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 \[ -\frac {g \sin (e+f x) \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b)}{f (a-b) (c \cos (e+f x)+c) \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} F\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f (a-b) \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \sqrt {a+b \sec (e+f x)}} \]

[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*e+1/2*f*x)^2)^(1/2)/cos(1/2*e+1/2*f*x)*EllipticE(sin(1/2*e+1/2*f*x),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*e+1/2*f*x)^2)
^(1/2)/cos(1/2*e+1/2*f*x)*EllipticF(sin(1/2*e+1/2*f*x),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)

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Rubi [A]  time = 0.44, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, 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 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]

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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 8.46, 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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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {g \sec \left (f x + e\right )} g \sec \left (f x + e\right )}{b c \sec \left (f x + e\right )^{2} + {\left (a + b\right )} c \sec \left (f x + e\right ) + a c}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \]

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)

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maple [C]  time = 2.31, size = 222, normalized size = 0.97 \[ \frac {i \left (2 a \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )-a \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )-b \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}}{c f \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (a -b \right )} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]

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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