3.637 \(\int \frac{\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=687 \[ \frac{b^2 \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{\tan (e+f x) \sqrt [3]{d \sec (e+f x)} F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )} \]

[Out]

(5*a*b^(2/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x
])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)
*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*
(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e +
f*x])^(1/3))/(3*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) + (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a
^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^
2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e +
f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]
^2)^(1/6)) + (AppellF1[1/2, 2, 5/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan
[e + f*x])/(a^2*f*(Sec[e + f*x]^2)^(1/6)) + (b^2*AppellF1[3/2, 2, 5/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e +
f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^3)/(3*a^4*f*(Sec[e + f*x]^2)^(1/6)) - (a*b*(d*Sec[e + f*x])^(1/3)
)/((a^2 + b^2)*f*(a^2 - b^2*Tan[e + f*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.883918, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3512, 757, 429, 444, 51, 63, 210, 634, 618, 204, 628, 208, 510} \[ \frac{b^2 \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{\tan (e+f x) \sqrt [3]{d \sec (e+f x)} F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt{3} \sqrt [6]{a^2+b^2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \sqrt [3]{d \sec (e+f x)} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 f \left (a^2+b^2\right )^{11/6} \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x])^2,x]

[Out]

(5*a*b^(2/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x
])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)
*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*
(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e +
f*x])^(1/3))/(3*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) + (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a
^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^
2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e +
f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]
^2)^(1/6)) + (AppellF1[1/2, 2, 5/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan
[e + f*x])/(a^2*f*(Sec[e + f*x]^2)^(1/6)) + (b^2*AppellF1[3/2, 2, 5/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e +
f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^3)/(3*a^4*f*(Sec[e + f*x]^2)^(1/6)) - (a*b*(d*Sec[e + f*x])^(1/3)
)/((a^2 + b^2)*f*(a^2 - b^2*Tan[e + f*x]^2))

Rule 3512

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(d^(2
*IntPart[m/2])*(d*Sec[e + f*x])^(2*FracPart[m/2]))/(b*f*(Sec[e + f*x]^2)^FracPart[m/2]), Subst[Int[(a + x)^n*(
1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&
!IntegerQ[m/2]

Rule 757

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^2)^p, (d/(d
^2 - e^2*x^2) - (e*x)/(d^2 - e^2*x^2))^(-m), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&
!IntegerQ[p] && ILtQ[m, 0]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 210

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r +
s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 634

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

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 208

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

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{d \sec (e+f x)}}{(a+b \tan (e+f x))^2} \, dx &=\frac{\sqrt [3]{d \sec (e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{\sqrt [3]{d \sec (e+f x)} \operatorname{Subst}\left (\int \left (\frac{a^2}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}}-\frac{2 a x}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}}+\frac{x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{\sqrt [3]{d \sec (e+f x)} \operatorname{Subst}\left (\int \frac{x^2}{\left (-a^2+x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (2 a \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}+\frac{\left (a^2 \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x^2\right )^2 \left (1+\frac{x^2}{b^2}\right )^{5/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (a \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right )^2 \left (1+\frac{x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (5 a \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-x\right ) \left (1+\frac{x}{b^2}\right )^{5/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{6 b \left (a^2+b^2\right ) f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-b^2 x^6} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a^2+b^2}-\frac{\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a^2+b^2}+\frac{\sqrt [3]{b} x}{2}}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}-b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{3 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}}\\ &=-\frac{5 a b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac{\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{b} \sqrt [6]{a^2+b^2}+2 b^{2/3} x}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{\left (5 a b \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} x+b^{2/3} x^2} \, dx,x,\sqrt [6]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^{5/3} f \sqrt [6]{\sec ^2(e+f x)}}\\ &=-\frac{5 a b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac{\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{\left (5 a b^{2/3} \sqrt [3]{d \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}\\ &=\frac{5 a b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt{3}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt{3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}}{\sqrt{3}}\right ) \sqrt [3]{d \sec (e+f x)}}{2 \sqrt{3} \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [3]{d \sec (e+f x)}}{3 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}-\frac{5 a b^{2/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [3]{d \sec (e+f x)}}{12 \left (a^2+b^2\right )^{11/6} f \sqrt [6]{\sec ^2(e+f x)}}+\frac{F_1\left (\frac{1}{2};2,\frac{5}{6};\frac{3}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan (e+f x)}{a^2 f \sqrt [6]{\sec ^2(e+f x)}}+\frac{b^2 F_1\left (\frac{3}{2};2,\frac{5}{6};\frac{5}{2};\frac{b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 a^4 f \sqrt [6]{\sec ^2(e+f x)}}-\frac{a b \sqrt [3]{d \sec (e+f x)}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 25.4959, size = 4485, normalized size = 6.53 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x])^2,x]

