Optimal. Leaf size=31 \[ \frac {1}{5} \sqrt {\sin (2 x)} \sec ^3(x)+\frac {4}{5} \sqrt {\sin (2 x)} \sec (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4299, 4291} \[ \frac {1}{5} \sqrt {\sin (2 x)} \sec ^3(x)+\frac {4}{5} \sqrt {\sin (2 x)} \sec (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4291
Rule 4299
Rubi steps
\begin {align*} \int \frac {\sec ^3(x)}{\sqrt {\sin (2 x)}} \, dx &=\frac {1}{5} \sec ^3(x) \sqrt {\sin (2 x)}+\frac {4}{5} \int \frac {\sec (x)}{\sqrt {\sin (2 x)}} \, dx\\ &=\frac {4}{5} \sec (x) \sqrt {\sin (2 x)}+\frac {1}{5} \sec ^3(x) \sqrt {\sin (2 x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 20, normalized size = 0.65 \[ \frac {1}{5} \sqrt {\sin (2 x)} \sec (x) \left (\sec ^2(x)+4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^3(x)}{\sqrt {\sin (2 x)}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 32, normalized size = 1.03 \[ \frac {4 \, \cos \relax (x)^{3} + \sqrt {2} {\left (4 \, \cos \relax (x)^{2} + 1\right )} \sqrt {\cos \relax (x) \sin \relax (x)}}{5 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \relax (x)^{3} \sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.21, size = 286, normalized size = 9.23
method | result | size |
default | \(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (5 \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+15 \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-14 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+15 \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+5 \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+14 \tan \left (\frac {x}{2}\right )\right )}{12 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \relax (x)^{3} \sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.39, size = 20, normalized size = 0.65 \[ \frac {\sqrt {\sin \left (2\,x\right )}\,\left (2\,\cos \left (2\,x\right )+3\right )}{5\,{\cos \relax (x)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________