3.211 \(\int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=744 \[ -\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a \cos (e+f x)+b}}{\sqrt{a+b} \sqrt{c \cos (e+f x)+d}}\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c
- a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[
ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a -
b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f
*x]]) - (2*Sqrt[a + b]*(b*c - a*(2*c - d))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x
])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[
e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a
+ b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]
*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e +
f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*C
sc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqr
t[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*Sqrt[c + d]*f*Sqr
t[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.10629, antiderivative size = 744, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {3942, 2798, 2811, 2998, 2818, 2996} \[ -\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - b)*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c
- a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[
ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a -
b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f
*x]]) - (2*Sqrt[a + b]*(b*c - a*(2*c - d))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x
])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[
e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a
+ b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]
*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e +
f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*C
sc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqr
t[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*Sqrt[c + d]*f*Sqr
t[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])

Rule 3942

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Dist
[(Sqrt[d + c*Sin[e + f*x]]*Sqrt[a + b*Csc[e + f*x]])/(Sqrt[b + a*Sin[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[
((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}
, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]

Rule 2798

Int[((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)/((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> D
ist[d^2/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b^2, Int[Simp[b*
c + a*d + 2*b*d*Sin[e + f*x], x]/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2811

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

Rule 2998

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 2818

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Si
mp[(2*(c + d*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c
- a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a +
b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a
+ b), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c
^2 - d^2, 0] && PosQ[(c + d)/(a + b)]

Rule 2996

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*
x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*
EllipticE[ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d)
)/((a + b)*(c - d))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rubi steps

\begin{align*} \int \frac{(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx &=\frac{\left (\sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{(b+a \cos (e+f x))^{3/2}}{(d+c \cos (e+f x))^{3/2}} \, dx}{\sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}\\ &=\frac{\left (a^2 \sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{\sqrt{d+c \cos (e+f x)}}{\sqrt{b+a \cos (e+f x)}} \, dx}{c^2 \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{\left ((b c-a d) \sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{b c+a d+2 a c \cos (e+f x)}{\sqrt{b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{c^2 \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}\\ &=-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{a+b \sec (e+f x)}}{c^2 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{\left ((b c-a (2 c-d)) (b c-a d) \sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{b+a \cos (e+f x)} \sqrt{d+c \cos (e+f x)}} \, dx}{c^2 (c-d) \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{\left ((b c-a d)^2 \sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{1+\cos (e+f x)}{\sqrt{b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{c (c-d) \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}\\ &=-\frac{2 (a-b) \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{a+b \sec (e+f x)}}{c (c-d) \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} (b c-a (2 c-d)) \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{a+b \sec (e+f x)}}{c^2 (c-d) \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 a \sqrt{a+b} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{a+b \sec (e+f x)}}{c^2 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 9.45425, size = 1720, normalized size = 2.31 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)*(-(b*c*Sin[e + f*x]) + a*d*Sin[e + f*x]))/((-c^2 + d^2)*f*(
b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(3/2)) + ((d + c*Cos[e + f*x])^(3/2)*(a + b*Sec[e + f*x])^(3/2)*((4*(
b*c - a*d)*(a*b*c - b^2*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(
e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]*Csc[e + f*x
]*EllipticF[ArcSin[Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]/Sqrt[2]], (2*(b*c -
a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]
]) + 4*(a^2*c - b^2*c)*(b*c - a*d)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f
*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]*C
sc[e + f*x]*EllipticF[ArcSin[Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]/Sqrt[2]],
(2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Co
s[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]
^2)/(b*c - a*d)]*Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]*Csc[e + f*x]*EllipticP
i[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]/Sqrt[
2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos
[e + f*x]])) + 2*(-(a*b*c) + a^2*d)*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x]]
*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*x])/(a + b)]], (2*(b*c - a*d
))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x])]*Sqrt[b + a*Cos[e + f*x]]*
Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b + a*Cos[e + f*x]))]) - (2*(
b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*C
sc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]*Csc[e +
f*x]*EllipticF[ArcSin[Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]/Sqrt[2]], (2*(b*
c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e +
f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f
*x)/2]^2)/(b*c - a*d)]*Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]*Csc[e + f*x]*Ell
ipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[-(((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d))]
/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d +
c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e + f*x]]))))/((c - d)*(
c + d)*f*(b + a*Cos[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(3/2))

