\(\int \frac {\sec ^2(x) \tan (x) (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x))}{(1-3 \sec ^2(x))^{5/6} (1-\sqrt {1-3 \sec ^2(x)})} \, dx\) [446]

Optimal result

Integrand size = 61, antiderivative size = 133 \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{1-3 \sec ^2(x)}}{\sqrt {3}}\right )+\frac {1}{4} \log \left (\sec ^2(x)\right )-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{2 \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \]

[Out]

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*s
ec(x)^2)^(2/3)+arctan(1/3*(1+2*(1-3*sec(x)^2)^(1/6))*3^(1/2))*3^(1/2)+1/2/(1-(1-3*sec(x)^2)^(1/2))

Rubi [A] (verified)

Time = 3.75 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.31, number of steps used = 29, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {4446, 6874, 6816, 267, 6829, 348, 59, 632, 210, 31, 6820, 272, 43, 65, 212, 25} \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt {3}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {1-3 \sec ^2(x)}\right )+\frac {\cos ^2(x)}{6}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right ) \]

[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*S
ec[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

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[u*((
a + b*x^n)^(m + p)/x^(n*p)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 348

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 4446

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, Dist[-(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])

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6820

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

Rule 6829

Int[(u_.)*(v_)^(m_.)*((a_.) + (b_.)*(y_)^(n_))^(p_.), x_Symbol] :> Module[{q, r}, Dist[q*r, Subst[Int[x^m*(a +
b*x^n)^p, x], x, y], x] /;  !FalseQ[r = Divides[y^m, v^m, x]] &&  !FalseQ[q = DerivativeDivides[y, u, x]]] /;
FreeQ[{a, b, m, n, p}, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (3+\sqrt [3]{1-\frac {3}{x^2}} x^2\right )}{\left (1-\sqrt {1-\frac {3}{x^2}}\right ) \left (1-\frac {3}{x^2}\right )^{5/6} x^5} \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {-3-x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}+\frac {3+x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \frac {-3-x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {3+x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {1}{\sqrt {1-\frac {3}{x^2}} x^3 \left (1-\sqrt {\frac {-3+x^2}{x^2}}\right )}-\frac {3}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \left (\frac {3}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}+\frac {1}{\sqrt {1-\frac {3}{x^2}} x \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right ) \\ & = 3 \text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-3 \text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x^3 \left (1-\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right ) \\ & = \frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-1+\sqrt {x}\right ) x^{5/6}} \, dx,x,\left (-3+\cos ^2(x)\right ) \sec ^2(x)\right )+3 \text {Subst}\left (\int \left (-\frac {1}{3 \left (1-\frac {3}{x^2}\right )^{5/6} x^3}-\frac {1}{3 \sqrt [3]{1-\frac {3}{x^2}} x^3}\right ) \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{-\frac {3}{x}+x-\sqrt {1-\frac {3}{x^2}} x} \, dx,x,\cos (x)\right ) \\ & = \frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )-\text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {3}{x^2}} x^3} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{(-1+x) x^{2/3}} \, dx,x,\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\text {Subst}\left (\int \left (-\frac {x}{3}-\frac {1}{3} \sqrt {1-\frac {3}{x^2}} x+\frac {\sqrt {1-\frac {3}{x^2}} x}{3-x^2}\right ) \, dx,x,\cos (x)\right ) \\ & = \frac {\cos ^2(x)}{6}+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \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}{3} \text {Subst}\left (\int \sqrt {1-\frac {3}{x^2}} x \, dx,x,\cos (x)\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\text {Subst}\left (\int \frac {\sqrt {1-\frac {3}{x^2}} x}{3-x^2} \, dx,x,\cos (x)\right ) \\ & = \frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \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}{6} \text {Subst}\left (\int \frac {\sqrt {1-3 x}}{x^2} \, dx,x,\sec ^2(x)\right )-3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x} \, dx,x,\cos (x)\right ) \\ & = \sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}}{\sqrt {3}}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-3 x} x} \, dx,x,\sec ^2(x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-3 x} x} \, dx,x,\sec ^2(x)\right ) \\ & = \sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}}{\sqrt {3}}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {x^2}{3}} \, dx,x,\sqrt {1-3 \sec ^2(x)}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {x^2}{3}} \, dx,x,\sqrt {1-3 \sec ^2(x)}\right ) \\ & = \sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}}{\sqrt {3}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {1-3 \sec ^2(x)}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 25.30 (sec) , antiderivative size = 1447, normalized size of antiderivative = 10.88 \[ \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 (6+\sqrt [3]{\frac {-5+\cos (2 x)}{1+\cos (2 x)}}+\cos (2 x) \sqrt [3]{\frac {-5+\cos (2 x)}{1+\cos (2 x)}}\right ) \left (3 \sec ^2(x)+\sqrt [3]{1-3 \sec ^2(x)}\right ) \tan (x) \left (-2-3 \tan ^2(x)\right )^{5/6} \left (2+3 \tan ^2(x)\right ) \sqrt {-\left (2+3 \tan ^2(x)\right )^2} \left (-1+\sqrt [3]{-2-3 \tan ^2(x)}\right ) \left (1+\sqrt [3]{-2-3 \tan ^2(x)}+\left (-2-3 \tan ^2(x)\right )^{2/3}\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 (-1+\sqrt {\frac {-5+\cos (2 x)}{1+\cos (2 x)}}\right ) \left (1-3 \sec ^2(x)\right )^{5/6} \left (6+\sqrt [3]{1-3 \sec ^2(x)}+\cos (2 x) \sqrt [3]{1-3 \sec ^2(x)}\right ) \left (12 \csc (x) \sec (x) \left (-2-3 \tan ^2(x)\right )^{5/6}+12 \cos (2 x) \csc (x) \sec (x) \left (-2-3 \tan ^2(x)\right )^{5/6}-88 \sin (2 x) \left (-2-3 \tan ^2(x)\right )^{5/6}-16 \cot ^2(x) \sin (2 x) \left (-2-3 \tan ^2(x)\right )^{5/6}+48 \sec ^2(x) \tan (x) \left (-2-3 \tan ^2(x)\right )^{5/6}+48 \cos (2 x) \sec ^2(x) \tan (x) \left (-2-3 \tan ^2(x)\right )^{5/6}-180 \sin (2 x) \tan ^2(x) \left (-2-3 \tan ^2(x)\right )^{5/6}+63 \sec ^2(x) \tan ^3(x) \left (-2-3 \tan ^2(x)\right )^{5/6}+63 \cos (2 x) \sec ^2(x) \tan ^3(x) \left (-2-3 \tan ^2(x)\right )^{5/6}-162 \sin (2 x) \tan ^4(x) \left (-2-3 \tan ^2(x)\right )^{5/6}+27 \sec ^2(x) \tan ^5(x) \left (-2-3 \tan ^2(x)\right )^{5/6}+27 \cos (2 x) \sec ^2(x) \tan ^5(x) \left (-2-3 \tan ^2(x)\right )^{5/6}-54 \sin (2 x) \tan ^6(x) \left (-2-3 \tan ^2(x)\right )^{5/6}-24 \sec ^2(x) \tan (x) \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-36 \sec ^2(x) \tan ^3(x) \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+4 \csc (x) \sec (x) \sqrt [3]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+6 \sec ^2(x) \tan (x) \sqrt [3]{-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-24 \sec ^2(x) \tan (x) \sqrt {-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-36 \sec ^2(x) \tan ^3(x) \sqrt {-2-3 \tan ^2(x)} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-20 \cot (x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+12 \csc (x) \sec (x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+10 \sin (2 x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+4 \cot ^2(x) \sin (2 x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-50 \tan (x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+18 \sec ^2(x) \tan (x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}+6 \sin (2 x) \tan ^2(x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}-30 \tan ^3(x) \left (-2-3 \tan ^2(x)\right )^{5/6} \sqrt {2+3 \tan ^2(x)} \sqrt {-\left (2+3 \tan ^2(x)\right )^2}\right )} \]

[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]

