3.412 \(\int \sqrt{\sec ^4(x) \tan (x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{2}{3} \sin (x) \cos (x) \sqrt{\tan (x) \sec ^4(x)} \]

[Out]

(2*Cos[x]*Sin[x]*Sqrt[Sec[x]^4*Tan[x]])/3

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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]

(2*Sec[x]^2*Tan[x]^2)/(3*Sqrt[Tan[x] + 2*Tan[x]^3 + Tan[x]^5])

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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]

2*sqrt(sin(x)/cos(x)**5)*sin(x)*cos(x)/3

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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]

(2*Cos[x]*Sin[x]*Sqrt[Sec[x]^4*Tan[x]])/3

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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]

2/3*(sin(x)/cos(x)^5)^(1/2)*sin(x)*cos(x)

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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]

2/3*tan(x)^(3/2)

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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]

2/3*sqrt(sin(x)/cos(x)^5)*cos(x)*sin(x)

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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]

Timed 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]

integrate(sqrt(sin(x)/cos(x)^5), x)