Integrand size = 61, antiderivative size = 133 \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{1-3 \sec ^2(x)}}{\sqrt {3}}\right )+\frac {1}{4} \log \left (\sec ^2(x)\right )-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{2 \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \] Output:
1/4*ln(sec(x)^2)-3/2*ln(1-(1-3*sec(x)^2)^(1/6))+1/3*ln(1-(1-3*sec(x)^2)^(1 /2))-(1-3*sec(x)^2)^(1/6)-1/4*(1-3*sec(x)^2)^(2/3)+arctan(1/3*(1+2*(1-3*se c(x)^2)^(1/6))*3^(1/2))*3^(1/2)+1/2/(1-(1-3*sec(x)^2)^(1/2))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 18.10 (sec) , antiderivative size = 785, normalized size of antiderivative = 5.90 \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\frac {2 (-5+\cos (2 x)) \sin ^2(x) \left (1+\sqrt [6]{-2-3 \tan ^2(x)}\right ) \left (1-\sqrt [6]{-2-3 \tan ^2(x)}+\sqrt [3]{-2-3 \tan ^2(x)}\right ) \left (6+\sqrt [3]{-2-3 \tan ^2(x)}+\cos (2 x) \sqrt [3]{-2-3 \tan ^2(x)}\right ) \left (-6 \arctan \left (\sqrt {2+3 \tan ^2(x)}\right ) \sqrt {-2-3 \tan ^2(x)}+5 \sqrt {2+3 \tan ^2(x)}+4 \text {arctanh}\left (\sqrt {-2-3 \tan ^2(x)}\right ) \sqrt {2+3 \tan ^2(x)}-\cos (2 x) \sqrt {2+3 \tan ^2(x)}-5 \log \left (\sec ^2(x)\right ) \sqrt {2+3 \tan ^2(x)}+9 \log \left (1-\sqrt [3]{-2-3 \tan ^2(x)}\right ) \sqrt {2+3 \tan ^2(x)}+12 \sqrt [6]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)}-36 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},-2-3 \tan ^2(x)\right ) \sqrt [6]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)}+3 \left (-2-3 \tan ^2(x)\right )^{2/3} \sqrt {2+3 \tan ^2(x)}-\sqrt {-\left (2+3 \tan ^2(x)\right )^2}-\cos (2 x) \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+6 \arctan \left (\frac {1+2 \sqrt [3]{-2-3 \tan ^2(x)}}{\sqrt {3}}\right ) \sqrt {6+9 \tan ^2(x)}\right )}{3 \left (132 \sqrt {2+3 \tan ^2(x)}-18 \sqrt [3]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)}+10 \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)}+\cos (6 x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)}+132 \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-8 \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+\cos (6 x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-2 \cos (4 x) \left (-6 \sqrt {2+3 \tan ^2(x)}-\sqrt [3]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)}+5 \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)}-6 \sqrt {-\left (2+3 \tan ^2(x)\right )^2}\right )-\cos (2 x) \left (144 \sqrt {2+3 \tan ^2(x)}+16 \sqrt [3]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)}+\left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)}+144 \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+9 \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}\right )\right )} \] Input:
Integrate[(Sec[x]^2*Tan[x]*((1 - 3*Sec[x]^2)^(1/3)*Sin[x]^2 + 3*Tan[x]^2)) /((1 - 3*Sec[x]^2)^(5/6)*(1 - Sqrt[1 - 3*Sec[x]^2])),x]
Output:
(2*(-5 + Cos[2*x])*Sin[x]^2*(1 + (-2 - 3*Tan[x]^2)^(1/6))*(1 - (-2 - 3*Tan [x]^2)^(1/6) + (-2 - 3*Tan[x]^2)^(1/3))*(6 + (-2 - 3*Tan[x]^2)^(1/3) + Cos [2*x]*(-2 - 3*Tan[x]^2)^(1/3))*(-6*ArcTan[Sqrt[2 + 3*Tan[x]^2]]*Sqrt[-2 - 3*Tan[x]^2] + 5*Sqrt[2 + 3*Tan[x]^2] + 4*ArcTanh[Sqrt[-2 - 3*Tan[x]^2]]*Sq rt[2 + 3*Tan[x]^2] - Cos[2*x]*Sqrt[2 + 3*Tan[x]^2] - 5*Log[Sec[x]^2]*Sqrt[ 2 + 3*Tan[x]^2] + 9*Log[1 - (-2 - 3*Tan[x]^2)^(1/3)]*Sqrt[2 + 3*Tan[x]^2] + 12*(-2 - 3*Tan[x]^2)^(1/6)*Sqrt[2 + 3*Tan[x]^2] - 36*Hypergeometric2F1[1 /6, 1, 7/6, -2 - 3*Tan[x]^2]*(-2 - 3*Tan[x]^2)^(1/6)*Sqrt[2 + 3*Tan[x]^2] + 3*(-2 - 3*Tan[x]^2)^(2/3)*Sqrt[2 + 3*Tan[x]^2] - Sqrt[-(2 + 3*Tan[x]^2)^ 2] - Cos[2*x]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 6*ArcTan[(1 + 2*(-2 - 3*Tan[x]^2 )^(1/3))/Sqrt[3]]*Sqrt[6 + 9*Tan[x]^2]))/(3*(132*Sqrt[2 + 3*Tan[x]^2] - 18 *(-2 - 3*Tan[x]^2)^(1/3)*Sqrt[2 + 3*Tan[x]^2] + 10*(-2 - 