\(\int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx\) [277]

Optimal result

Integrand size = 39, antiderivative size = 229 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {g (b+a \cos (e+f x)) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{(a-b) c f \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{(a-b) f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}} \]

[Out]

-g*(b+a*cos(f*x+e))*sin(f*x+e)*(g*sec(f*x+e))^(1/2)/(a-b)/f/(c+c*cos(f*x+e))/(a+b*sec(f*x+e))^(1/2)+g*(b+a*cos
(f*x+e))*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2)*(a/(a+b))^(1/2))
*(g*sec(f*x+e))^(1/2)/(a-b)/c/f/((b+a*cos(f*x+e))/(a+b))^(1/2)/(a+b*sec(f*x+e))^(1/2)+g*(cos(1/2*f*x+1/2*e)^2)
^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(f*x+e))/(a+b))^(1/2)
*(g*sec(f*x+e))^(1/2)/c/f/(a+b*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4060, 2847, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {g \sin (e+f x) \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \sqrt {a+b \sec (e+f x)}} \]

[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 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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 2847

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] && Ne
Q[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 4060

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]

Rubi steps \begin{align*} \text {integral}& = \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}} \operatorname {EllipticF}\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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 8.39 (sec) , antiderivative size = 1019, normalized size of antiderivative = 4.45 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (b+a \cos (e+f x)) (g \sec (e+f x))^{3/2} \left (\frac {2 \csc (e)}{(-a+b) 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 )}{(-a+b) f}\right )}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\csc (e) \left (b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))\right )}{a \sqrt {1+\cot ^2(e)} \left (1+\frac {b \csc (e)}{a \sqrt {1+\cot ^2(e)}}\right )},\frac {\csc (e) \left (b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))\right )}{a \sqrt {1+\cot ^2(e)} \left (-1+\frac {b \csc (e)}{a \sqrt {1+\cot ^2(e)}}\right )}\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\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 (f x-\arctan (\cot (e))) \sqrt {\frac {a \sqrt {1+\cot ^2(e)}-a \sqrt {1+\cot ^2(e)} \sin (f x-\arctan (\cot (e)))}{a \sqrt {1+\cot ^2(e)}-b \csc (e)}} \sqrt {\frac {a \sqrt {1+\cot ^2(e)}+a \sqrt {1+\cot ^2(e)} \sin (f x-\arctan (\cot (e)))}{a \sqrt {1+\cot ^2(e)}+b \csc (e)}} \sqrt {b-a \sqrt {1+\cot ^2(e)} \sin (e) \sin (f x-\arctan (\cot (e)))}}{(-a+b) f \sqrt {1+\cot ^2(e)} \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))}+\frac {a \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\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} \left (\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {\sec (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a \sqrt {1+\tan ^2(e)} \left (1-\frac {b \sec (e)}{a \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a \sqrt {1+\tan ^2(e)} \left (-1-\frac {b \sec (e)}{a \sqrt {1+\tan ^2(e)}}\right )}\right ) \sin (f x+\arctan (\tan (e))) \tan (e)}{\sqrt {1+\tan ^2(e)} \sqrt {\frac {a \sqrt {1+\tan ^2(e)}-a \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{b \sec (e)+a \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {a \sqrt {1+\tan ^2(e)}+a \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{-b \sec (e)+a \sqrt {1+\tan ^2(e)}}} \sqrt {b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}-\frac {\frac {\sin (f x+\arctan (\tan (e))) \tan (e)}{\sqrt {1+\tan ^2(e)}}+\frac {2 a \cos (e) \left (b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{a^2 \cos ^2(e)+a^2 \sin ^2(e)}}{\sqrt {b+a \cos (e) \cos (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}\right )}{2 (-a+b) f \sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \]

[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]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.70 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {i g \cos \left (f x +e \right ) \sqrt {g \sec \left (f x +e \right )}\, \sqrt {a +b \sec \left (f x +e \right )}\, \left (2 \operatorname {EllipticF}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right ) a -a \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}}{c f \left (a -b \right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\)	\(207\)

[In]

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

[Out]

-I*g/c/f/(a-b)*cos(f*x+e)*(g*sec(f*x+e))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(2*EllipticF(I*(-cot(f*x+e)+csc(f*x+e)),
(-(a-b)/(a+b))^(1/2))*a-a*EllipticE(I*(-cot(f*x+e)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2))-b*EllipticE(I*(-cot(f*x+e
)+csc(f*x+e)),(-(a-b)/(a+b))^(1/2)))*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)/(b+a*cos(f*x+e))/(1/(cos(
f*x+e)+1))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.26 \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=-\frac {6 \, a g \sqrt {\frac {a \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, {\left (3 \, a - 2 \, b\right )} g \cos \left (f x + e\right ) + i \, {\left (3 \, a - 2 \, b\right )} g\right )} \sqrt {a g} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) + 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (-i \, {\left (3 \, a - 2 \, b\right )} g \cos \left (f x + e\right ) - i \, {\left (3 \, a - 2 \, b\right )} g\right )} \sqrt {a g} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) - 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (i \, a g \cos \left (f x + e\right ) + i \, a g\right )} \sqrt {a g} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) + 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (-i \, a g \cos \left (f x + e\right ) - i \, a g\right )} \sqrt {a g} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (f x + e\right ) - 3 i \, a \sin \left (f x + e\right ) + 2 \, b}{3 \, a}\right )\right )}{6 \, {\left ({\left (a^{2} - a b\right )} c f \cos \left (f x + e\right ) + {\left (a^{2} - a b\right )} c f\right )}} \]

[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]

-1/6*(6*a*g*sqrt((a*cos(f*x + e) + b)/cos(f*x + e))*sqrt(g/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + sqrt(2)*(
I*(3*a - 2*b)*g*cos(f*x + e) + I*(3*a - 2*b)*g)*sqrt(a*g)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(
9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) + 3*I*a*sin(f*x + e) + 2*b)/a) + sqrt(2)*(-I*(3*a - 2*b)*g*cos(f*x
+ e) - I*(3*a - 2*b)*g)*sqrt(a*g)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1
/3*(3*a*cos(f*x + e) - 3*I*a*sin(f*x + e) + 2*b)/a) - 3*sqrt(2)*(I*a*g*cos(f*x + e) + I*a*g)*sqrt(a*g)*weierst
rassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8
/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) + 3*I*a*sin(f*x + e) + 2*b)/a)) - 3*sqrt(2)*(-I*a*g*cos(f*x +
e) - I*a*g)*sqrt(a*g)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInver
se(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(f*x + e) - 3*I*a*sin(f*x + e) + 2*b)/a))
)/((a^2 - a*b)*c*f*cos(f*x + e) + (a^2 - a*b)*c*f)

Sympy [F]

\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]

[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

Maxima [F]

\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]

[In]

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

[Out]

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