3.414 \(\int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx\)

Optimal. Leaf size=47 \[ \frac{3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac{3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]

[Out]

(3*Cos[x]^3*Sin[x]*(Sec[x]^12*Tan[x]^2)^(1/3))/5 + (3*Cos[x]*Sin[x]^3*(Sec[x]^12
*Tan[x]^2)^(1/3))/11

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Rubi [A]  time = 0.234184, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac{3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]

Antiderivative was successfully verified.

[In]  Int[(Sec[x]^12*Tan[x]^2)^(1/3),x]

[Out]

(3*Cos[x]^3*Sin[x]*(Sec[x]^12*Tan[x]^2)^(1/3))/5 + (3*Cos[x]*Sin[x]^3*(Sec[x]^12
*Tan[x]^2)^(1/3))/11

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Rubi in Sympy [A]  time = 1.75774, size = 44, normalized size = 0.94 \[ \frac{3 \sqrt [3]{\frac{\sin ^{2}{\left (x \right )}}{\cos ^{14}{\left (x \right )}}} \sin{\left (x \right )} \cos ^{5}{\left (x \right )}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{17}{6} \\ \frac{11}{6} \end{matrix}\middle |{\sin ^{2}{\left (x \right )}} \right )}}{5 \sqrt [6]{\cos ^{2}{\left (x \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((sin(x)**2/cos(x)**14)**(1/3),x)

[Out]

3*(sin(x)**2/cos(x)**14)**(1/3)*sin(x)*cos(x)**5*hyper((5/6, 17/6), (11/6,), sin
(x)**2)/(5*(cos(x)**2)**(1/6))

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Mathematica [A]  time = 0.0346558, size = 29, normalized size = 0.62 \[ \frac{3}{55} \sin (x) \cos (x) \left (6 \cos ^2(x)+5\right ) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sec[x]^12*Tan[x]^2)^(1/3),x]

[Out]

(3*Cos[x]*(5 + 6*Cos[x]^2)*Sin[x]*(Sec[x]^12*Tan[x]^2)^(1/3))/55

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Maple [F]  time = 0.542, size = 0, normalized size = 0. \[ \int \sqrt [3]{{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) \right ) ^{14}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((sin(x)^2/cos(x)^14)^(1/3),x)

[Out]

int((sin(x)^2/cos(x)^14)^(1/3),x)

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Maxima [A]  time = 1.8147, size = 18, normalized size = 0.38 \[ \frac{3}{11} \, \tan \left (x\right )^{\frac{11}{3}} + \frac{3}{5} \, \tan \left (x\right )^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sin(x)^2/cos(x)^14)^(1/3),x, algorithm="maxima")

[Out]

3/11*tan(x)^(11/3) + 3/5*tan(x)^(5/3)

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Fricas [A]  time = 0.21367, size = 39, normalized size = 0.83 \[ \frac{3}{55} \,{\left (6 \, \cos \left (x\right )^{3} + 5 \, \cos \left (x\right )\right )} \left (-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )^{14}}\right )^{\frac{1}{3}} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sin(x)^2/cos(x)^14)^(1/3),x, algorithm="fricas")

[Out]

3/55*(6*cos(x)^3 + 5*cos(x))*(-(cos(x)^2 - 1)/cos(x)^14)^(1/3)*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)**2/cos(x)**14)**(1/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{\sin \left (x\right )^{2}}{\cos \left (x\right )^{14}}\right )^{\frac{1}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sin(x)^2/cos(x)^14)^(1/3),x, algorithm="giac")

[Out]

integrate((sin(x)^2/cos(x)^14)^(1/3), x)