Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 216} \[ \sin ^{-1}(\coth (x)) \]
Antiderivative was successfully verified.
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Rule 216
Rule 4122
Rubi steps
\begin {align*} \int \sqrt {-\text {csch}^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\coth (x)\right )\\ &=\sin ^{-1}(\coth (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt {-\text {csch}^2(x)} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.98, size = 15, normalized size = 5.00 \[ -i \, \log \left (e^{x} + 1\right ) + i \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 27, normalized size = 9.00 \[ {\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 67, normalized size = 22.33 \[ {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )-{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.56, size = 19, normalized size = 6.33 \[ i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.33 \[ \int \sqrt {-\frac {1}{{\mathrm {sinh}\relax (x)}^2}} \,d x \]
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 \[ \int \sqrt {- \operatorname {csch}^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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