Integrand size = 28, antiderivative size = 125 \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=-\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9}{4} \sec ^8(x) \left (\cos ^5(x) \sin (x)\right )^{4/3}+\frac {3}{2} \sqrt [3]{\cos ^5(x) \sin (x)} \sqrt [3]{\sec ^6(x) \tan (x)}+\frac {3}{4} \sqrt [3]{\cos ^5(x) \sin (x)} \tan ^2(x) \sqrt [3]{\sec ^6(x) \tan (x)}+\frac {3}{14} \sqrt [3]{\cos ^5(x) \sin (x)} \tan ^4(x) \sqrt [3]{\sec ^6(x) \tan (x)} \] Output:
-9/10*sin(x)^4/(cos(x)^5*sin(x))^(2/3)-9/4*sec(x)^8*(cos(x)^5*sin(x))^(4/3 )+3/2*(cos(x)^5*sin(x))^(1/3)*(sec(x)^6*tan(x))^(1/3)+3/4*(cos(x)^5*sin(x) )^(1/3)*tan(x)^2*(sec(x)^6*tan(x))^(1/3)+3/14*(cos(x)^5*sin(x))^(1/3)*tan( x)^4*(sec(x)^6*tan(x))^(1/3)
Time = 0.60 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.46 \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=-\frac {3 \sin (x) \left (924 \sin (x)+252 \sin (3 x)-5 (158 \cos (x)+57 \cos (3 x)+9 \cos (5 x)) \sqrt [3]{\sec ^6(x) \tan (x)}\right )}{2240 \left (\cos ^5(x) \sin (x)\right )^{2/3}} \] Input:
Integrate[(-3*Tan[x] + (Sec[x]^6*Tan[x])^(1/3))/(Cos[x]^5*Sin[x])^(2/3),x]
Output:
(-3*Sin[x]*(924*Sin[x] + 252*Sin[3*x] - 5*(158*Cos[x] + 57*Cos[3*x] + 9*Co s[5*x])*(Sec[x]^6*Tan[x])^(1/3)))/(2240*(Cos[x]^5*Sin[x])^(2/3))
Time = 0.99 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4889, 25, 7270, 2035, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt [3]{\tan (x) \sec ^6(x)}-3 \tan (x)}{\left (\sin (x) \cos ^5(x)\right )^{2/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt [3]{\tan (x) \sec (x)^6}-3 \tan (x)}{\left (\sin (x) \cos (x)^5\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \int -\frac {3 \tan (x)-\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3}}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )}d\tan (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {3 \tan (x)-\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3}}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )}d\tan (x)\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle -\frac {\tan ^{\frac {2}{3}}(x) \int \frac {\left (\tan ^2(x)+1\right ) \left (3 \tan (x)-\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3}\right )}{\tan ^{\frac {2}{3}}(x)}d\tan (x)}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )^2}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \tan ^{\frac {2}{3}}(x) \int \left (\tan ^2(x)+1\right ) \left (3 \tan (x)-\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3}\right )d\sqrt [3]{\tan (x)}}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \tan ^{\frac {2}{3}}(x) \int \left (\left (3 \tan (x)-\sqrt [3]{\left (\tan ^{\frac {7}{3}}(x)+\sqrt [3]{\tan (x)}\right )^3}\right ) \tan ^2(x)+3 \tan (x)-\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3}\right )d\sqrt [3]{\tan (x)}}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \tan ^{\frac {2}{3}}(x) \left (\frac {3}{10} \tan ^{\frac {10}{3}}(x)+\frac {3}{4} \tan ^{\frac {4}{3}}(x)-\frac {\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3} \sqrt [3]{\tan (x)}}{2 \left (\tan ^2(x)+1\right )}-\frac {\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3} \tan ^{\frac {13}{3}}(x)}{14 \left (\tan ^2(x)+1\right )}-\frac {\sqrt [3]{\tan (x) \left (\tan ^2(x)+1\right )^3} \tan ^{\frac {7}{3}}(x)}{4 \left (\tan ^2(x)+1\right )}\right )}{\left (\frac {\tan (x)}{\left (\tan ^2(x)+1\right )^3}\right )^{2/3} \left (\tan ^2(x)+1\right )^2}\) |
Input:
Int[(-3*Tan[x] + (Sec[x]^6*Tan[x])^(1/3))/(Cos[x]^5*Sin[x])^(2/3),x]
Output:
