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

Optimal. Leaf size=229 \[ \frac{g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (e+f x),\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}}-\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f (a-b) \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}} \]

[Out]

(g*(b + a*Cos[e + f*x])*EllipticE[(e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(b + a*C
os[e + f*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + (g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(e + f*x)/2,
(2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*Cos[e + f*x])*Sqrt[g*Sec[e +
f*x]]*Sin[e + f*x])/((a - b)*f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])

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Rubi [A]  time = 0.439518, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {3975, 2768, 2752, 2663, 2661, 2655, 2653} \[ -\frac{g \sin (e+f x) \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} F\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f \sqrt{a+b \sec (e+f x)}}+\frac{g \sqrt{g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{c f (a-b) \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \sqrt{a+b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(b + a*Cos[e + f*x])*EllipticE[(e + f*x)/2, (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(b + a*C
os[e + f*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + (g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(e + f*x)/2,
(2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*Cos[e + f*x])*Sqrt[g*Sec[e +
f*x]]*Sin[e + f*x])/((a - b)*f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])

Rule 3975

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

Rule 2768

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b
^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x])), x] + Dist[d/(a*(b*c - a*
d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*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]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

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

Mathematica [C]  time = 8.38203, size = 1019, normalized size = 4.45 \[ \frac{(b+a \cos (e+f x)) (g \sec (e+f x))^{3/2} \left (\frac{2 \csc (e)}{(b-a) f}+\frac{2 \sec \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}+\frac{f x}{2}\right ) \sin \left (\frac{f x}{2}\right )}{(b-a) f}\right ) \cos ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{\sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)}+\frac{a \sqrt{b+a \cos (e+f x)} \csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) (g \sec (e+f x))^{3/2} \left (\frac{F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\sec (e) \left (b+a \cos (e) \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{a \sqrt{\tan ^2(e)+1} \left (1-\frac{b \sec (e)}{a \sqrt{\tan ^2(e)+1}}\right )},-\frac{\sec (e) \left (b+a \cos (e) \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{a \sqrt{\tan ^2(e)+1} \left (-\frac{b \sec (e)}{a \sqrt{\tan ^2(e)+1}}-1\right )}\right ) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \tan (e)}{\sqrt{\tan ^2(e)+1} \sqrt{\frac{a \sqrt{\tan ^2(e)+1}-a \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} a+b \sec (e)}} \sqrt{\frac{\cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1} a+\sqrt{\tan ^2(e)+1} a}{a \sqrt{\tan ^2(e)+1}-b \sec (e)}} \sqrt{b+a \cos (e) \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}}-\frac{\frac{\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \tan (e)}{\sqrt{\tan ^2(e)+1}}+\frac{2 a \cos (e) \left (b+a \cos (e) \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{a^2 \cos ^2(e)+a^2 \sin ^2(e)}}{\sqrt{b+a \cos (e) \cos \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}}\right ) \cos ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 (b-a) f \sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)}+\frac{F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{\csc (e) \left (b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )\right )}{a \sqrt{\cot ^2(e)+1} \left (\frac{b \csc (e)}{a \sqrt{\cot ^2(e)+1}}+1\right )},\frac{\csc (e) \left (b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )\right )}{a \sqrt{\cot ^2(e)+1} \left (\frac{b \csc (e)}{a \sqrt{\cot ^2(e)+1}}-1\right )}\right ) \sqrt{b+a \cos (e+f x)} \csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) (g \sec (e+f x))^{3/2} \sec \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\frac{a \sqrt{\cot ^2(e)+1}-a \sqrt{\cot ^2(e)+1} \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{a \sqrt{\cot ^2(e)+1}-b \csc (e)}} \sqrt{\frac{\sqrt{\cot ^2(e)+1} \sin \left (f x-\tan ^{-1}(\cot (e))\right ) a+\sqrt{\cot ^2(e)+1} a}{\sqrt{\cot ^2(e)+1} a+b \csc (e)}} \sqrt{b-a \sqrt{\cot ^2(e)+1} \sin (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )} \cos ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{(b-a) f \sqrt{\cot ^2(e)+1} \sqrt{a+b \sec (e+f x)} (\sec (e+f x) c+c)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[e/2 + (f*x)/2]^2*(b + a*Cos[e + f*x])*(g*Sec[e + f*x])^(3/2)*((2*Csc[e])/((-a + b)*f) + (2*Sec[e/2]*Sec[e
/2 + (f*x)/2]*Sin[(f*x)/2])/((-a + b)*f)))/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (AppellF1[1/2, 1/
2, 1/2, 3/2, (Csc[e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]))/(a*Sqrt[1 + Cot[e]^2]*(1 + (
b*Csc[e])/(a*Sqrt[1 + Cot[e]^2]))), (Csc[e]*(b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*x - ArcTan[Cot[e]]]))/(a*Sq
rt[1 + Cot[e]^2]*(-1 + (b*Csc[e])/(a*Sqrt[1 + Cot[e]^2])))]*Cos[e/2 + (f*x)/2]^2*Sqrt[b + a*Cos[e + f*x]]*Csc[
e/2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*Sec[f*x - ArcTan[Cot[e]]]*Sqrt[(a*Sqrt[1 + Cot[e]^2] - a*Sqrt[1 + Cot[e]^
2]*Sin[f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] - b*Csc[e])]*Sqrt[(a*Sqrt[1 + Cot[e]^2] + a*Sqrt[1 + Cot[e
]^2]*Sin[f*x - ArcTan[Cot[e]]])/(a*Sqrt[1 + Cot[e]^2] + b*Csc[e])]*Sqrt[b - a*Sqrt[1 + Cot[e]^2]*Sin[e]*Sin[f*
x - ArcTan[Cot[e]]]])/((-a + b)*f*Sqrt[1 + Cot[e]^2]*Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])) + (a*Cos[e
/2 + (f*x)/2]^2*Sqrt[b + a*Cos[e + f*x]]*Csc[e/2]*Sec[e/2]*(g*Sec[e + f*x])^(3/2)*((AppellF1[-1/2, -1/2, -1/2,
1/2, -((Sec[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt[1 + Tan[e]^2]*(1 - (b*Sec
[e])/(a*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a*Sqrt
[1 + Tan[e]^2]*(-1 - (b*Sec[e])/(a*Sqrt[1 + Tan[e]^2]))))]*Sin[f*x + ArcTan[Tan[e]]]*Tan[e])/(Sqrt[1 + Tan[e]^
2]*Sqrt[(a*Sqrt[1 + Tan[e]^2] - a*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(b*Sec[e] + a*Sqrt[1 + Tan[e]^
2])]*Sqrt[(a*Sqrt[1 + Tan[e]^2] + a*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(b*Sec[e]) + a*Sqrt[1 + Ta
n[e]^2])]*Sqrt[b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]]) - ((Sin[f*x + ArcTan[Tan[e]]]*Tan[e
])/Sqrt[1 + Tan[e]^2] + (2*a*Cos[e]*(b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(a^2*Cos[e]^2
+ a^2*Sin[e]^2))/Sqrt[b + a*Cos[e]*Cos[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]]))/(2*(-a + b)*f*Sqrt[a + b*S
ec[e + f*x]]*(c + c*Sec[e + f*x]))

