\(\int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx\) [612]

Optimal result

Integrand size = 25, antiderivative size = 477 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx=\frac {a \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {a^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}+\frac {a^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

-(cos(1/2*arctan(tan(f*x+e)))^2)^(1/2)/cos(1/2*arctan(tan(f*x+e)))*EllipticE(sin(1/2*arctan(tan(f*x+e))),2^(1/
2))*(d*sec(f*x+e))^(3/2)/(a^2+b^2)/f/(sec(f*x+e)^2)^(3/4)+cos(f*x+e)*(d*sec(f*x+e))^(3/2)*sin(f*x+e)/(a^2+b^2)
/f+1/2*a*arctan((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(d*sec(f*x+e))^(3/2)/(a^2+b^2)^(5/4)/f/(sec(f*x+
e)^2)^(3/4)/b^(1/2)-1/2*a*arctanh((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(d*sec(f*x+e))^(3/2)/(a^2+b^2)
^(5/4)/f/(sec(f*x+e)^2)^(3/4)/b^(1/2)-1/2*a^2*cot(f*x+e)*EllipticPi((sec(f*x+e)^2)^(1/4),-b/(a^2+b^2)^(1/2),I)
*(d*sec(f*x+e))^(3/2)*(-tan(f*x+e)^2)^(1/2)/b/(a^2+b^2)^(3/2)/f/(sec(f*x+e)^2)^(3/4)+1/2*a^2*cot(f*x+e)*Ellipt
icPi((sec(f*x+e)^2)^(1/4),b/(a^2+b^2)^(1/2),I)*(d*sec(f*x+e))^(3/2)*(-tan(f*x+e)^2)^(1/2)/b/(a^2+b^2)^(3/2)/f/
(sec(f*x+e)^2)^(3/4)-b*(d*sec(f*x+e))^(3/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3593, 759, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx=-\frac {a^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 b f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}+\frac {a^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right )}{2 b f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}+\frac {a (d \sec (e+f x))^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}-\frac {(d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right )}{f \left (a^2+b^2\right ) \sec ^2(e+f x)^{3/4}}-\frac {a (d \sec (e+f x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 \sqrt {b} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{f \left (a^2+b^2\right )} \]

[In]

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

[Out]

(a*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(2*Sqrt[b]*(a^2 + b^2)^(
5/4)*f*(Sec[e + f*x]^2)^(3/4)) - (a*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x
])^(3/2))/(2*Sqrt[b]*(a^2 + b^2)^(5/4)*f*(Sec[e + f*x]^2)^(3/4)) - (EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(d*Se
c[e + f*x])^(3/2))/((a^2 + b^2)*f*(Sec[e + f*x]^2)^(3/4)) + (Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])
/((a^2 + b^2)*f) - (a^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*S
ec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(2*b*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) + (a^2*Cot[e + f*x]
*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2
])/(2*b*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) - (b*(d*Sec[e + f*x])^(3/2))/((a^2 + b^2)*f*(a + b*Tan[e +
f*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 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 211

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

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 233

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)), x] - Dist[a, Int[1/(a + b*x^2)^(5
/4), x], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 408

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/x), Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 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 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 760

