3.418 \(\int \frac{-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{(\cos ^5(x) \sin (x))^{2/3}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{9 \sin ^4(x)}{10 \left (\sin (x) \cos ^5(x)\right )^{2/3}}-\frac{9}{4} \sec ^8(x) \left (\sin (x) \cos ^5(x)\right )^{4/3}+\frac{3}{14} \tan ^4(x) \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)}+\frac{3}{4} \tan ^2(x) \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)}+\frac{3}{2} \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)} \]

[Out]

(-9*Sin[x]^4)/(10*(Cos[x]^5*Sin[x])^(2/3)) - (9*Sec[x]^8*(Cos[x]^5*Sin[x])^(4/3))/4 + (3*(Cos[x]^5*Sin[x])^(1/
3)*(Sec[x]^6*Tan[x])^(1/3))/2 + (3*(Cos[x]^5*Sin[x])^(1/3)*Tan[x]^2*(Sec[x]^6*Tan[x])^(1/3))/4 + (3*(Cos[x]^5*
Sin[x])^(1/3)*Tan[x]^4*(Sec[x]^6*Tan[x])^(1/3))/14

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Rubi [A]  time = 1.01891, antiderivative size = 141, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6719, 6733, 6742, 14} \[ -\frac{9 \sin ^2(x) \cos ^2(x)}{4 \left (\sin (x) \cos ^5(x)\right )^{2/3}}-\frac{9 \sin ^4(x)}{10 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac{3 \sin ^3(x) \cos ^3(x) \sqrt [3]{\tan (x) \sec ^6(x)}}{4 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac{3 \sin ^5(x) \cos (x) \sqrt [3]{\tan (x) \sec ^6(x)}}{14 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac{3 \sin (x) \cos ^5(x) \sqrt [3]{\tan (x) \sec ^6(x)}}{2 \left (\sin (x) \cos ^5(x)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(-3*Tan[x] + (Sec[x]^6*Tan[x])^(1/3))/(Cos[x]^5*Sin[x])^(2/3),x]

[Out]

(-9*Cos[x]^2*Sin[x]^2)/(4*(Cos[x]^5*Sin[x])^(2/3)) - (9*Sin[x]^4)/(10*(Cos[x]^5*Sin[x])^(2/3)) + (3*Cos[x]^5*S
in[x]*(Sec[x]^6*Tan[x])^(1/3))/(2*(Cos[x]^5*Sin[x])^(2/3)) + (3*Cos[x]^3*Sin[x]^3*(Sec[x]^6*Tan[x])^(1/3))/(4*
(Cos[x]^5*Sin[x])^(2/3)) + (3*Cos[x]*Sin[x]^5*(Sec[x]^6*Tan[x])^(1/3))/(14*(Cos[x]^5*Sin[x])^(2/3))

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 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 \frac{-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx &=\operatorname{Subst}\left (\int \frac{-3 x+\sqrt [3]{x \left (1+x^2\right )^3}}{\left (\frac{x}{\left (1+x^2\right )^3}\right )^{2/3} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac{\left (\cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int \frac{\left (1+x^2\right ) \left (-3 x+\sqrt [3]{x \left (1+x^2\right )^3}\right )}{x^{2/3}} \, dx,x,\tan (x)\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int \left (1+x^6\right ) \left (-3 x^3+\sqrt [3]{x^3 \left (1+x^6\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int \left (-3 x^3+\sqrt [3]{x^3 \left (1+x^6\right )^3}-x^6 \left (3 x^3-\sqrt [3]{\left (x+x^7\right )^3}\right )\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int \sqrt [3]{x^3 \left (1+x^6\right )^3} \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int x^6 \left (3 x^3-\sqrt [3]{\left (x+x^7\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int \left (3 x^9-x^6 \sqrt [3]{x^3 \left (1+x^6\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname{Subst}\left (\int x \left (1+x^6\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^4(x) \tan ^{\frac{2}{3}}(x)\right ) \operatorname{Subst}\left (\int x^6 \sqrt [3]{x^3 \left (1+x^6\right )^3} \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname{Subst}\left (\int \left (x+x^7\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{8 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname{Subst}\left (\int x^7 \left (1+x^6\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{8 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname{Subst}\left (\int \left (x^7+x^{13}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac{9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac{9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac{3 \cos (x) \sin ^5(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{14 \left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.393317, size = 58, normalized size = 0.46 \[ -\frac{3 \sin (x) \left (924 \sin (x)+252 \sin (3 x)-5 (158 \cos (x)+57 \cos (3 x)+9 \cos (5 x)) \sqrt [3]{\tan (x) \sec ^6(x)}\right )}{2240 \left (\sin (x) \cos ^5(x)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3*Tan[x] + (Sec[x]^6*Tan[x])^(1/3))/(Cos[x]^5*Sin[x])^(2/3),x]