(-3*Tan[x]^(2/3)*((3*Tan[x]^(4/3))/4 + (3*Tan[x]^(10/3))/10 - (Tan[x]^(1/3 )*(Tan[x]*(1 + Tan[x]^2)^3)^(1/3))/(2*(1 + Tan[x]^2)) - (Tan[x]^(7/3)*(Tan [x]*(1 + Tan[x]^2)^3)^(1/3))/(4*(1 + Tan[x]^2)) - (Tan[x]^(13/3)*(Tan[x]*( 1 + Tan[x]^2)^3)^(1/3))/(14*(1 + Tan[x]^2))))/((Tan[x]/(1 + Tan[x]^2)^3)^( 2/3)*(1 + Tan[x]^2)^2)
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {\left (\frac {\sin \left (x \right )}{\cos \left (x \right )^{7}}\right )^{\frac {1}{3}}-3 \tan \left (x \right )}{\left (\sin \left (x \right ) \cos \left (x \right )^{5}\right )^{\frac {2}{3}}}d x\]
Input:
int(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(sin(x)*cos(x)^5)^(2/3),x)
Output:
int(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(sin(x)*cos(x)^5)^(2/3),x)
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.45 \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=-\frac {3 \, \left (\cos \left (x\right )^{5} \sin \left (x\right )\right )^{\frac {1}{3}} {\left (21 \, {\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) - 5 \, {\left (9 \, \cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{3} + 2 \, \cos \left (x\right )\right )} \left (\frac {\sin \left (x\right )}{\cos \left (x\right )^{7}}\right )^{\frac {1}{3}}\right )}}{140 \, \cos \left (x\right )^{5}} \] Input:
integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, al gorithm="fricas")
Output:
-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)/cos(x)^7)^(1/3))/cos(x)^5
Timed out. \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=\text {Timed out} \] Input:
integrate(((sin(x)/cos(x)**7)**(1/3)-3*tan(x))/(cos(x)**5*sin(x))**(2/3),x )
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.48 \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=-\frac {3}{20} \, \tan \left (x\right )^{\frac {20}{3}} - \frac {3}{7} \, \tan \left (x\right )^{\frac {14}{3}} - \frac {9}{10} \, \tan \left (x\right )^{\frac {10}{3}} - \frac {3}{8} \, \tan \left (x\right )^{\frac {8}{3}} - \frac {9}{4} \, \tan \left (x\right )^{\frac {4}{3}} + \frac {3 \, {\left (14 \, \tan \left (x\right )^{7} + 60 \, \tan \left (x\right )^{5} + 105 \, \tan \left (x\right )^{3} + 140 \, \tan \left (x\right )\right )}}{280 \, \tan \left (x\right )^{\frac {1}{3}}} \] Input:
integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, al gorithm="maxima")
Output:
-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)
\[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=\int { \frac {\left (\frac {\sin \left (x\right )}{\cos \left (x\right )^{7}}\right )^{\frac {1}{3}} - 3 \, \tan \left (x\right )}{\left (\cos \left (x\right )^{5} \sin \left (x\right )\right )^{\frac {2}{3}}} \,d x } \] Input:
integrate(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x, al gorithm="giac")
Output:
integrate(((sin(x)/cos(x)^7)^(1/3) - 3*tan(x))/(cos(x)^5*sin(x))^(2/3), x)
Timed out. \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=\int -\frac {3\,\mathrm {tan}\left (x\right )-{\left (\frac {\sin \left (x\right )}{{\cos \left (x\right )}^7}\right )}^{1/3}}{{\left ({\cos \left (x\right )}^5\,\sin \left (x\right )\right )}^{2/3}} \,d x \] Input:
int(-(3*tan(x) - (sin(x)/cos(x)^7)^(1/3))/(cos(x)^5*sin(x))^(2/3),x)
Output:
int(-(3*tan(x) - (sin(x)/cos(x)^7)^(1/3))/(cos(x)^5*sin(x))^(2/3), x)
\[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx=-3 \left (\int \frac {\tan \left (x \right )}{\sin \left (x \right )^{\frac {2}{3}} \cos \left (x \right )^{\frac {10}{3}}}d x \right )+\int \frac {1}{\sin \left (x \right )^{\frac {1}{3}} \cos \left (x \right )^{\frac {17}{3}}}d x \] Input:
int(((sin(x)/cos(x)^7)^(1/3)-3*tan(x))/(cos(x)^5*sin(x))^(2/3),x)
Output:
- 3*int(tan(x)/(sin(x)**(2/3)*cos(x)**(1/3)*cos(x)**3),x) + int(1/(sin(x) **(1/3)*cos(x)**(2/3)*cos(x)**5),x)