\(\int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx\) [634]

Optimal result

Integrand size = 25, antiderivative size = 579 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {b^{4/3} \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) \sqrt [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}} \]

[Out]

3*b/(a^2+b^2)/f/(d*sec(f*x+e))^(1/3)-b^(4/3)*arctanh(b^(1/3)*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(1/6))*(sec(f*x+e)
^2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+1/4*b^(4/3)*ln((a^2+b^2)^(1/3)-b^(1/3)*(a^2+b^2)^(1/6)*(sec(f
*x+e)^2)^(1/6)+b^(2/3)*(sec(f*x+e)^2)^(1/3))*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)-1/4*b
^(4/3)*ln((a^2+b^2)^(1/3)+b^(1/3)*(a^2+b^2)^(1/6)*(sec(f*x+e)^2)^(1/6)+b^(2/3)*(sec(f*x+e)^2)^(1/3))*(sec(f*x+
e)^2)^(1/6)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+1/2*b^(4/3)*arctan(-1/3*3^(1/2)+2/3*b^(1/3)*(sec(f*x+e)^2)^
(1/6)/(a^2+b^2)^(1/6)*3^(1/2))*(sec(f*x+e)^2)^(1/6)*3^(1/2)/(a^2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+1/2*b^(4/3)
*arctan(1/3*3^(1/2)+2/3*b^(1/3)*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(1/6)*3^(1/2))*(sec(f*x+e)^2)^(1/6)*3^(1/2)/(a^
2+b^2)^(7/6)/f/(d*sec(f*x+e))^(1/3)+AppellF1(1/2,1,7/6,3/2,b^2*tan(f*x+e)^2/a^2,-tan(f*x+e)^2)*(sec(f*x+e)^2)^
(1/6)*tan(f*x+e)/a/f/(d*sec(f*x+e))^(1/3)

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3593, 771, 440, 455, 53, 65, 302, 648, 632, 210, 642, 214} \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\frac {\tan (e+f x) \sqrt [6]{\sec ^2(e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a f \sqrt [3]{d \sec (e+f x)}}-\frac {\sqrt {3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}+\frac {\sqrt {3} b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}+\frac {3 b}{f \left (a^2+b^2\right ) \sqrt [3]{d \sec (e+f x)}}+\frac {b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \sqrt [6]{\sec ^2(e+f x)} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{4 f \left (a^2+b^2\right )^{7/6} \sqrt [3]{d \sec (e+f x)}} \]

[In]

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

[Out]

(3*b)/((a^2 + b^2)*f*(d*Sec[e + f*x])^(1/3)) - (Sqrt[3]*b^(4/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)
^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(1/6))/(2*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) +
(Sqrt[3]*b^(4/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f
*x]^2)^(1/6))/(2*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) - (b^(4/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6
))/(a^2 + b^2)^(1/6)]*(Sec[e + f*x]^2)^(1/6))/((a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) + (b^(4/3)*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)]*(Sec[e + f*
x]^2)^(1/6))/(4*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) - (b^(4/3)*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)]*(Sec[e + f*x]^2)^(1/6))/(4*(a^2 + b^2)^(7/
6)*f*(d*Sec[e + f*x])^(1/3)) + (AppellF1[1/2, 1, 7/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e +
f*x]^2)^(1/6)*Tan[e + f*x])/(a*f*(d*Sec[e + f*x])^(1/3))

Rule 53

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 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 214

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

Rule 302

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

Rule 440

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 455

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 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 642

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 648

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 771

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 3593

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [6]{\sec ^2(e+f x)} \text {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {\sqrt [6]{\sec ^2(e+f x)} \text {Subst}\left (\int \left (\frac {a}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{7/6}}+\frac {x}{\left (-a^2+x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{7/6}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {\sqrt [6]{\sec ^2(e+f x)} \text {Subst}\left (\int \frac {x}{\left (-a^2+x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (a \sqrt [6]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{7/6}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}+\frac {\sqrt [6]{\sec ^2(e+f x)} \text {Subst}\left (\int \frac {1}{\left (-a^2+x\right ) \left (1+\frac {x}{b^2}\right )^{7/6}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 b f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (b \sqrt [6]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2+x\right ) \sqrt [6]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{2 \left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (3 b^3 \sqrt [6]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x^4}{-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 [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}-\frac {\left (b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {-\frac {1}{2} \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 )}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {\left (b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {-\frac {1}{2} \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 )}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {\left (b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 )}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 )}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {\left (b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 )}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (3 b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 ) f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (3 b^{5/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 ) f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {b^{4/3} \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 [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \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 [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}}+\frac {\left (3 b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {\left (3 b^{4/3} \sqrt [6]{\sec ^2(e+f x)}\right ) \text {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 )^{7/6} f \sqrt [3]{d \sec (e+f x)}} \\ & = \frac {3 b}{\left (a^2+b^2\right ) f \sqrt [3]{d \sec (e+f x)}}-\frac {\sqrt {3} b^{4/3} \arctan \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 [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\sqrt {3} b^{4/3} \arctan \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 [6]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) \sqrt [6]{\sec ^2(e+f x)}}{\left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {b^{4/3} \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 [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}-\frac {b^{4/3} \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 [6]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^{7/6} f \sqrt [3]{d \sec (e+f x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,\frac {7}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) \sqrt [6]{\sec ^2(e+f x)} \tan (e+f x)}{a f \sqrt [3]{d \sec (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 52.39 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=-\frac {60 d \operatorname {AppellF1}\left (\frac {7}{3},\frac {7}{6},\frac {7}{6},\frac {10}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a \cos (e+f x)+b \sin (e+f x))}{7 b f (d \sec (e+f x))^{4/3} \left (7 (a+i b) \operatorname {AppellF1}\left (\frac {10}{3},\frac {7}{6},\frac {13}{6},\frac {13}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+7 (a-i b) \operatorname {AppellF1}\left (\frac {10}{3},\frac {13}{6},\frac {7}{6},\frac {13}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right )+20 \operatorname {AppellF1}\left (\frac {7}{3},\frac {7}{6},\frac {7}{6},\frac {10}{3},\frac {a-i b}{a+b \tan (e+f x)},\frac {a+i b}{a+b \tan (e+f x)}\right ) (a+b \tan (e+f x))\right )} \]

[In]

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

[Out]

(-60*d*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a*Cos[e
+ f*x] + b*Sin[e + f*x]))/(7*b*f*(d*Sec[e + f*x])^(4/3)*(7*(a + I*b)*AppellF1[10/3, 7/6, 13/6, 13/3, (a - I*b)
/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 7*(a - I*b)*AppellF1[10/3, 13/6, 7/6, 13/3, (a - I*b)
/(a + b*Tan[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])] + 20*AppellF1[7/3, 7/6, 7/6, 10/3, (a - I*b)/(a + b*Tan
[e + f*x]), (a + I*b)/(a + b*Tan[e + f*x])]*(a + b*Tan[e + f*x])))

Maple [F]

\[\int \frac {1}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +b \tan \left (f x +e \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\int \frac {1}{\sqrt [3]{d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\int { \frac {1}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))} \, dx=\int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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