-1/3*((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))*Tan[x]*(-2 - 3*Tan[x]^2)^(5/6)*(2 + 3*Tan[x]^2)*Sqrt[-(2 + 3*Tan[x]^2)^2]*(-1 + (
-2 - 3*Tan[x]^2)^(1/3))*(1 + (-2 - 3*Tan[x]^2)^(1/3) + (-2 - 3*Tan[x]^2)^(2/3))*(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]]*Sqrt[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]]*Sq
rt[6 + 9*Tan[x]^2]))/((-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))*(12*Csc[x]*Sec[x]*(-2 - 3*Tan[x]^2)^(5/6) + 12*Cos[2*x]*Csc[x]*Sec[x
]*(-2 - 3*Tan[x]^2)^(5/6) - 88*Sin[2*x]*(-2 - 3*Tan[x]^2)^(5/6) - 16*Cot[x]^2*Sin[2*x]*(-2 - 3*Tan[x]^2)^(5/6)
+ 48*Sec[x]^2*Tan[x]*(-2 - 3*Tan[x]^2)^(5/6) + 48*Cos[2*x]*Sec[x]^2*Tan[x]*(-2 - 3*Tan[x]^2)^(5/6) - 180*Sin[
2*x]*Tan[x]^2*(-2 - 3*Tan[x]^2)^(5/6) + 63*Sec[x]^2*Tan[x]^3*(-2 - 3*Tan[x]^2)^(5/6) + 63*Cos[2*x]*Sec[x]^2*Ta
n[x]^3*(-2 - 3*Tan[x]^2)^(5/6) - 162*Sin[2*x]*Tan[x]^4*(-2 - 3*Tan[x]^2)^(5/6) + 27*Sec[x]^2*Tan[x]^5*(-2 - 3*
Tan[x]^2)^(5/6) + 27*Cos[2*x]*Sec[x]^2*Tan[x]^5*(-2 - 3*Tan[x]^2)^(5/6) - 54*Sin[2*x]*Tan[x]^6*(-2 - 3*Tan[x]^
2)^(5/6) - 24*Sec[x]^2*Tan[x]*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 36*Sec[x]^2*Tan[x]^3*Sqrt[2 + 3
*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 4*Csc[x]*Sec[x]*(-2 - 3*Tan[x]^2)^(1/3)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 +
3*Tan[x]^2)^2] + 6*Sec[x]^2*Tan[x]*(-2 - 3*Tan[x]^2)^(1/3)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 2
4*Sec[x]^2*Tan[x]*Sqrt[-2 - 3*Tan[x]^2]*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 36*Sec[x]^2*Tan[x]^3*
Sqrt[-2 - 3*Tan[x]^2]*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 20*Cot[x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[
2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 12*Csc[x]*Sec[x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2]*Sqrt
[-(2 + 3*Tan[x]^2)^2] + 10*Sin[2*x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 4
*Cot[x]^2*Sin[2*x]*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 50*Tan[x]*(-2 - 3*
Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 18*Sec[x]^2*Tan[x]*(-2 - 3*Tan[x]^2)^(5/6)*Sq
rt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^2] + 6*Sin[2*x]*Tan[x]^2*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2
]*Sqrt[-(2 + 3*Tan[x]^2)^2] - 30*Tan[x]^3*(-2 - 3*Tan[x]^2)^(5/6)*Sqrt[2 + 3*Tan[x]^2]*Sqrt[-(2 + 3*Tan[x]^2)^
2]))

Maple [F]

\[\int \frac {\tan \left (x \right ) \left ({\left (1-3 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {1}{3}} \left (\sin ^{2}\left (x \right )\right )+3 \left (\tan ^{2}\left (x \right )\right )\right )}{\cos \left (x \right )^{2} {\left (1-3 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {5}{6}} \left (1-\sqrt {1-3 \left (\sec ^{2}\left (x \right )\right )}\right )}d x\]

[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
)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Exception raised: TypeError} \]

[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, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   Curve not irreducible after change of va
riable 0 -> infinity

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Timed out} \]

[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

Maxima [F(-1)]

Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\text {Timed out} \]

[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, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=\int { -\frac {{\left ({\left (-3 \, \sec \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} \sin \left (x\right )^{2} + 3 \, \tan \left (x\right )^{2}\right )} \tan \left (x\right )}{{\left (-3 \, \sec \left (x\right )^{2} + 1\right )}^{\frac {5}{6}} {\left (\sqrt {-3 \, \sec \left (x\right )^{2} + 1} - 1\right )} \cos \left (x\right )^{2}} \,d x } \]

[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, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx=-\int \frac {\mathrm {tan}\left (x\right )\,\left ({\sin \left (x\right )}^2\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{1/3}+3\,{\mathrm {tan}\left (x\right )}^2\right )}{{\cos \left (x\right )}^2\,\left (\sqrt {1-\frac {3}{{\cos \left (x\right )}^2}}-1\right )\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{5/6}} \,d x \]

[In]

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)

[Out]

-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)