3*Tan[x]^2)^(5/6) *Sqrt[2 + 3*Tan[x]^2] + Cos[6*x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x] ^2] + 132*Sqrt[-(2 + 3*Tan[x]^2)^2] - 8*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[-(2 + 3*Tan[x]^2)^2] + Cos[6*x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[-(2 + 3*Tan[x]^2)^ 2] - 2*Cos[4*x]*(-6*Sqrt[2 + 3*Tan[x]^2] - (-2 - 3*Tan[x]^2)^(1/3)*Sqrt[2 + 3*Tan[x]^2] + 5*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2] - 6*Sqrt[-( 2 + 3*Tan[x]^2)^2]) - Cos[2*x]*(144*Sqrt[2 + 3*Tan[x]^2] + 16*(-2 - 3*Tan[ x]^2)^(1/3)*Sqrt[2 + 3*Tan[x]^2] + (-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[ x]^2] + 144*Sqrt[-(2 + 3*Tan[x]^2)^2] + 9*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[...
Time = 4.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.50, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {3042, 4861, 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.
| \(\displaystyle \int \frac {\tan (x) \sec ^2(x) \left (3 \tan ^2(x)+\sin ^2(x) \sqrt [3]{1-3 \sec ^2(x)}\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (x) \sec (x)^2 \left (3 \tan (x)^2+\sin (x)^2 \sqrt [3]{1-3 \sec (x)^2}\right )}{\left (1-3 \sec (x)^2\right )^{5/6} \left (1-\sqrt {1-3 \sec (x)^2}\right )}dx\) |
| \(\Big \downarrow \) 4861 |
| \(\displaystyle -\int \frac {\left (1-\cos ^2(x)\right ) \sec ^5(x) \left (\sqrt [3]{1-3 \sec ^2(x)} \cos ^2(x)+3\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )}d\cos (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {\left (-\sqrt [3]{\left (\cos ^2(x)-3\right ) \sec ^2(x)} \cos ^2(x)-3\right ) \sec ^5(x)}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (\sqrt {\left (\cos ^2(x)-3\right ) \sec ^2(x)}-1\right )}+\frac {\left (\sqrt [3]{\left (\cos ^2(x)-3\right ) \sec ^2(x)} \cos ^2(x)+3\right ) \sec ^3(x)}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (\sqrt {\left (\cos ^2(x)-3\right ) \sec ^2(x)}-1\right )}\right )d\cos (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {3-\cos ^2(x)} \sec (x) \arcsin \left (\frac {\cos (x)}{\sqrt {3}}\right )}{2 \sqrt {1-3 \sec ^2(x)}}+\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt {3}}\right )+\frac {\cos ^2(x)}{6}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )-\frac {3-\cos ^2(x)}{6 \sqrt {1-3 \sec ^2(x)}}+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )\) |
Input:
Int[(Sec[x]^2*Tan[x]*((1 - 3*Sec[x]^2)^(1/3)*Sin[x]^2 + 3*Tan[x]^2))/((1 - 3*Sec[x]^2)^(5/6)*(1 - Sqrt[1 - 3*Sec[x]^2])),x]
Output:
Sqrt[3]*ArcTan[(1 + 2*(1 - 3*Sec[x]^2)^(1/6))/Sqrt[3]] + Cos[x]^2/6 + Log[ 1 - Sqrt[-((3 - Cos[x]^2)*Sec[x]^2)]]/3 - (3*Log[1 - (1 - 3*Sec[x]^2)^(1/6 )])/2 + Log[1 - Sqrt[1 - 3*Sec[x]^2]]/2 - (3 - Cos[x]^2)/(6*Sqrt[1 - 3*Sec [x]^2]) + (ArcSin[Cos[x]/Sqrt[3]]*Sqrt[3 - Cos[x]^2]*Sec[x])/(2*Sqrt[1 - 3 *Sec[x]^2]) - (1 - 3*Sec[x]^2)^(1/6) - (1 - 3*Sec[x]^2)^(2/3)/4
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto rs[Cos[c*(a + b*x)], x]}, Simp[-(b*c)^(-1) Subst[Int[SubstFor[1/x, Cos[c* (a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])
\[\int \frac {\tan \left (x \right ) \left (\left (1-3 \sec \left (x \right )^{2}\right )^{\frac {1}{3}} \sin \left (x \right )^{2}+3 \tan \left (x \right )^{2}\right )}{\cos \left (x \right )^{2} \left (1-3 \sec \left (x \right )^{2}\right )^{\frac {5}{6}} \left (1-\sqrt {1-3 \sec \left (x \right )^{2}}\right )}d x\]
Input:
int(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3*sec(x) ^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x)
Output:
int(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3*sec(x) ^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x)
Exception generated. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3* sec(x)^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: Curv e not irreducible after change of variable 0 -> infinity
Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Timed out} \] Input:
integrate(tan(x)*((1-3*sec(x)**2)**(1/3)*sin(x)**2+3*tan(x)**2)/cos(x)**2/ (1-3*sec(x)**2)**(5/6)/(1-(1-3*sec(x)**2)**(1/2)),x)
Output:
Timed out
Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Timed out} \] Input:
integrate(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3* sec(x)^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x, algorithm="maxima")
Output:
Timed out
Exception generated. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3* sec(x)^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=-\int \frac {\mathrm {tan}\left (x\right )\,\left ({\sin \left (x\right )}^2\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{1/3}+3\,{\mathrm {tan}\left (x\right )}^2\right )}{{\cos \left (x\right )}^2\,\left (\sqrt {1-\frac {3}{{\cos \left (x\right )}^2}}-1\right )\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{5/6}} \,d x \] Input:
int(-(tan(x)*(sin(x)^2*(1 - 3/cos(x)^2)^(1/3) + 3*tan(x)^2))/(cos(x)^2*((1 - 3/cos(x)^2)^(1/2) - 1)*(1 - 3/cos(x)^2)^(5/6)),x)
Output:
-int((tan(x)*(sin(x)^2*(1 - 3/cos(x)^2)^(1/3) + 3*tan(x)^2))/(cos(x)^2*((1 - 3/cos(x)^2)^(1/2) - 1)*(1 - 3/cos(x)^2)^(5/6)), x)
\[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\frac {\left (\int \frac {\sin \left (x \right )^{2} \tan \left (x \right )}{\cos \left (x \right )^{2} \sec \left (x \right )^{2}}d x \right )}{3}+\int \frac {\left (-3 \sec \left (x \right )^{2}+1\right )^{\frac {2}{3}} \tan \left (x \right )^{3}}{9 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{6}-6 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{4}+\sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{2}}d x -3 \left (\int \frac {\left (-3 \sec \left (x \right )^{2}+1\right )^{\frac {2}{3}} \tan \left (x \right )^{3}}{9 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{4}-6 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{2}+\sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2}}d x \right )-\left (\int \frac {\left (-3 \sec \left (x \right )^{2}+1\right )^{\frac {1}{6}} \tan \left (x \right )^{3}}{3 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{4}-\sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{2}}d x \right )+3 \left (\int \frac {\left (-3 \sec \left (x \right )^{2}+1\right )^{\frac {1}{6}} \tan \left (x \right )^{3}}{3 \sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{2}-\sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2}}d x \right )+\frac {\left (\int \frac {\sin \left (x \right )^{2} \tan \left (x \right )}{\sqrt {-3 \sec \left (x \right )^{2}+1}\, \cos \left (x \right )^{2} \sec \left (x \right )^{2}}d x \right )}{3} \] Input:
int(tan(x)*((1-3*sec(x)^2)^(1/3)*sin(x)^2+3*tan(x)^2)/cos(x)^2/(1-3*sec(x) ^2)^(5/6)/(1-(1-3*sec(x)^2)^(1/2)),x)
Output:
(int((( - 3*sec(x)**2 + 1)**(5/6)*sin(x)**2*tan(x))/(( - 3*sec(x)**2 + 1)* *(5/6)*cos(x)**2*sec(x)**2),x) + 3*int((( - 3*sec(x)**2 + 1)**(2/3)*tan(x) **3)/(9*sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)**6 - 6*sqrt( - 3*sec(x)* *2 + 1)*cos(x)**2*sec(x)**4 + sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)**2 ),x) - 9*int((( - 3*sec(x)**2 + 1)**(2/3)*tan(x)**3)/(9*sqrt( - 3*sec(x)** 2 + 1)*cos(x)**2*sec(x)**4 - 6*sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)** 2 + sqrt( - 3*sec(x)**2 + 1)*cos(x)**2),x) - 3*int((( - 3*sec(x)**2 + 1)** (1/6)*tan(x)**3)/(3*sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)**4 - sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)**2),x) + 9*int((( - 3*sec(x)**2 + 1)**( 1/6)*tan(x)**3)/(3*sqrt( - 3*sec(x)**2 + 1)*cos(x)**2*sec(x)**2 - sqrt( - 3*sec(x)**2 + 1)*cos(x)**2),x) + int((( - 3*sec(x)**2 + 1)**(1/3)*sin(x)** 2*tan(x))/(( - 3*sec(x)**2 + 1)**(5/6)*cos(x)**2*sec(x)**2),x))/3