Optimal. Leaf size=42 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3661, 402, 217, 206, 377} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {\cot ^2(x)-1}}\right )-\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {\cot ^2(x)-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 402
Rule 3661
Rubi steps
\begin {align*} \int \sqrt {-1+\cot ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,\cot (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )\\ &=-\tanh ^{-1}\left (\frac {\cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {-1+\cot ^2(x)}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 1.43 \[ \frac {\sin (x) \sqrt {\cot ^2(x)-1} \left (\sqrt {2} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )-\tanh ^{-1}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right )\right )}{\sqrt {\cos (2 x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.56, size = 123, normalized size = 2.93 \[ \frac {1}{2} \, \sqrt {2} \log \left (-2 \, \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 1\right ) - \frac {1}{2} \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right ) + 1}\right ) + \frac {1}{2} \, \log \left (\frac {\sqrt {2} \sqrt {-\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1}{\cos \left (2 \, x\right ) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 35, normalized size = 0.83 \[ -\ln \left (\cot \relax (x )+\sqrt {-1+\cot ^{2}\relax (x )}\right )+\arctanh \left (\frac {\cot \relax (x ) \sqrt {2}}{\sqrt {-1+\cot ^{2}\relax (x )}}\right ) \sqrt {2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 34, normalized size = 0.81 \[ \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\mathrm {cot}\relax (x)}{\sqrt {{\mathrm {cot}\relax (x)}^2-1}}\right )-\ln \left (\mathrm {cot}\relax (x)+\sqrt {{\mathrm {cot}\relax (x)}^2-1}\right ) \]
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 \[ \int \sqrt {\cot ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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