3.446 \(\int \frac{\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt{1-3 \sec ^2(x)}\right )} \, dx\)
Optimal. Leaf size=133 \[ -\frac{1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}+\frac{1}{2 \left (1-\sqrt{1-3 \sec ^2(x)}\right )}+\frac{1}{4} \log \left (\sec ^2(x)\right )-\frac{3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{1-3 \sec ^2(x)}\right )+\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt{3}}\right ) \]
[Out]
Sqrt[3]*ArcTan[(1 + 2*(1 - 3*Sec[x]^2)^(1/6))/Sqrt[3]] + Log[Sec[x]^2]/4 - (3*Lo
g[1 - (1 - 3*Sec[x]^2)^(1/6)])/2 + Log[1 - Sqrt[1 - 3*Sec[x]^2]]/3 - (1 - 3*Sec[
x]^2)^(1/6) - (1 - 3*Sec[x]^2)^(2/3)/4 + 1/(2*(1 - Sqrt[1 - 3*Sec[x]^2]))
_______________________________________________________________________________________
Rubi [A] time = 8.93239, antiderivative size = 174, normalized size of antiderivative =
1.31, number of steps used = 29, number of rules used = 16, integrand size = 61, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.262
\[ \frac{\cos ^2(x)}{6}-\frac{1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}-\frac{3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac{1}{2} \log \left (1-\sqrt{1-3 \sec ^2(x)}\right )+\frac{1}{6} \cos ^2(x) \sqrt{1-3 \sec ^2(x)}+\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt{3}}\right )+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-3 \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right ) \]
Antiderivative was successfully verified.
[In] 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]
[Out]
Sqrt[3]*ArcTan[(1 + 2*(1 - 3*Sec[x]^2)^(1/6))/Sqrt[3]] + ArcTanh[Sqrt[1 - 3*Sec[
x]^2]]/2 + 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 - (1 - 3*Sec[x]^2)
^(1/6) + (Cos[x]^2*Sqrt[1 - 3*Sec[x]^2])/6 - (1 - 3*Sec[x]^2)^(2/3)/4
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_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)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [C] time = 71.1464, size = 4397, normalized size = 33.06 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] 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]
[Out]
(-3*(6 + ((-5 + Cos[2*x])/(1 + Cos[2*x]))^(1/3) + Cos[2*x]*((-5 + Cos[2*x])/(1 +
Cos[2*x]))^(1/3))*(3*Sec[x]^2 + (1 - 3*Sec[x]^2)^(1/3))*Sin[x]^2*Tan[x]*(-2 - 3
*Tan[x]^2)^(5/6)*(1 + Tan[x]^2)*(2 + 3*Tan[x]^2)*(-8*AppellF1[1, 1/2, 1, 2, (-3*
Tan[x]^2)/2, -Tan[x]^2] + 4*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*T
an[x]^2 + 3*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^2)^2*((4*A
ppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2] + 3*AppellF1[2, 3/2, 1, 3, (-3
*Tan[x]^2)/2, -Tan[x]^2])*Tan[x]^2*(30*3^(2/3)*Hypergeometric2F1[1/3, 1/3, 4/3,
(3 + 3*Tan[x]^2)^(-1)]*Sqrt[-2 - 3*Tan[x]^2]*(1 + Tan[x]^2)*((2 + 3*Tan[x]^2)/(1
+ Tan[x]^2))^(1/3) + 12*3^(1/6)*Hypergeometric2F1[5/6, 5/6, 11/6, (3 + 3*Tan[x]
^2)^(-1)]*(1 + Tan[x]^2)*((2 + 3*Tan[x]^2)/(1 + Tan[x]^2))^(5/6) + 5*(2*Log[1 +
Tan[x]^2]*(-2 - 3*Tan[x]^2)^(5/6)*(1 + Tan[x]^2) + 9*Tan[x]^4*(4 + Sqrt[-2 - 3*T
an[x]^2]) + 3*Tan[x]^2*(20 - 2*(-2 - 3*Tan[x]^2)^(1/3) + 5*Sqrt[-2 - 3*Tan[x]^2]
) + 2*(12 - 2*(-2 - 3*Tan[x]^2)^(1/3) + 3*Sqrt[-2 - 3*Tan[x]^2] + (-2 - 3*Tan[x]
^2)^(5/6)))) - 8*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*(30*3^(2/3)*
Hypergeometric2F1[1/3, 1/3, 4/3, (3 + 3*Tan[x]^2)^(-1)]*Sqrt[-2 - 3*Tan[x]^2]*(1
+ Tan[x]^2)*((2 + 3*Tan[x]^2)/(1 + Tan[x]^2))^(1/3) + 12*3^(1/6)*Hypergeometric
2F1[5/6, 5/6, 11/6, (3 + 3*Tan[x]^2)^(-1)]*(1 + Tan[x]^2)*((2 + 3*Tan[x]^2)/(1 +
Tan[x]^2))^(5/6) + 5*(2*Log[1 + Tan[x]^2]*(-2 - 3*Tan[x]^2)^(5/6)*(1 + Tan[x]^2
) + 9*Tan[x]^4*(4 + Sqrt[-2 - 3*Tan[x]^2]) + Tan[x]^2*(60 - 7*(-2 - 3*Tan[x]^2)^
(1/3) + 15*Sqrt[-2 - 3*Tan[x]^2]) + 2*(12 - 2*(-2 - 3*Tan[x]^2)^(1/3) + 3*Sqrt[-
2 - 3*Tan[x]^2] + (-2 - 3*Tan[x]^2)^(5/6))))))/(10*2^(1/6)*(-1 + Sqrt[(-5 + Cos[
2*x])/(1 + Cos[2*x])])*(1 - 3*Sec[x]^2)^(5/6)*(6 + (1 - 3*Sec[x]^2)^(1/3) + Cos[
2*x]*(1 - 3*Sec[x]^2)^(1/3))*(-4 - 6*Tan[x]^2)^(5/6)*(-8*AppellF1[1, 1/2, 1, 2,
(-3*Tan[x]^2)/2, -Tan[x]^2] + (4*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]
^2] + 3*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2])*Tan[x]^2)*(1152*Appe
llF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^3 + 2880*AppellF1[1, 1/2
, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^5 - 1152*AppellF1[1, 1/2, 1, 2, (-3
*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[
x]^5 - 864*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1
, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^5 + 1728*AppellF1[1, 1/2, 1, 2, (-3*Tan[
x]^2)/2, -Tan[x]^2]^2*Tan[x]^7 - 2880*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -T
an[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7 + 288*Appel
lF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7 - 2160*AppellF1[1, 1/2,
