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 = 0.144341, 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, Rules used = {6719, 14} \[ \frac{3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac{3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt [3]{x^2 \left (1+x^2\right )^6}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac{\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \operatorname{Subst}\left (\int x^{2/3} \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac{2}{3}}(x)}\\ &=\frac{\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \operatorname{Subst}\left (\int \left (x^{2/3}+x^{8/3}\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac{2}{3}}(x)}\\ &=\frac{3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac{3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.146176, size = 63, normalized size = 1.34 \[ \frac{3 \sin (x) \cos (x) \left (8 \left (-\tan ^2(x)\right )^{5/6}+3 \cos (2 x) \left (\left (-\tan ^2(x)\right )^{5/6}-1\right )-3\right ) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}}{55 \left (-\tan ^2(x)\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Cos[x]*Sin[x]*(Sec[x]^12*Tan[x]^2)^(1/3)*(-3 + 8*(-Tan[x]^2)^(5/6) + 3*Cos[2*x]*(-1 + (-Tan[x]^2)^(5/6))))/
(55*(-Tan[x]^2)^(5/6))

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Maple [F]  time = 0.348, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) \right ) ^{14}}}}\, dx \end{align*}

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 = 1.52159, size = 18, normalized size = 0.38 \begin{align*} \frac{3}{11} \, \tan \left (x\right )^{\frac{11}{3}} + \frac{3}{5} \, \tan \left (x\right )^{\frac{5}{3}} \end{align*}

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 = 2.87204, size = 96, normalized size = 2.04 \begin{align*} \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 ) \end{align*}

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. \begin{align*} \text{Timed out} \end{align*}

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

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)