\(\int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx\) [636]

Optimal result

Integrand size = 25, antiderivative size = 687 \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=-\frac {a \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \log \left (\sqrt [3]{a^2+b^2}-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \log \left (\sqrt [3]{a^2+b^2}+\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )} \]

[Out]

-1/3*a*arctanh(b^(1/3)*(sec(f*x+e)^2)^(1/6)/(a^2+b^2)^(1/6))*(d*sec(f*x+e))^(5/3)/b^(2/3)/(a^2+b^2)^(7/6)/f/(s
ec(f*x+e)^2)^(5/6)+1/12*a*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))*(d*sec(f*x+e))^(5/3)/b^(2/3)/(a^2+b^2)^(7/6)/f/(sec(f*x+e)^2)^(5/6)-1/12*a*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))*(d*sec(f*x+e))^(5/3)/b^(2/3)/(a^2+b^2)^(7
/6)/f/(sec(f*x+e)^2)^(5/6)+1/6*a*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))
*(d*sec(f*x+e))^(5/3)/b^(2/3)/(a^2+b^2)^(7/6)/f/(sec(f*x+e)^2)^(5/6)*3^(1/2)+1/6*a*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))*(d*sec(f*x+e))^(5/3)/b^(2/3)/(a^2+b^2)^(7/6)/f/(sec(f*x+e)^2
)^(5/6)*3^(1/2)+AppellF1(1/2,2,1/6,3/2,b^2*tan(f*x+e)^2/a^2,-tan(f*x+e)^2)*(d*sec(f*x+e))^(5/3)*tan(f*x+e)/a^2
/f/(sec(f*x+e)^2)^(5/6)+1/3*b^2*AppellF1(3/2,2,1/6,5/2,b^2*tan(f*x+e)^2/a^2,-tan(f*x+e)^2)*(d*sec(f*x+e))^(5/3
)*tan(f*x+e)^3/a^4/f/(sec(f*x+e)^2)^(5/6)-a*b*(d*sec(f*x+e))^(5/3)/(a^2+b^2)/f/(a^2-b^2*tan(f*x+e)^2)

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3593, 771, 440, 455, 44, 65, 302, 648, 632, 210, 642, 214, 524} \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\frac {\tan (e+f x) (d \sec (e+f x))^{5/3} \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{a^2 f \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}+\frac {a (d \sec (e+f x))^{5/3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt {3} \sqrt [6]{a^2+b^2}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right )}{3 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{f \left (a^2+b^2\right ) \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {a (d \sec (e+f x))^{5/3} \log \left (-\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}-\frac {a (d \sec (e+f x))^{5/3} \log \left (\sqrt [3]{b} \sqrt [6]{a^2+b^2} \sqrt [6]{\sec ^2(e+f x)}+\sqrt [3]{a^2+b^2}+b^{2/3} \sqrt [3]{\sec ^2(e+f x)}\right )}{12 b^{2/3} f \left (a^2+b^2\right )^{7/6} \sec ^2(e+f x)^{5/6}}+\frac {b^2 \tan ^3(e+f x) (d \sec (e+f x))^{5/3} \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right )}{3 a^4 f \sec ^2(e+f x)^{5/6}} \]

[In]

Int[(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x])^2,x]

[Out]

-1/2*(a*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])^(5
/3))/(Sqrt[3]*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) + (a*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])^(5/3))/(2*Sqrt[3]*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Se
c[e + f*x]^2)^(5/6)) - (a*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e + f*x])^(5/3))/
(3*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) + (a*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])^(5/3))/(12*b^(2/3)*(a^2 + b^2)^(7/6)
*f*(Sec[e + f*x]^2)^(5/6)) - (a*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])^(5/3))/(12*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) +
(AppellF1[1/2, 2, 1/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(5/3)*Tan[e + f*x])/(
a^2*f*(Sec[e + f*x]^2)^(5/6)) + (b^2*AppellF1[3/2, 2, 1/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*
Sec[e + f*x])^(5/3)*Tan[e + f*x]^3)/(3*a^4*f*(Sec[e + f*x]^2)^(5/6)) - (a*b*(d*Sec[e + f*x])^(5/3))/((a^2 + b^
2)*f*(a^2 - b^2*Tan[e + f*x]^2))