1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[
x]^2]*Tan[x]^7 + 432*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1
[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7 + 162*AppellF1[2, 3/2, 1, 3,
(-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7 - 1728*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]
^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9 +
720*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^9 - 1296*AppellF
1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)
/2, -Tan[x]^2]*Tan[x]^9 + 1080*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2
]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9 + 405*AppellF1[2,
3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^9 + 432*AppellF1[2, 1/2, 2, 3, (
-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^11 + 648*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2
)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^11 + 2
43*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^11 + 720*AppellF1
[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^3*(-2 - 3*Tan[x]^2)^(1/3) -
192*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-
3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^3*(-2 - 3*Tan[x]^2)^(1/3) - 144*AppellF1[1, 1/2
, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan
[x]^2]*Tan[x]^3*(-2 - 3*Tan[x]^2)^(1/3) + 1008*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]
^2)/2, -Tan[x]^2]^2*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) - 1032*AppellF1[1, 1/2, 1,
2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2
]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) - 774*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2,
-Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^5*(-2 - 3*
Tan[x]^2)^(1/3) + 128*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF
1[3, 1/2, 3, 4, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) + 9
6*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3, 3/2, 2, 4, (-3*
Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) + 108*AppellF1[1, 1/2,
1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3, 5/2, 1, 4, (-3*Tan[x]^2)/2, -Tan[x
]^2]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) - 1080*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2
)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2
- 3*Tan[x]^2)^(1/3) + 96*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Ta
n[x]^7*(-2 - 3*Tan[x]^2)^(1/3) - 810*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Ta
n[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[
x]^2)^(1/3) + 144*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2,
3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[x]^2)^(1/3) + 54*Ap
pellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7*(-2 - 3*Tan[x]^2)^(1
/3) + 320*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3, 1/2, 3,
4, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[x]^2)^(1/3) + 240*AppellF1[
1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3, 3/2, 2, 4, (-3*Tan[x]^2)/2
, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[x]^2)^(1/3) + 270*AppellF1[1, 1/2, 1, 2, (-3*T
an[x]^2)/2, -Tan[x]^2]*AppellF1[3, 5/2, 1, 4, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]
^7*(-2 - 3*Tan[x]^2)^(1/3) + 144*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]
^2]^2*Tan[x]^9*(-2 - 3*Tan[x]^2)^(1/3) + 216*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2
)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*(-2
- 3*Tan[x]^2)^(1/3) + 81*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Ta
n[x]^9*(-2 - 3*Tan[x]^2)^(1/3) + 192*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Ta
n[x]^2]*AppellF1[3, 1/2, 3, 4, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*(-2 - 3*Tan[
x]^2)^(1/3) + 144*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3,
3/2, 2, 4, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*(-2 - 3*Tan[x]^2)^(1/3) + 162*A
ppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[3, 5/2, 1, 4, (-3*Tan
[x]^2)/2, -Tan[x]^2]*Tan[x]^9*(-2 - 3*Tan[x]^2)^(1/3) + 1152*AppellF1[1, 1/2, 1,
2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^3*Sqrt[-2 - 3*Tan[x]^2] + 2880*AppellF1
[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^5*Sqrt[-2 - 3*Tan[x]^2] - 11
52*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3
*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^5*Sqrt[-2 - 3*Tan[x]^2] - 864*AppellF1[1, 1/2, 1
, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]
^2]*Tan[x]^5*Sqrt[-2 - 3*Tan[x]^2] + 1728*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2
, -Tan[x]^2]^2*Tan[x]^7*Sqrt[-2 - 3*Tan[x]^2] - 2880*AppellF1[1, 1/2, 1, 2, (-3*
Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x
]^7*Sqrt[-2 - 3*Tan[x]^2] + 288*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^