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

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
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))^{3/2} \text {Subst}\left (\int \frac {1}{(a+x)^2 \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {-a-\frac {x}{2}}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = \frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(d \sec (e+f x))^{3/2} \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a^2 (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\left (a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt [4]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (1+\frac {a^2}{b^2}-x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\left (a b (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x^2}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}-b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (a^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = -\frac {E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\left (a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a (d \sec (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (a^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (a^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}} \\ & = \frac {a \arctan \left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {E\left (\left .\frac {1}{2} \arctan (\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {a^2 \cot (e+f x) \operatorname {EllipticPi}\left (-\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}+\frac {a^2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {b}{\sqrt {a^2+b^2}},\arcsin \left (\sqrt [4]{\sec ^2(e+f x)}\right ),-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 27.97 (sec) , antiderivative size = 1125, normalized size of antiderivative = 2.36 \[ \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx=\frac {\sec (e+f x) (d \sec (e+f x))^{3/2} (a \cos (e+f x)+b \sin (e+f x))^2 \left (\frac {b \cos (e+f x)}{a (a-i b) (a+i b)}+\frac {\sin (e+f x)}{(a-i b) (a+i b)}-\frac {b}{(a-i b) (a+i b) (a \cos (e+f x)+b \sin (e+f x))}\right )}{f (a+b \tan (e+f x))^2}+\frac {\sqrt {\sec (e+f x)} (d \sec (e+f x))^{3/2} (a \cos (e+f x)+b \sin (e+f x))^2 \left (-\frac {a E\left (\left .\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )\right |-1\right ) \sqrt {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}}{\sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}}-\frac {-2 \sqrt {2} a b \sqrt {a^2+b^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {1+i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}+\sqrt {2} a^2 \sqrt {a^2+b^2} \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {1+i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}+a^2 \left (a+i b+\sqrt {a^2+b^2}\right ) \operatorname {EllipticPi}\left (\frac {(1+i) \left (a+i \left (-b+\sqrt {a^2+b^2}\right )\right )}{a+b-\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}-a^3 \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}-i a^2 b \operatorname {EllipticPi}\left (\frac {(1+i) \left (a-i \left (b+\sqrt {a^2+b^2}\right )\right )}{a+b+\sqrt {a^2+b^2}},\arcsin \left (\frac {\sqrt {\frac {(1+i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {2}}\right ),2\right ) \sqrt {-\frac {2+2 i \tan \left (\frac {1}{2} (e+f x)\right )}{i+\tan \left (\frac {1}{2} (e+f x)\right )}}+2 b^2 \sqrt {a^2+b^2} \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}+2 a b \sqrt {a^2+b^2} \tan \left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}}{2 b \sqrt {a^2+b^2} \sqrt {\frac {-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{\left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}}}\right )}{a \left (a^2+b^2\right ) f \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} (a+b \tan (e+f x))^2} \]

[In]

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

[Out]

(Sec[e + f*x]*(d*Sec[e + f*x])^(3/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*((b*Cos[e + f*x])/(a*(a - I*b)*(a + I
*b)) + Sin[e + f*x]/((a - I*b)*(a + I*b)) - b/((a - I*b)*(a + I*b)*(a*Cos[e + f*x] + b*Sin[e + f*x]))))/(f*(a
+ b*Tan[e + f*x])^2) + (Sqrt[Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(-((a*El
lipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 + Tan[(e + f*x)/2]^2])/Sqrt[1 - Tan[(e + f*x)/2]^2]) - (-2*Sqrt[2
]*a*b*Sqrt[a^2 + b^2]*EllipticF[ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]],
2]*Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] + Sqrt[2]*a^2*Sqrt[a^2 + b^2]*EllipticPi[((1 + I)
*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + T
an[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((1 + I*Tan[(e + f*x)/2])/(I + Tan[(e + f*x)/2]))] + a^2*(a + I*b + Sqrt[
a^2 + b^2])*EllipticPi[((1 + I)*(a + I*(-b + Sqrt[a^2 + b^2])))/(a + b - Sqrt[a^2 + b^2]), ArcSin[Sqrt[((1 + I
)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*x)/2])/(I + Tan[(e
+ f*x)/2]))] - a^3*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), ArcSin[Sqrt[
((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*x)/2])/(I +
Tan[(e + f*x)/2]))] - I*a^2*b*EllipticPi[((1 + I)*(a - I*(b + Sqrt[a^2 + b^2])))/(a + b + Sqrt[a^2 + b^2]), A
rcSin[Sqrt[((1 + I)*(1 + Tan[(e + f*x)/2]))/(I + Tan[(e + f*x)/2])]/Sqrt[2]], 2]*Sqrt[-((2 + (2*I)*Tan[(e + f*
x)/2])/(I + Tan[(e + f*x)/2]))] + 2*b^2*Sqrt[a^2 + b^2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^
2] + 2*a*b*Sqrt[a^2 + b^2]*Tan[(e + f*x)/2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^2])/(2*b*Sqr
t[a^2 + b^2]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)/(I + Tan[(e + f*x)/2])^2])))/(a*(a^2 + b^2)*f*Sqrt[(1 + Tan[(e + f
*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a + b*Tan[e + f*x])^2)

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 16453 vs. \(2 (436 ) = 872\).

Time = 8.87 (sec) , antiderivative size = 16454, normalized size of antiderivative = 34.49

method	result	size
default	\(\text {Expression too large to display}\)	\(16454\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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