Optimal. Leaf size=34 \[ \frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)}-\sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4121, 3658, 3473, 3475} \[ \frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)}-\sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
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 \left (-\tanh ^2(x)\right )^{3/2} \, dx\\ &=-\left (\left (\coth (x) \sqrt {-\tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\right )\\ &=\frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)}-\left (\coth (x) \sqrt {-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=-\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}+\frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.79 \[ -\frac {1}{2} \sqrt {-\tanh ^2(x)} (\text {csch}(x) \text {sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 1, normalized size = 0.03 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 83, normalized size = 2.44 \[ -i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - \frac {i \, {\left (3 \, e^{\left (4 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) + 3 \, \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right )\right )}}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 123, normalized size = 3.62 \[ \frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}-\frac {2 \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right ) \left (1+{\mathrm e}^{2 x}\right )}-\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.49, size = 33, normalized size = 0.97 \[ i \, x + \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
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 \[ \int {\left (\frac {1}{{\mathrm {cosh}\relax (x)}^2}-1\right )}^{3/2} \,d x \]
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 \[ \int \left (\operatorname {sech}^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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