2]^2*Tan[x]^7*Sqrt[-2 - 3*Tan[x]^2] - 2160*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/
2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*Sqrt[-
2 - 3*Tan[x]^2] + 432*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF
1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*Sqrt[-2 - 3*Tan[x]^2] + 162
*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7*Sqrt[-2 - 3*Tan[x
]^2] - 1728*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2,
2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*Sqrt[-2 - 3*Tan[x]^2] + 720*AppellF1[
2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^9*Sqrt[-2 - 3*Tan[x]^2] - 129
6*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*
Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*Sqrt[-2 - 3*Tan[x]^2] + 1080*AppellF1[2, 1/2, 2
, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]
^2]*Tan[x]^9*Sqrt[-2 - 3*Tan[x]^2] + 405*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2,
-Tan[x]^2]^2*Tan[x]^9*Sqrt[-2 - 3*Tan[x]^2] + 432*AppellF1[2, 1/2, 2, 3, (-3*Ta
n[x]^2)/2, -Tan[x]^2]^2*Tan[x]^11*Sqrt[-2 - 3*Tan[x]^2] + 648*AppellF1[2, 1/2, 2
, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]
^2]*Tan[x]^11*Sqrt[-2 - 3*Tan[x]^2] + 243*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2
, -Tan[x]^2]^2*Tan[x]^11*Sqrt[-2 - 3*Tan[x]^2] + 384*AppellF1[1, 1/2, 1, 2, (-3*
Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^3*(-2 - 3*Tan[x]^2)^(5/6) + 576*AppellF1[1, 1/2
, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^5*(-2 - 3*Tan[x]^2)^(5/6) - 384*App
ellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x
]^2)/2, -Tan[x]^2]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(5/6) - 288*AppellF1[1, 1/2, 1, 2,
(-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*
Tan[x]^5*(-2 - 3*Tan[x]^2)^(5/6) - 576*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -
Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Ta
n[x]^2)^(5/6) + 96*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7
*(-2 - 3*Tan[x]^2)^(5/6) - 432*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2
]*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[x]^2)^
(5/6) + 144*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3/2,
1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^7*(-2 - 3*Tan[x]^2)^(5/6) + 54*AppellF1
[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^7*(-2 - 3*Tan[x]^2)^(5/6) +
144*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^9*(-2 - 3*Tan[x]
^2)^(5/6) + 216*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF1[2, 3
/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^9*(-2 - 3*Tan[x]^2)^(5/6) + 81*Appe
llF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^9*(-2 - 3*Tan[x]^2)^(5/6
)))
_______________________________________________________________________________________
Maple [F] time = 4.865, size = 0, normalized size = 0. \[ \int{\frac{\tan \left ( x \right ) }{ \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \sqrt [3]{1-3\, \left ( \sec \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}+3\, \left ( \tan \left ( x \right ) \right ) ^{2} \right ) \left ( 1-3\, \left ( \sec \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{5}{6}}} \left ( 1-\sqrt{1-3\, \left ( \sec \left ( x \right ) \right ) ^{2}} \right ) ^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] 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)
[Out]
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)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((-3*sec(x)^2 + 1)^(1/3)*sin(x)^2 + 3*tan(x)^2)*tan(x)/((-3*sec(x)^2 + 1)^(5/6)*(sqrt(-3*sec(x)^2 + 1) - 1)*cos(x)^2),x, algorithm="maxima")
[Out]
Timed out
_______________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((-3*sec(x)^2 + 1)^(1/3)*sin(x)^2 + 3*tan(x)^2)*tan(x)/((-3*sec(x)^2 + 1)^(5/6)*(sqrt(-3*sec(x)^2 + 1) - 1)*cos(x)^2),x, algorithm="fricas")
[Out]
Timed out
_______________________________________________________________________________________
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(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)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left ({\left (-3 \, \sec \left (x\right )^{2} + 1\right )}^{\frac{1}{3}} \sin \left (x\right )^{2} + 3 \, \tan \left (x\right )^{2}\right )} \tan \left (x\right )}{{\left (-3 \, \sec \left (x\right )^{2} + 1\right )}^{\frac{5}{6}}{\left (\sqrt{-3 \, \sec \left (x\right )^{2} + 1} - 1\right )} \cos \left (x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-((-3*sec(x)^2 + 1)^(1/3)*sin(x)^2 + 3*tan(x)^2)*tan(x)/((-3*sec(x)^2 + 1)^(5/6)*(sqrt(-3*sec(x)^2 + 1) - 1)*cos(x)^2),x, algorithm="giac")
[Out]
integrate(-((-3*sec(x)^2 + 1)^(1/3)*sin(x)^2 + 3*tan(x)^2)*tan(x)/((-3*sec(x)^2
+ 1)^(5/6)*(sqrt(-3*sec(x)^2 + 1) - 1)*cos(x)^2), x)