Optimal. Leaf size=19 \[ \frac{2}{3} \sin (x) \cos (x) \sqrt{\tan (x) \sec ^4(x)} \]
[Out]
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Rubi [A] time = 0.22063, antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{2 \tan ^2(x) \sec ^2(x)}{3 \sqrt{\tan ^5(x)+2 \tan ^3(x)+\tan (x)}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[Sec[x]^4*Tan[x]],x]
[Out]
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Rubi in Sympy [A] time = 1.65225, size = 20, normalized size = 1.05 \[ \frac{2 \sqrt{\frac{\sin{\left (x \right )}}{\cos ^{5}{\left (x \right )}}} \sin{\left (x \right )} \cos{\left (x \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((sin(x)/cos(x)**5)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0127513, size = 19, normalized size = 1. \[ \frac{2}{3} \sin (x) \cos (x) \sqrt{\tan (x) \sec ^4(x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[Sec[x]^4*Tan[x]],x]
[Out]
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Maple [A] time = 0.201, size = 16, normalized size = 0.8 \[{\frac{2\,\cos \left ( x \right ) \sin \left ( x \right ) }{3}\sqrt{{\frac{\sin \left ( x \right ) }{ \left ( \cos \left ( x \right ) \right ) ^{5}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((sin(x)/cos(x)^5)^(1/2),x)
[Out]
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Maxima [A] time = 1.5454, size = 8, normalized size = 0.42 \[ \frac{2}{3} \, \tan \left (x\right )^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sin(x)/cos(x)^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221691, size = 20, normalized size = 1.05 \[ \frac{2}{3} \, \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right )^{5}}} \cos \left (x\right ) \sin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sin(x)/cos(x)^5),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sin(x)/cos(x)**5)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right )^{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sin(x)/cos(x)^5),x, algorithm="giac")
[Out]