac {1}{3 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}\) | \(35\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\tanh \left (x \right )}}-\frac {1}{3 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}\) | \(35\) |
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. 69 vs.
\(2 (34) = 68\).
time = 0.48, size = 69, normalized size = 1.41 \begin {gather*} -\frac {1}{12} \, \sqrt {2} {\left (\frac {3}{e^{\left (-2 \, x\right )} + 1} + 1\right )} {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}} - \frac {1}{8} \, \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. 166 vs.
\(2 (34) = 68\).
time = 0.37, size = 166, normalized size = 3.39 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\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 )}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\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 {3}{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. 95 vs.
\(2 (34) = 68\).
time = 0.41, size = 95, normalized size = 1.94 \begin {gather*} -\frac {1}{24} \, \sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 3 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 3 \, e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3}} + 3 \, \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 = 32, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{4}-\frac {\frac {\mathrm {tanh}\left (x\right )}{2}+\frac {5}{6}}{{\left (\mathrm {tanh}\left (x\right )+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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