[Out]

((d*Sec[e + f*x])^(1/3)*((5*(-1)^(5/6)*a*b^(2/3)*(-2*ArcTan[Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3)
)/(a^2 + b^2)^(1/6)] + 2*ArcTan[Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] + 4*Arc
Tan[((-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6)] - Sqrt[3]*Log[(a^2 + b^2)^(1/3) - (-1)^(1/6)*Sq
rt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)] + Sqrt[3]*Log[(a^2
+ b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f
*x]^(2/3)]))/(12*(a - I*b)*(a + I*b)*(a^2 + b^2)^(5/6)) + 3*((-2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^
2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(10/3))/(21*(a^2 + b^2)^2*Sqrt[1 -
Sec[e + f*x]^2]) + (Sec[e + f*x]^(1/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^2 + b^2) + (7*
(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e +
f*x]^2]*Sec[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Se
c[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 +
b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x
]^2))))/(3*(a^2 - b^2*(-1 + Sec[e + f*x]^2))))))/(f*(a + b*Tan[e + f*x])^2*((5*(-1)^(5/6)*a*b^(2/3)*((4*(-1)^(
1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + (Sqrt[3] - (2*(-1)^(1/6)*b^(1/3)*Sec[e
+ f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^
(1/6)*(1 + (Sqrt[3] + (2*(-1)^(1/6)*b^(1/3)*Sec[e + f*x]^(1/3))/(a^2 + b^2)^(1/6))^2)) + (4*(-1)^(1/6)*b^(1/3)
*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(3*(a^2 + b^2)^(1/6)*(1 + ((-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3))/(a^2 + b^2
)^(1/3))) - (Sqrt[3]*(-(((-1)^(1/6)*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3]) + (2*(
-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3))/((a^2 + b^2)^(1/3) - (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 +
b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(2/3)) + (Sqrt[3]*(((-1)^(1/6)*b^(1/3)*(a^2 +
b^2)^(1/6)*Sec[e + f*x]^(4/3)*Sin[e + f*x])/Sqrt[3] + (2*(-1)^(1/3)*b^(2/3)*Sec[e + f*x]^(5/3)*Sin[e + f*x])/3
))/((a^2 + b^2)^(1/3) + (-1)^(1/6)*Sqrt[3]*b^(1/3)*(a^2 + b^2)^(1/6)*Sec[e + f*x]^(1/3) + (-1)^(1/3)*b^(2/3)*S
ec[e + f*x]^(2/3))))/(12*(a - I*b)*(a + I*b)*(a^2 + b^2)^(5/6)) + 3*((-2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e
+ f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^(19/3)*Sin[e + f*x])/(21*(a
^2 + b^2)^2*(1 - Sec[e + f*x]^2)^(3/2)) - (2*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]
^2)/(a^2 + b^2)]*Sec[e + f*x]^(7/3)*Sin[e + f*x])/(21*(a^2 + b^2)^2*Sqrt[1 - Cos[e + f*x]^2]*Sqrt[1 - Sec[e +
f*x]^2]) - (20*b^2*AppellF1[7/6, 1/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[
e + f*x]^2]*Sec[e + f*x]^(13/3)*Sin[e + f*x])/(63*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]) + (2*b^2*Sec[e + f*x
]^(10/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6,
1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/((-1 + S
ec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3
*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7
/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)))*Sin[e + f*x])/(3*(a^2 -
b^2*(-1 + Sec[e + f*x]^2))^2) + (Sec[e + f*x]^(4/3)*((-(a*b) + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x])/(a^
2 + b^2) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqr
t[1 - Cos[e + f*x]^2]*Sec[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f
*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f
*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)
])*Sec[e + f*x]^2)))*Sin[e + f*x])/(9*(a^2 - b^2*(-1 + Sec[e + f*x]^2))) - (2*b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec
[e + f*x]^(10/3)*((14*b^2*AppellF1[13/6, 1/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e
+ f*x]^2*Tan[e + f*x])/(13*(a^2 + b^2)) + (7*AppellF1[13/6, 3/2, 1, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2
)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/13))/(21*(a^2 + b^2)^2*Sqrt[1 - Sec[e + f*x]^2]) + (Sec[e + f*x]^(
