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

Optimal. Leaf size=477 \[ \frac {a \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{2 \sqrt {b} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {E\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{\left (a^2+b^2\right ) f}-\frac {a^2 \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}+\frac {a^2 \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{2 b \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}-\frac {b (d \sec (e+f x))^{3/2}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[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]
time = 0.29, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3593, 759, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \begin {gather*} -\frac {a^2 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\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} \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\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} \text {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} \text {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} \tanh ^{-1}\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 )} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(d \sec (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx &=\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} \tan ^{-1}(\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} \tan ^{-1}(\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} \tan ^{-1}(\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 \tan ^{-1}\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 \tanh ^{-1}\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} \tan ^{-1}(\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) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\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) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 69.39, size = 4487, normalized size = 9.41 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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*(Sqrt[Se
c[e + f*x]]/(4*a*(a*Cos[e + f*x] + b*Sin[e + f*x])) - (Cos[2*(e + f*x)]*Sqrt[Sec[e + f*x]])/(4*a*(a*Cos[e + f*
x] + b*Sin[e + f*x])))*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(4*b + 4*a*Tan[(e + f*x)/2] - 4
*b*Tan[(e + f*x)/2]^2 - 4*a*Tan[(e + f*x)/2]^3 - (a^2*Sqrt[b - Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b - Sqrt[a^2 + b^
2])*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]]*Sqrt[a^2 + b*(b -
Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4])/(Sqrt[b]*Sqrt[a^2 + b*(b - Sqr
t[a^2 + b^2])]) - (a^2*Sqrt[b + Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b + Sqrt[a^2 + b^2])*Tan[(e + f*x)/2]^2 + a^2*(-
1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[a^2 + b*(b + Sqrt[a^2 + b^2])]*Sqrt[1 - Tan
[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4])/(Sqrt[b]*Sqrt[a^2 + b*(b + Sqrt[a^2 + b^2])]) + 4*a*EllipticE
[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^
2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*EllipticPi[a^2/(a^2 + 2*(b^2 + Sq
rt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4]))/(4*a*(a^2 + b^2)*f*(1 + Ta
n[(e + f*x)/2]^2)*((Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)
]*(4*b + 4*a*Tan[(e + f*x)/2] - 4*b*Tan[(e + f*x)/2]^2 - 4*a*Tan[(e + f*x)/2]^3 - (a^2*Sqrt[b - Sqrt[a^2 + b^2
]]*ArcTan[(2*b*(b - Sqrt[a^2 + b^2])*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b - S
qrt[a^2 + b^2]]*Sqrt[a^2 + b*(b - Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4
])/(Sqrt[b]*Sqrt[a^2 + b*(b - Sqrt[a^2 + b^2])]) - (a^2*Sqrt[b + Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b + Sqrt[a^2 +
b^2])*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[a^2 + b*(b
+ Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4])/(Sqrt[b]*Sqrt[a^2 + b*(b + S
qrt[a^2 + b^2])]) + 4*a*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*EllipticPi[
a^2/(a^2 + 2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*
EllipticPi[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2
]^4]))/(4*a*(a^2 + b^2)*(1 + Tan[(e + f*x)/2]^2)^2) - (((Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(1 - Tan[(e + f*
x)/2]^2) + (Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*(1 + Tan[(e + f*x)/2]^2))/(1 - Tan[(e + f*x)/2]^2)^2)*(4*b + 4
*a*Tan[(e + f*x)/2] - 4*b*Tan[(e + f*x)/2]^2 - 4*a*Tan[(e + f*x)/2]^3 - (a^2*Sqrt[b - Sqrt[a^2 + b^2]]*ArcTan[
(2*b*(b - Sqrt[a^2 + b^2])*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b - Sqrt[a^2 +
b^2]]*Sqrt[a^2 + b*(b - Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4])/(Sqrt[b
]*Sqrt[a^2 + b*(b - Sqrt[a^2 + b^2])]) - (a^2*Sqrt[b + Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b + Sqrt[a^2 + b^2])*Tan[
(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[a^2 + b*(b + Sqrt[a^
2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sqrt[1 - Tan[(e + f*x)/2]^4])/(Sqrt[b]*Sqrt[a^2 + b*(b + Sqrt[a^2 +
b^2])]) + 4*a*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*EllipticPi[a^2/(a^2 +
2*b^2 - 2*Sqrt[b^2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4] - 4*a*EllipticPi
[a^2/(a^2 + 2*(b^2 + Sqrt[b^2*(a^2 + b^2)])), ArcSin[Tan[(e + f*x)/2]], -1]*Sqrt[1 - Tan[(e + f*x)/2]^4]))/(8*
a*(a^2 + b^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]) - (Sqrt[(1 + T
an[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(2*a*Sec[(e + f*x)/2]^2 - 4*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]
- 6*a*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2 + (a^2*Sqrt[b - Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b - Sqrt[a^2 + b^2]
)*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]]*Sqrt[a^2 + b*(b - S
qrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^3)/(Sqrt[b]*Sqrt[a^2 + b*(
b - Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4]) + (a^2*Sqrt[b + Sqrt[a^2 + b^2]]*ArcTan[(2*b*(b + Sqrt[a^2
+ b^2])*Tan[(e + f*x)/2]^2 + a^2*(-1 + Tan[(e + f*x)/2]^2))/(2*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[a^2 + b
*(b + Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4])]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^3)/(Sqrt[b]*Sqrt[a^
2 + b*(b + Sqrt[a^2 + b^2])]*Sqrt[1 - Tan[(e + f*x)/2]^4]) - (4*a*EllipticE[ArcSin[Tan[(e + f*x)/2]], -1]*Sec[
(e + f*x)/2]^2*Tan[(e + f*x)/2]^3)/Sqrt[1 - Tan[(e + f*x)/2]^4] + (4*a*EllipticPi[a^2/(a^2 + 2*b^2 - 2*Sqrt[b^
2*(a^2 + b^2)]), ArcSin[Tan[(e + f*x)/2]], -1]*...

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 25421 vs. \(2 (436 ) = 872\).
time = 1.09, size = 25422, normalized size = 53.30

method	result	size
default	\(\text {Expression too large to display}\)	\(25422\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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