Optimal. Leaf size=16 \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {-\tanh ^2(x)}} \]
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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} \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {-\tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\text {sech}^2(x)}} \, dx &=\int \frac {1}{\sqrt {-\tanh ^2(x)}} \, dx\\ &=\frac {\tanh (x) \int \coth (x) \, dx}{\sqrt {-\tanh ^2(x)}}\\ &=\frac {\log (\sinh (x)) \tanh (x)}{\sqrt {-\tanh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {-\tanh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 37, normalized size = 2.31 \[ -\frac {i \, x}{\mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right )} + \frac {i \, \log \left (-i \, e^{\left (2 \, x\right )} + i\right )}{\mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 81, normalized size = 5.06 \[ -\frac {\left ({\mathrm e}^{2 x}-1\right ) x}{\sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{2 x}-1\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.44, size = 22, normalized size = 1.38 \[ i \, x + i \, \log \left (e^{\left (-x\right )} + 1\right ) + i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
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 \[ \int \frac {1}{\sqrt {\frac {1}{{\mathrm {cosh}\relax (x)}^2}-1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {sech}^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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