Optimal. Leaf size=14 \[ -\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 14, 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 = {3738, 4207,
223, 212} \begin {gather*} -\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 3738
Rule 4207
Rubi steps
\begin {align*} \int \sqrt {-1+\coth ^2(x)} \, dx &=\int \sqrt {\text {csch}^2(x)} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right )\\ &=-\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.29 \begin {gather*} \sqrt {\text {csch}^2(x)} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 15, normalized size = 1.07
method | result | size |
derivativedivides | \(-\ln \left (\coth \left (x \right )+\sqrt {-1+\coth ^{2}\left (x \right )}\right )\) | \(15\) |
default | \(-\ln \left (\coth \left (x \right )+\sqrt {-1+\coth ^{2}\left (x \right )}\right )\) | \(15\) |
risch | \(-\sqrt {\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right )+\sqrt {\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 17, normalized size = 1.21 \begin {gather*} \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 1.21 \begin {gather*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \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 \sqrt {\coth ^{2}{\left (x \right )} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 23, normalized size = 1.64 \begin {gather*} -{\left (\log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 14, normalized size = 1.00 \begin {gather*} -\ln \left (\mathrm {coth}\left (x\right )+\sqrt {{\mathrm {coth}\left (x\right )}^2-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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