Optimal. Leaf size=25 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {\coth ^2(x)+1}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 25, 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 = {3661, 377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {\coth ^2(x)+1}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 377
Rule 3661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\coth ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx,x,\coth (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {1+\coth ^2(x)}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1+\coth ^2(x)}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 1.76 \[ \frac {\sqrt {\cosh (2 x)} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )}{\sqrt {2} \sqrt {\coth ^2(x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 547, normalized size = 21.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 69, normalized size = 2.76 \[ \frac {\sqrt {2} {\left (\log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )}}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 62, normalized size = 2.48 \[ \frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \coth \relax (x )+2\right ) \sqrt {2}}{4 \sqrt {\left (\coth \relax (x )-1\right )^{2}+2 \coth \relax (x )}}\right )}{4}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2-2 \coth \relax (x )\right ) \sqrt {2}}{4 \sqrt {\left (1+\coth \relax (x )\right )^{2}-2 \coth \relax (x )}}\right )}{4} \]
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 \[ \int \frac {1}{\sqrt {\coth \relax (x)^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 63, normalized size = 2.52 \[ \frac {\sqrt {2}\,\left (\ln \left (\mathrm {coth}\relax (x)+\sqrt {2}\,\sqrt {{\mathrm {coth}\relax (x)}^2+1}+1\right )-\ln \left (\mathrm {coth}\relax (x)-1\right )\right )}{4}+\frac {\sqrt {2}\,\left (\ln \left (\mathrm {coth}\relax (x)+1\right )-\ln \left (\sqrt {2}\,\sqrt {{\mathrm {coth}\relax (x)}^2+1}-\mathrm {coth}\relax (x)+1\right )\right )}{4} \]
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 \[ \int \frac {1}{\sqrt {\coth ^{2}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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