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

Optimal. Leaf size=598 \[ -\frac{2 d (a-b) \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{c^2 f \sqrt{a+b}}-\frac{2 d \sqrt{a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}} \]

[Out]

(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])
/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e
+ f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))
]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c^2*f) - (2*Sqrt[a + b]*d*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))]*(1 + Sec[e
+ f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[
-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]) - (2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*E
llipticF[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)*f)

________________________________________________________________________________________

Rubi [A]  time = 0.90203, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3939, 3936, 3986, 3984, 3994} \[ -\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{c^2 f \sqrt{a+b}}-\frac{2 d (a-b) \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} (b c-a d)}-\frac{2 d \sqrt{a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt{c+d} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])
/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e
+ f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))
]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c^2*f) - (2*Sqrt[a + b]*d*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))]*(1 + Sec[e
+ f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[
-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]) - (2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*E
llipticF[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)*f)

Rule 3939

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

Rule 3936

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[(
2*(a + b*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 + Csc[e + f*x]))/((c - d)*(a + b*Csc[e + f*x]))]*Sqrt[-(((b*c - a*
d)*(1 - Csc[e + f*x]))/((c + d)*(a + b*Csc[e + f*x])))]*EllipticPi[(a*(c + d))/(c*(a + b)), ArcSin[(Rt[(a + b)
/(c + d), 2]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(c*f*R
t[(a + b)/(c + d), 2]*Cot[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]

Rule 3986

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(3/2), x_Symbol] :> Dist[(a - b)/(c - d), Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]
]), x], x] + Dist[(b*c - a*d)/(c - d), Int[(Csc[e + f*x]*(1 + Csc[e + f*x]))/(Sqrt[a + b*Csc[e + f*x]]*(c + d*
Csc[e + f*x])^(3/2)), 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 3984

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (
c_)]), x_Symbol] :> Simp[(-2*(c + d*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 - Csc[e + f*x]))/((a + b)*(c + d*Csc[e
+ f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Csc[e + f*x]))/((a - b)*(c + d*Csc[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)
/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b
*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[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]

Rule 3994

Int[(sec[(e_.) + (f_.)*(x_)]*((A_) + (B_.)*sec[(e_.) + (f_.)*(x_)]))/(Sqrt[(a_) + (b_.)*sec[(e_.) + (f_.)*(x_)
]]*((c_) + (d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)), x_Symbol] :> Simp[(2*A*(1 + Sec[e + f*x])*Sqrt[((b*c - a*d)*
(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*EllipticE[ArcSin[(Rt[(c + d)/(a + b), 2]*Sqrt[a + b*Sec[e
+ f*x]])/Sqrt[c + d*Sec[e + f*x]]], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2
]*Tan[e + f*x]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[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]

Rubi steps

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

Mathematica [B]  time = 9.25764, size = 1678, normalized size = 2.81 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.484, size = 2847, normalized size = 4.8 \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))^(1/2)/(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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e
)+b)/(1+cos(f*x+e)))^(1/2)*a*c*d-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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(
1+cos(f*x+e)))^(1/2)*b*c*d+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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1
/2)*a*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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*
c*d+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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*b*c*d-b*
c*d*((a-b)/(a+b))^(1/2)-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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)
*b*c^2+cos(f*x+e)^2*((a-b)/(a+b))^(1/2)*a*c*d-cos(f*x+e)*((a-b)/(a+b))^(1/2)*a*c*d+cos(f*x+e)*((a-b)/(a+b))^(1
/2)*b*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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*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/(c+d)*(d+c*co
s(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*b*d^2-2*sin(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/(c+d)*
(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*c^2+2*sin(f*x+e)*Elli
pticPi((-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
/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*d^2-cos(f*x+e)
^2*((a-b)/(a+b))^(1/2)*a*d^2+cos(f*x+e)*((a-b)/(a+b))^(1/2)*a*d^2-cos(f*x+e)*((a-b)/(a+b))^(1/2)*b*d^2+b*d^2*(
(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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))
^(1/2)*a*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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)
*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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*a*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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*b*d^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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))
^(1/2)*a*c^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/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+
b)/(1+cos(f*x+e)))^(1/2)*a*d^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*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/
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))*b*c*d*
(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*sin(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))*a*c*d*(1/(c+d)*(d+c*cos
(f*x+e))/(1+cos(f*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(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*(1/(c+d)*(d+c*cos(f*x+e))/(1+cos(f
*x+e)))^(1/2)*(1/(a+b)*(a*cos(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e))*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{\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((a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

[Out]

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

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

[Out]

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