Optimal. Leaf size=28 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3742, 385, 209}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 385
Rule 3742
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx &=-\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {1-\cot ^2(x)}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 42, normalized size = 1.50 \begin {gather*} -\frac {\sqrt {\cos (2 x)} \csc (x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{\sqrt {2-2 \cot ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 31, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}\, \cot \left (x \right )}{-1+\cot ^{2}\left (x \right )}\right )}{2}\) | \(31\) |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\left (\cot ^{2}\left (x \right )\right )}\, \cot \left (x \right )}{-1+\cot ^{2}\left (x \right )}\right )}{2}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (22) = 44\).
time = 0.55, size = 90, normalized size = 3.21 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left ({\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ), {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \cos \left (2 \, x\right )\right ) \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. 56 vs.
\(2 (22) = 44\).
time = 3.13, size = 56, normalized size = 2.00 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right )^{2} + \cos \left (2 \, x\right )\right )}}\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 \frac {1}{\sqrt {1 - \cot ^{2}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{2} i \, \sqrt {2} \log \left (i \, \sqrt {2} + i\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {\sqrt {2} \arcsin \left (\sqrt {2} \cos \left (x\right )\right )}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 85, normalized size = 3.04 \begin {gather*} -\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )-\mathrm {i}}\right )\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )+1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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