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Maple [C]  time = 0.332, size = 222, normalized size = 1. \begin{align*}{\frac{i \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{fc \left ( a-b \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) } \left ( 2\,a{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) -a{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) -b{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) \right ) \sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}{\frac{1}{\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I/c/f/(a-b)*(2*a*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b))^(1/2))-a*EllipticE(I*(-1+cos(f*x+e))/si
n(f*x+e),(-(a-b)/(a+b))^(1/2))-b*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),(-(a-b)/(a+b))^(1/2)))*(1/(a+b)*(a*cos
(f*x+e)+b)/(1+cos(f*x+e)))^(1/2)*(g/cos(f*x+e))^(3/2)*cos(f*x+e)^2*(1/cos(f*x+e)*(a*cos(f*x+e)+b))^(1/2)/(a*co
s(f*x+e)+b)/(1/(1+cos(f*x+e)))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right ) + a} \sqrt{g \sec \left (f x + e\right )} g \sec \left (f x + e\right )}{b c \sec \left (f x + e\right )^{2} +{\left (a + b\right )} c \sec \left (f x + e\right ) + a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(f*x + e) + a)*sqrt(g*sec(f*x + e))*g*sec(f*x + e)/(b*c*sec(f*x + e)^2 + (a + b)*c*sec(f*x
+ e) + a*c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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