Optimal. Leaf size=31 \[ -\sinh ^{-1}(\coth (x))+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1+\coth ^2(x)}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 399, 221,
385, 212} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {\coth ^2(x)+1}}\right )-\sinh ^{-1}(\coth (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 385
Rule 399
Rule 3742
Rubi steps
\begin {align*} \int \sqrt {1+\coth ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {1+x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=2 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\coth (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\coth (x)\right )\\ &=-\sinh ^{-1}(\coth (x))+2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {1+\coth ^2(x)}}\right )\\ &=-\sinh ^{-1}(\coth (x))+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1+\coth ^2(x)}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 60, normalized size = 1.94 \begin {gather*} \frac {\sqrt {1+\coth ^2(x)} \left (-\tanh ^{-1}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right )+\sqrt {2} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh (x)}{\sqrt {\cosh (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs.
\(2(25)=50\).
time = 0.95, size = 97, normalized size = 3.13
method | result | size |
derivativedivides | \(\frac {\sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}{2}-\arcsinh \left (\coth \left (x \right )\right )-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}{2}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}\right )}{2}\) | \(97\) |
default | \(\frac {\sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}{2}-\arcsinh \left (\coth \left (x \right )\right )-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (1+\coth \left (x \right )\right )^{2}-2 \coth \left (x \right )}}\right )}{2}-\frac {\sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}{2}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {\left (\coth \left (x \right )-1\right )^{2}+2 \coth \left (x \right )}}\right )}{2}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 683 vs.
\(2 (25) = 50\).
time = 0.36, size = 683, normalized size = 22.03 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {2 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \, {\left (14 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} + 15 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} + 45 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} + 9 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{4} + 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \sqrt {2} \cosh \left (x\right )^{4} + 18 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, \sqrt {2}\right )} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{5} + 6 \, \sqrt {2} \cosh \left (x\right )^{3} + 4 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 4\right )}}{\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac {1}{2} \, \log \left (\frac {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (25) = 50\).
time = 0.40, size = 119, normalized size = 3.84 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\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 = 1.34, size = 68, normalized size = 2.19 \begin {gather*} \frac {\sqrt {2}\,\left (\ln \left (\mathrm {coth}\left (x\right )+\sqrt {2}\,\sqrt {{\mathrm {coth}\left (x\right )}^2+1}+1\right )-\ln \left (\mathrm {coth}\left (x\right )-1\right )\right )}{2}-\mathrm {asinh}\left (\mathrm {coth}\left (x\right )\right )+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {coth}\left (x\right )+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {coth}\left (x\right )}^2+1}-\mathrm {coth}\left (x\right )+1\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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