Optimal. Leaf size=32 \[ \tan ^{-1}\left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right )-\frac {\sin ^{-1}(\tan (x))}{\sqrt {2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {12, 402, 216, 377, 203} \[ \tan ^{-1}\left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right )-\frac {\sin ^{-1}(\tan (x))}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 203
Rule 216
Rule 377
Rule 402
Rubi steps
\begin {align*} \int \sqrt {\cot (2 x) \tan (x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {2}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tan (x)\right )}{\sqrt {2}}+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac {\sin ^{-1}(\tan (x))}{\sqrt {2}}+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {1-\tan ^2(x)}}\right )\\ &=-\frac {\sin ^{-1}(\tan (x))}{\sqrt {2}}+\tan ^{-1}\left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 52, normalized size = 1.62 \[ \frac {\cos (x) \sqrt {\tan (x) \cot (2 x)} \left (\sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin (x)\right )-\tan ^{-1}\left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )\right )}{\sqrt {\cos (2 x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.25, size = 115, normalized size = 3.59 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (3 \, \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right )^{2} + \sqrt {2} \cos \left (2 \, x\right ) - \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.83, size = 138, normalized size = 4.31 \[ \frac {1}{2} \, {\left (\pi - \sqrt {2} \arctan \left (-i\right ) - \sqrt {2} \arctan \left (\sqrt {2}\right ) - i \, \log \left (2 \, \sqrt {2} + 3\right )\right )} \mathrm {sgn}\left (\sin \left (2 \, x\right )\right ) - \frac {\sqrt {2} {\left (-i \, \sqrt {2} \log \left (2 i \, \sqrt {2} + 3 i\right ) - 2 \, \arctan \left (-i\right )\right )} \mathrm {sgn}\left (\cos \relax (x)\right ) + 2 \, {\left (\sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (\frac {3 \, {\left (2 \, \sqrt {2} \sqrt {-2 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1} - 1\right )}}{4 \, \cos \relax (x)^{2} - 3} - 1\right )}\right ) + \arcsin \left (4 \, \cos \relax (x)^{2} - 3\right )\right )} \mathrm {sgn}\left (\cos \relax (x)\right )}{4 \, \mathrm {sgn}\left (\cos \relax (x)\right ) \mathrm {sgn}\left (\sin \left (2 \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.76, size = 242, normalized size = 7.56
method | result | size |
default | \(\frac {\sqrt {2}\, \left (4 \EllipticPi \left (\frac {\sqrt {3+2 \sqrt {2}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, -\frac {1}{3+2 \sqrt {2}}, \frac {\sqrt {3-2 \sqrt {2}}}{\sqrt {3+2 \sqrt {2}}}\right )-2 \EllipticPi \left (\frac {\sqrt {3+2 \sqrt {2}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{3+2 \sqrt {2}}, \frac {\sqrt {3-2 \sqrt {2}}}{\sqrt {3+2 \sqrt {2}}}\right )-\EllipticF \left (\frac {\left (-1+\cos \relax (x )\right ) \left (1+\sqrt {2}\right )}{\sin \relax (x )}, 3-2 \sqrt {2}\right )\right ) \left (2+\sqrt {2}\right ) \cos \relax (x ) \left (\sin ^{2}\relax (x )\right ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\cos \relax (x )^{2}}}\, \sqrt {-\frac {2 \left (\cos \relax (x ) \sqrt {2}-\sqrt {2}-2 \cos \relax (x )+1\right )}{1+\cos \relax (x )}}\, \sqrt {\frac {\cos \relax (x ) \sqrt {2}-\sqrt {2}+2 \cos \relax (x )-1}{1+\cos \relax (x )}}}{2 \sqrt {3+2 \sqrt {2}}\, \left (1+\sqrt {2}\right ) \left (-1+\cos \relax (x )\right ) \left (2 \left (\cos ^{2}\relax (x )\right )-1\right )}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {\frac {\mathrm {cot}\left (2\,x\right )}{\mathrm {cot}\relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {\cot {\left (2 x \right )}}{\cot {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________