1/3)*((7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sin[e +
f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2
, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2
)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Se
c[e + f*x]^2)) - (14*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^
2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]^3*Tan[e + f*x])/((-1 + Sec[e + f*x]^2)^2*(7*(a^2 + b^2)*AppellF1[1/6
, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e
+ f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec
[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) + (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b
^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/((-1 + Sec[e + f*x]^2)*(7*
(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7
/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6,
Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)) + ((b^2*Sin[e + f*x])/Sqrt[1 - Cos[e + f*x
]^2] + b^2*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x])/(a^2 + b^2) + (7*(3*a^2 - 2*b^2)*Sqrt[1 - Cos[e
+ f*x]^2]*Sec[e + f*x]*((2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*
Sec[e + f*x]^2*Tan[e + f*x])/(7*(a^2 + b^2)) + (AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^
2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/7))/((-1 + Sec[e + f*x]^2)*(7*(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7
/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (
b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2
)/(a^2 + b^2)])*Sec[e + f*x]^2)) - (7*(3*a^2 - 2*b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e +
f*x]^2)/(a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2]*Sec[e + f*x]*(6*(2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*x]^
2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*
x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2*Tan[e + f*x] + 7*(a^2 + b^2)*((2*b^2*AppellF1[7/6, 1/2, 2, 13/6, Sec[e + f*
x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(7*(a^2 + b^2)) + (AppellF1[7/6, 3/2, 1,
13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/7) + 3*Sec[e + f*x]^2*(2*
b^2*((28*b^2*AppellF1[13/6, 1/2, 3, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan
[e + f*x])/(13*(a^2 + b^2)) + (7*AppellF1[13/6, 3/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)
]*Sec[e + f*x]^2*Tan[e + f*x])/13) + (a^2 + b^2)*((14*b^2*AppellF1[13/6, 3/2, 2, 19/6, Sec[e + f*x]^2, (b^2*Se
c[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/(13*(a^2 + b^2)) + (21*AppellF1[13/6, 5/2, 1, 19/6, Se
c[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*Sec[e + f*x]^2*Tan[e + f*x])/13))))/((-1 + Sec[e + f*x]^2)*(7*
(a^2 + b^2)*AppellF1[1/6, 1/2, 1, 7/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + 3*(2*b^2*AppellF1[7
/6, 1/2, 2, 13/6, Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)] + (a^2 + b^2)*AppellF1[7/6, 3/2, 1, 13/6,
Sec[e + f*x]^2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])*Sec[e + f*x]^2)^2)))/(3*(a^2 - b^2*(-1 + Sec[e + f*x]^2))))
))

________________________________________________________________________________________

Maple [F]  time = 0.274, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}\sqrt [3]{d\sec \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(f*x+e))^(1/3)/(a+b*tan(f*x+e))^2,x)

[Out]

int((d*sec(f*x+e))^(1/3)/(a+b*tan(f*x+e))^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((d*sec(f*x+e))^(1/3)/(a+b*tan(f*x+e))^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((d*sec(f*x+e))^(1/3)/(a+b*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{d \sec{\left (e + f x \right )}}}{\left (a + b \tan{\left (e + f x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))**(1/3)/(a+b*tan(f*x+e))**2,x)

[Out]

Integral((d*sec(e + f*x))**(1/3)/(a + b*tan(e + f*x))**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(f*x+e))^(1/3)/(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e))^(1/3)/(b*tan(f*x + e) + a)^2, x)