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Maple [B]  time = 0.442, size = 4298, normalized size = 5.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/c/(c+d)/(c-d)/((a-b)/(a+b))^(1/2)*(sin(f*x+e)*cos(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin
(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x
+e))/(1+cos(f*x+e)))^(1/2)*a^2*c*d-2*sin(f*x+e)*cos(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f
*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e
))/(1+cos(f*x+e)))^(1/2)*a*b*c*d+2*sin(f*x+e)*cos(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x
+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))
/(1+cos(f*x+e)))^(1/2)*a*b*c*d+EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+
d))^(1/2))*a^2*c^2*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(
1/2)*sin(f*x+e)+cos(f*x+e)*((a-b)/(a+b))^(1/2)*a*b*c^2+EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e
),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f
*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-a*b*c*d*((a-b)/(a+b))^(1/2)+a*b*d^2*((a-b)/(a+b))^(1/2)-b^2*c*d*((a-b)
/(a+b))^(1/2)-cos(f*x+e)^2*((a-b)/(a+b))^(1/2)*a^2*d^2+cos(f*x+e)*((a-b)/(a+b))^(1/2)*a^2*d^2-cos(f*x+e)*((a-b
)/(a+b))^(1/2)*b^2*c^2-cos(f*x+e)*((a-b)/(a+b))^(1/2)*a*b*d^2+cos(f*x+e)*((a-b)/(a+b))^(1/2)*b^2*c*d+cos(f*x+e
)^2*((a-b)/(a+b))^(1/2)*a^2*c*d-cos(f*x+e)^2*((a-b)/(a+b))^(1/2)*a*b*c^2+b^2*c^2*((a-b)/(a+b))^(1/2)-2*sin(f*x
+e)*cos(f*x+e)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-
b)/(a+b))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/
2)*a^2*c^2-sin(f*x+e)*cos(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(
c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*b
^2*c*d-2*sin(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*
(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*c*d+2*sin(
f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*c
os(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*c*d-2*sin(f*x+e)*cos(f*
x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos
(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*c^2+sin(f*x+e)*cos(f*x+e)
*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x
+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*b^2*c*d-sin(f*x+e)*cos(f*x+e)*Ell
ipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+
b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^2*c*d+sin(f*x+e)*cos(f*x+e)*Ellipti
cE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(
1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*c^2+sin(f*x+e)*cos(f*x+e)*EllipticE((
-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+co
s(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*d^2+sin(f*x+e)*EllipticF((-1+cos(f*x+e))*
((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2
)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*b^2*c*d-sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(
1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c
*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^2*c*d+sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e)
,((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1
+cos(f*x+e)))^(1/2)*a*b*c^2+EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))
^(1/2))*a*b*d^2*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2
)*sin(f*x+e)+2*sin(f*x+e)*cos(f*x+e)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((
c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e
))/(1+cos(f*x+e)))^(1/2)*a^2*d^2-EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(
c+d))^(1/2))*b^2*c*d*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))
^(1/2)*sin(f*x+e)-EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a^
2*d^2*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+
e)-EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b^2*c^2*(1/(a+b)*
(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)-2*EllipticPi
((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a^2*c^2*
(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+2*s
in(f*x+e)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a
+b))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^
2*d^2+cos(f*x+e)^2*((a-b)/(a+b))^(1/2)*a*b*c*d-cos(f*x+e)*((a-b)/(a+b))^(1/2)*a^2*c*d+sin(f*x+e)*cos(f*x+e)*El
lipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)
+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*b^2*c^2-sin(f*x+e)*cos(f*x+e)*Ellipt
icE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/
(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^2*d^2-sin(f*x+e)*cos(f*x+e)*EllipticE(
(-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+c
os(f*x+e)))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*b^2*c^2+sin(f*x+e)*EllipticF((-1+cos(f*x+e))
*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/
2)*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^2*c*d-2*sin(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b)
)^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(
d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a*b*c^2+sin(f*x+e)*cos(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1
/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(1/(c+d)*(d+c*
cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*a^2*c^2)*cos(f*x+e)*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)*((d+c*cos(f*x+e))/
cos(f*x+e))^(1/2)/sin(f*x+e)/(d+c*cos(f*x+e))/(a*cos(f*x+e)+b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Sympy [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((a+b*sec(f*x+e))**(3/2)/(c+d*sec(f*x+e))**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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