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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])

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 {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {1}{(a+x)^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}} \\ & = \frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \left (\frac {a^2}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}-\frac {2 a x}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}+\frac {x^2}{\left (-a^2+x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}}\right ) \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}} \\ & = \frac {(d \sec (e+f x))^{5/3} \text {Subst}\left (\int \frac {x^2}{\left (-a^2+x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}-\frac {\left (2 a (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}}+\frac {\left (a^2 (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right )^2 \sqrt [6]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right )^2 \sqrt [6]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{b f \sec ^2(e+f x)^{5/6}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{6 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a b (d \sec (e+f x))^{5/3}\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 \sec ^2(e+f x)^{5/6}} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{3 \sqrt [3]{b} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{3 \sqrt [3]{b} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{3 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}} \\ & = -\frac {a \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 )}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\left (a (d \sec (e+f x))^{5/3}\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 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}}+\frac {\left (a (d \sec (e+f x))^{5/3}\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 \sqrt [3]{b} \left (a^2+b^2\right ) f \sec ^2(e+f x)^{5/6}} \\ & = -\frac {a \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )}+\frac {\left (a (d \sec (e+f x))^{5/3}\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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {\left (a (d \sec (e+f x))^{5/3}\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 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}} \\ & = -\frac {a \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 ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \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 ) (d \sec (e+f x))^{5/3}}{2 \sqrt {3} b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \text {arctanh}\left (\frac {\sqrt [3]{b} \sqrt [6]{\sec ^2(e+f x)}}{\sqrt [6]{a^2+b^2}}\right ) (d \sec (e+f x))^{5/3}}{3 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}-\frac {a \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 ) (d \sec (e+f x))^{5/3}}{12 b^{2/3} \left (a^2+b^2\right )^{7/6} f \sec ^2(e+f x)^{5/6}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {1}{6},\frac {3}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan (e+f x)}{a^2 f \sec ^2(e+f x)^{5/6}}+\frac {b^2 \operatorname {AppellF1}\left (\frac {3}{2},2,\frac {1}{6},\frac {5}{2},\frac {b^2 \tan ^2(e+f x)}{a^2},-\tan ^2(e+f x)\right ) (d \sec (e+f x))^{5/3} \tan ^3(e+f x)}{3 a^4 f \sec ^2(e+f x)^{5/6}}-\frac {a b (d \sec (e+f x))^{5/3}}{\left (a^2+b^2\right ) f \left (a^2-b^2 \tan ^2(e+f x)\right )} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 129.13 (sec) , antiderivative size = 8003, normalized size of antiderivative = 11.65 \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\text {Result too large to show} \]

[In]

Integrate[(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x])^2,x]

[Out]

Result too large to show

Maple [F]

\[\int \frac {\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}{\left (a +b \tan \left (f x +e \right )\right )^{2}}d x\]

[In]

int((d*sec(f*x+e))^(5/3)/(a+b*tan(f*x+e))^2,x)

[Out]

int((d*sec(f*x+e))^(5/3)/(a+b*tan(f*x+e))^2,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate((d*sec(f*x+e))^(5/3)/(a+b*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate((d*sec(f*x+e))**(5/3)/(a+b*tan(f*x+e))**2,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate((d*sec(f*x+e))^(5/3)/(a+b*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*sec(f*x+e))^(5/3)/(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e))^(5/3)/(b*tan(f*x + e) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \sec (e+f x))^{5/3}}{(a+b \tan (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]

[In]

int((d/cos(e + f*x))^(5/3)/(a + b*tan(e + f*x))^2,x)

[Out]

int((d/cos(e + f*x))^(5/3)/(a + b*tan(e + f*x))^2, x)