∫ 1____________∘ a+b sec(e+fx) (c+d sec(e+fx))3∕2 dx

3.220 \(\int \frac{1}{\sqrt{a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=622 \[ -\frac{2 d \sqrt{a+b} (2 c-d) \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt{c+d}}-\frac{2 d^2 (a-b) \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} (b c-a d)^2} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*d^2*Cot[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*

Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b

)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e

+ f*x]))/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) - (2*Sqrt[a + b]*(2*c - d)*d*Cot[e + f*x]*EllipticF[ArcSin[(

Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c +

d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f

*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c^2*(c - d)*Sqrt[c + d]*(b*c - a*d)*f) - (2*Sqrt

[a + b]*Cot[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a

+ b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/(

(a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d

*Sec[e + f*x]))/(a*c^2*Sqrt[c + d]*f)

________________________________________________________________________________________

Rubi [A] time = 1.31597, antiderivative size = 763, normalized size of antiderivative = 1.23, 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, 3054, 2811, 2998, 2818, 2996} \[ -\frac{2 d \sqrt{a+b} (2 c-d) \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f (c-d) \sqrt{c+d} (b c-a d) \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \Pi \left (\frac{(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt{c+d} \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}}-\frac{2 d^2 (a-b) \sqrt{a+b} \csc (e+f x) \sqrt{a+b \sec (e+f x)} (c \cos (e+f x)+d)^{3/2} \sqrt{-\frac{(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt{-\frac{(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{b+a \cos (e+f x)}}{\sqrt{a+b} \sqrt{d+c \cos (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} (b c-a d)^2 \sqrt{a \cos (e+f x)+b} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]

[Out]

(-2*(a - b)*Sqrt[a + b]*d^2*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]*Ellipt

icE[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]*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sq

rt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(2*c - d)*d*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*C

os[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]*(b*c - a*d)*f*Sq

rt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*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*C

os[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), 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]])/(a

*c^2*Sqrt[c + d]*f*Sqrt[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 3054

Int[((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.

)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x

], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C - 2*a*b*C*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, C}, 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{1}{\sqrt{a+b \sec (e+f x)} (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{\cos ^2(e+f x)}{\sqrt{b+a \cos (e+f x)} (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 (\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 (\sqrt{d+c \cos (e+f x)} \sqrt{a+b \sec (e+f x)}\right ) \int \frac{-d^2-2 c d \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 \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)}}{a c^2 \sqrt{c+d} f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{\left ((2 c-d) 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 (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} d^2 \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} (b c-a d)^2 f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \sqrt{a+b} (2 c-d) 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} (b c-a d) f \sqrt{b+a \cos (e+f x)} \sqrt{c+d \sec (e+f x)}}-\frac{2 \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)}}{a 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 = 9.53524, size = 1731, normalized size = 2.78 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]

[Out]

(Sqrt[b + a*Cos[e + f*x]]*(d + c*Cos[e + f*x])^(3/2)*Sec[e + f*x]^2*((-4*b*c*d*(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]*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*(b*c - a*d)*(b*c^2 - a*c*d - 2*b*d^2)*(

(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))]*S

in[(e + f*x)/2]^4)/((a + b)*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[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]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sq

rt[-(((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*d^2*((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])*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]]) - ((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]*EllipticPi[(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)*(b*c - a*d)*f*Sqrt[a + b*Sec[e + f*x]]*(c

+ d*Sec[e + f*x])^(3/2)) + (2*d^2*(b + a*Cos[e + f*x])*(d + c*Cos[e + f*x])*Sec[e + f*x]*Tan[e + f*x])/((-(b*c

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

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Maple [B] time = 0.463, size = 3451, normalized size = 5.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x)

[Out]

2/f/c/(c+d)/(c-d)/(a*d-b*c)/((a-b)/(a+b))^(1/2)*(-b*c*d^2*((a-b)/(a+b))^(1/2)-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*c^3*(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)+sin(f*x+e)*cos(f*x+e)*(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)*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*c^2*d-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*d^3*(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)+sin(f*x+e)*cos(f*x+e)*(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)*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*c*d^2-sin(f*x+e)*cos(f*x+e)*(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)*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*c^2*d-sin(f*x+e)*cos(f*x+e)*(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)*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*c*d^2+sin(f*x+e)*cos(f*x+e)*(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)*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*c*d^2-2*sin(f*x+e)*cos(f*x+e)*(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)*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*c^2*d-2*sin(f*x+e)*cos(f*x+e)*(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)*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))*b*c*d^2+cos(f*x+e)^2*((a-b)/(a+b))^(1/2)*a*c*d^2-cos(f

*x+e)*((a-b)/(a+b))^(1/2)*a*c*d^2+cos(f*x+e)*((a-b)/(a+b))^(1/2)*b*c*d^2+b*d^3*((a-b)/(a+b))^(1/2)-cos(f*x+e)^

2*((a-b)/(a+b))^(1/2)*a*d^3+cos(f*x+e)*((a-b)/(a+b))^(1/2)*a*d^3-cos(f*x+e)*((a-b)/(a+b))^(1/2)*b*d^3-sin(f*x+

e)*cos(f*x+e)*(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)*

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*c^3-sin(f*x+e)*cos

(f*x+e)*(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)*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))*a*d^3+sin(f*x+e)*cos(f*x+e

)*(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)*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*d^3+2*sin(f*x+e)*cos(f*x+e)*(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)*EllipticPi((-1+c

os(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*d^3+2*sin(f*

x+e)*cos(f*x+e)*(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

)*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))*b*c^3+sin(f*x+e)*(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)*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*c^2*d+Ellip

ticF((-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*c*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)-sin(f*x+e)*(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)*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*c^2*d-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*c*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*c*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*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*c^2*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)-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))*b*c*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)/s

in(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*b*d^3*(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*d^3*(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))*b*c^3*(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)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(1

/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)/sin(f*x+e)/(d+c*cos(f*x+e))/(a*cos(f*x+e)+b)

________________________________________________________________________________________

Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right ) + a}{\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(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sec(f*x + e) + a)*(d*sec(f*x + e) + c)^(3/2)), x)

________________________________________________________________________________________

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(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \sec{\left (e + f x \right )}} \left (c + d \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*sec(f*x+e))**(3/2)/(a+b*sec(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))**(3/2)), x)

________________________________________________________________________________________

Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right ) + a}{\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(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sec(f*x + e) + a)*(d*sec(f*x + e) + c)^(3/2)), x)

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