Optimal. Leaf size=16 \[ \frac {\coth (x) \log (\cosh (x))}{\sqrt {-\coth ^2(x)}} \]
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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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4121, 3658, 3475} \[ \frac {\coth (x) \log (\cosh (x))}{\sqrt {-\coth ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-\text {csch}^2(x)}} \, dx &=\int \frac {1}{\sqrt {-\coth ^2(x)}} \, dx\\ &=\frac {\coth (x) \int \tanh (x) \, dx}{\sqrt {-\coth ^2(x)}}\\ &=\frac {\coth (x) \log (\cosh (x))}{\sqrt {-\coth ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\coth (x) \log (\cosh (x))}{\sqrt {-\coth ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.42, 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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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 = 0.32, size = 81, normalized size = 5.06 \[ -\frac {\left (1+{\mathrm e}^{2 x}\right ) x}{\sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )}{\sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, 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 \[ \int \frac {1}{\sqrt {-\frac {1}{{\mathrm {sinh}\relax (x)}^2}-1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \operatorname {csch}^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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