Optimal. Leaf size=34 \[ -\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}+\frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)} \]
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Rubi [A]
time = 0.02, 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 = {4206, 3739,
3554, 3556} \begin {gather*} \frac {1}{2} \tanh (x) \sqrt {-\tanh ^2(x)}-\sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
Rule 4206
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 \begin {gather*} -\frac {1}{2} (2 \coth (x) \log (\cosh (x))+\text {csch}(x) \text {sech}(x)) \sqrt {-\tanh ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs.
\(2(28)=56\).
time = 1.44, size = 123, normalized size = 3.62
method | result | size |
risch | \(\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}-\frac {2 \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, {\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right ) \left (1+{\mathrm e}^{2 x}\right )}-\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 33, normalized size = 0.97 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \left (\operatorname {sech}^{2}{\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 [C] Result contains complex when optimal does not.
time = 0.39, size = 83, normalized size = 2.44 \begin {gather*} -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}} \end {gather*}
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 \begin {gather*} \int {\left (\frac {1}{{\mathrm {cosh}\left (x\right )}^2}-1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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