[Out]

(-3*Sin[x]*(924*Sin[x] + 252*Sin[3*x] - 5*(158*Cos[x] + 57*Cos[3*x] + 9*Cos[5*x])*(Sec[x]^6*Tan[x])^(1/3)))/(2
240*(Cos[x]^5*Sin[x])^(2/3))

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Maple [F]  time = 0.75, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sqrt [3]{{\frac{\sin \left ( x \right ) }{ \left ( \cos \left ( x \right ) \right ) ^{7}}}}-3\,\tan \left ( x \right ) \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x)

[Out]

int(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x)

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Maxima [A]  time = 1.73825, size = 81, normalized size = 0.65 \begin{align*} -\frac{3}{20} \, \tan \left (x\right )^{\frac{20}{3}} - \frac{3}{7} \, \tan \left (x\right )^{\frac{14}{3}} - \frac{9}{10} \, \tan \left (x\right )^{\frac{10}{3}} - \frac{3}{8} \, \tan \left (x\right )^{\frac{8}{3}} - \frac{9}{4} \, \tan \left (x\right )^{\frac{4}{3}} + \frac{3 \,{\left (14 \, \tan \left (x\right )^{7} + 60 \, \tan \left (x\right )^{5} + 105 \, \tan \left (x\right )^{3} + 140 \, \tan \left (x\right )\right )}}{280 \, \tan \left (x\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, algorithm="maxima")

[Out]

-3/20*tan(x)^(20/3) - 3/7*tan(x)^(14/3) - 9/10*tan(x)^(10/3) - 3/8*tan(x)^(8/3) - 9/4*tan(x)^(4/3) + 3/280*(14
*tan(x)^7 + 60*tan(x)^5 + 105*tan(x)^3 + 140*tan(x))/tan(x)^(1/3)

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Fricas [A]  time = 2.9936, size = 182, normalized size = 1.46 \begin{align*} -\frac{3 \, \left (\cos \left (x\right )^{5} \sin \left (x\right )\right )^{\frac{1}{3}}{\left (21 \,{\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) - 5 \,{\left (9 \, \cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{3} + 2 \, \cos \left (x\right )\right )} \left (\frac{\sin \left (x\right )}{\cos \left (x\right )^{7}}\right )^{\frac{1}{3}}\right )}}{140 \, \cos \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, algorithm="fricas")

[Out]

-3/140*(cos(x)^5*sin(x))^(1/3)*(21*(3*cos(x)^2 + 2)*sin(x) - 5*(9*cos(x)^5 + 3*cos(x)^3 + 2*cos(x))*(sin(x)/co
s(x)^7)^(1/3))/cos(x)^5

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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)/cos(x)**7)**(1/3)-3*tan(x))/(cos(x)**5*sin(x))**(2/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{\sin \left (x\right )}{\cos \left (x\right )^{7}}\right )^{\frac{1}{3}} - 3 \, \tan \left (x\right )}{\left (\cos \left (x\right )^{5} \sin \left (x\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, algorithm="giac")

[Out]

integrate(((sin(x)/cos(x)^7)^(1/3) - 3*tan(x))/(cos(x)^5*sin(x))^(2/3), x)