3.446 \(\int \frac{\sec ^2(x) \tan (x) (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x))}{(1-3 \sec ^2(x))^{5/6} (1-\sqrt{1-3 \sec ^2(x)})} \, dx\)
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*Log[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 = 5.10991, 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, Rules
used = {4361, 6742, 6684, 261, 6697, 341, 57, 618, 204, 31, 6688, 266, 47, 63, 206, 25}
\[ \frac{\cos ^2(x)}{6}-\frac{1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}-\frac{3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac{1}{2} \log \left (1-\sqrt{1-3 \sec ^2(x)}\right )+\frac{1}{6} \cos ^2(x) \sqrt{1-3 \sec ^2(x)}+\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt{3}}\right )+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-3 \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right ) \]
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*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 4361
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 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rule 6684
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa
lseQ[q]]
Rule 261
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 6697
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 341
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 57
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 618
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 266
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 47
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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
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])
Rubi steps
\begin{align*} \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 &=-\operatorname{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 )\\ &=-\operatorname{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 )\\ &=-\operatorname{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 )-\operatorname{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 )\\ &=-\operatorname{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 )-\operatorname{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 \operatorname{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 \operatorname{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 )-\operatorname{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 )-\operatorname{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 (3-\cos ^2(x)\right ) \sec ^2(x)}\right )-\frac{1}{2} \operatorname{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 \operatorname{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 )-\operatorname{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 (3-\cos ^2(x)\right ) \sec ^2(x)}\right )-\operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{3}{x^2}\right )^{5/6} x^3} \, dx,x,\cos (x)\right )-\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-\frac{3}{x^2}} x^3} \, dx,x,\cos (x)\right )-\operatorname{Subst}\left (\int \frac{1}{(-1+x) x^{2/3}} \, dx,x,\sqrt{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\operatorname{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 (3-\cos ^2(x)\right ) \sec ^2(x)}\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} \operatorname{Subst}\left (\int \sqrt{1-\frac{3}{x^2}} x \, dx,x,\cos (x)\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\operatorname{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 (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\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} \operatorname{Subst}\left (\int \frac{\sqrt{1-3 x}}{x^2} \, dx,x,\sec ^2(x)\right )-3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{3}{x^2}} x} \, dx,x,\cos (x)\right )\\ &=\sqrt{3} \tan ^{-1}\left (\frac{1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt{3}}\right )+\frac{\cos ^2(x)}{6}-\frac{3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\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} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-3 x} x} \, dx,x,\sec ^2(x)\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-3 x} x} \, dx,x,\sec ^2(x)\right )\\ &=\sqrt{3} \tan ^{-1}\left (\frac{1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt{3}}\right )+\frac{\cos ^2(x)}{6}-\frac{3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\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} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{3}-\frac{x^2}{3}} \, dx,x,\sqrt{1-3 \sec ^2(x)}\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{3}-\frac{x^2}{3}} \, dx,x,\sqrt{1-3 \sec ^2(x)}\right )\\ &=\sqrt{3} \tan ^{-1}\left (\frac{1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt{3}}\right )+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-3 \sec ^2(x)}\right )+\frac{\cos ^2(x)}{6}-\frac{3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac{1}{3} \log \left (1-\sqrt{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\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] time = 56.1542, 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*AppellF
1[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]*Tan[x]^2 +
3*AppellF1[2, 3/2, 1, 3, (-3*Tan[x]^2)/2, -Tan[x]^2]*Tan[x]^2)^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*(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 + Ta
n[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, -Ta
n[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*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*AppellF1[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, -Tan[x]^2]*AppellF1[2, 1/2, 2, 3, (-3*Tan[x]^2)/2, -
Tan[x]^2]*Tan[x]^7 + 288*AppellF1[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*Ap
pellF1[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, -Ta
n[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*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 + 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 + 243*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]*Appe
llF1[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]*AppellF1[3, 1/2, 3, 4, (-3*Tan[x]^2)/
2, -Tan[x]^2]*Tan[x]^5*(-2 - 3*Tan[x]^2)^(1/3) + 96*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]*AppellF
1[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*Tan[x]^7*(-2 - 3*Tan[x
]^2)^(1/3) - 810*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, -T
an[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*AppellF1[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*Tan[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, -Tan[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, -T
an[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*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]^9*(-2 - 3*Ta
n[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] + 288
0*AppellF1[1, 1/2, 1, 2, (-3*Tan[x]^2)/2, -Tan[x]^2]^2*Tan[x]^5*Sqrt[-2 - 3*Tan[x]^2] - 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*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]*AppellF1[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] - 1296*AppellF1[1, 1/2, 1, 2, (-3*T
an[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] + 108
0*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*Tan[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] + 3
84*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*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)^(5/6) - 288*Ap
pellF1[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*Tan[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*AppellF1[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 = 180., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
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(-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(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
________________________________________________________________________________________
Fricas [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(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]
Timed out
________________________________________________________________________________________
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(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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
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, 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)