Optimal. Leaf size=16 \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4121, 3658, 3475} \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \sqrt {-1+\text {sech}^2(x)} \, dx &=\int \sqrt {-\tanh ^2(x)} \, dx\\ &=\left (\coth (x) \sqrt {-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.13, size = 31, normalized size = 1.94 \[ i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 81, normalized size = 5.06 \[ -\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.55, size = 13, normalized size = 0.81 \[ -i \, x - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {\frac {1}{{\mathrm {cosh}\relax (x)}^2}-1} \,d x \]
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 \[ \int \sqrt {\operatorname {sech}^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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