Optimal. Leaf size=29 \[ \sqrt {\tanh ^2(x)} \coth (x) \log (\cosh (x))-\frac {1}{2} \tanh ^2(x)^{3/2} \coth (x) \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4121, 3658, 3473, 3475} \[ \sqrt {\tanh ^2(x)} \coth (x) \log (\cosh (x))-\frac {1}{2} \tanh ^2(x)^{3/2} \coth (x) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \left (1-\text {sech}^2(x)\right )^{3/2} \, dx &=\int \tanh ^2(x)^{3/2} \, dx\\ &=\left (\coth (x) \sqrt {\tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\\ &=-\frac {1}{2} \coth (x) \tanh ^2(x)^{3/2}+\left (\coth (x) \sqrt {\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {\tanh ^2(x)}-\frac {1}{2} \coth (x) \tanh ^2(x)^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.86 \[ \frac {1}{2} \sqrt {\tanh ^2(x)} (\text {csch}(x) \text {sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 183, normalized size = 6.31 \[ -\frac {x \cosh \relax (x)^{4} + 4 \, x \cosh \relax (x) \sinh \relax (x)^{3} + x \sinh \relax (x)^{4} + 2 \, {\left (x - 1\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, x \cosh \relax (x)^{2} + x - 1\right )} \sinh \relax (x)^{2} - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (x \cosh \relax (x)^{3} + {\left (x - 1\right )} \cosh \relax (x)\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 72, normalized size = 2.48 \[ -x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 3 \, \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 120, normalized size = 4.14 \[ -\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}+\frac {2 \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right ) \left (1+{\mathrm e}^{2 x}\right )}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 33, normalized size = 1.14 \[ -x - \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (1-\frac {1}{{\mathrm {cosh}\relax (x)}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (1 - \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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