Optimal. Leaf size=28 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, 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 = {3661, 377, 203} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 377
Rule 3661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\cot ^2(x)}} \, dx &=-\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}{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*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 42, normalized size = 1.50 \[ -\frac {\sqrt {\cos (2 x)} \csc (x) \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right )}{\sqrt {2-2 \cot ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 56, normalized size = 2.00 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right ) + \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right )^{2} + \cos \left (2 \, x\right )\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.21, size = 34, normalized size = 1.21 \[ -\frac {1}{2} i \, \sqrt {2} \log \left (i \, \sqrt {2} + i\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) - \frac {\sqrt {2} \arcsin \left (\sqrt {2} \cos \relax (x)\right )}{2 \, \mathrm {sgn}\left (\sin \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 31, normalized size = 1.11 \[ \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\left (\cot ^{2}\relax (x )\right )}\, \cot \relax (x )}{-1+\cot ^{2}\relax (x )}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.51, size = 90, normalized size = 3.21 \[ \frac {1}{4} \, \sqrt {2} \arctan \left ({\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ), {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \cos \left (2 \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.63, size = 85, normalized size = 3.04 \[ -\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\relax (x)}^2}\,1{}\mathrm {i}}{\mathrm {cot}\relax (x)-\mathrm {i}}\right )\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\relax (x)}^2}\,1{}\mathrm {i}}{\mathrm {cot}\relax (x)+1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - \cot ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________