3.47 \(\int \frac{\sqrt{g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 c}{c+d}\right )}{f (a+b) \sqrt{c+d \sin (e+f x)}} \]

[Out]

(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e +
f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 0.205679, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2937} \[ \frac{2 \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 c}{c+d}\right )}{f (a+b) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

(2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e +
f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2937

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

Rubi steps

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

Mathematica [B]  time = 28.8557, size = 3429, normalized size = 30.08 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[g*Sin[e + f*x]]/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

-((c*Sqrt[-c^2 + d^2]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^
2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f
*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^
2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2
+ d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d
^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e
+ f*x]))/(c^2 - d^2)])/(Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*f*(a + b*Sin[e + f*x])*(c + d*Si
n[e + f*x])*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]*(-(c^2*Sqrt[-c^2 + d^2]*((-(a*c) + (b + Sqrt[
-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + S
qrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt
[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a
*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2]
+ c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sec[(e + f*x)
/2]^2*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(4*Sqrt[-a^2 + b^2]*(b
*c - a*d)*(d + Sqrt[-c^2 + d^2])^3*Sqrt[c + d*Sin[e + f*x]]*(-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2])))^(
3/2)) + (c*d*Sqrt[-c^2 + d^2]*Cos[e + f*x]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticP
i[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2
+ d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c +
(-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*
c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[
2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e
+ f*x]))/(c^2 - d^2)])/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*(c + d*Sin[e + f*x])^(3/2)*Sqr
t[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d^2]*Cos[e + f*x]*((-(a*c) + (b + Sqrt[-a^
2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + Sqrt
[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c
^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sq
rt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] +
c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])])*Sqrt[(c*Sec[(e +
f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)])/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt[S
in[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*Sqrt[-((c*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d
^2]*((-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) - Sqrt
[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2
+ d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^2 + d^2])] + (a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2
+ d^2]))*EllipticPi[(2*a*Sqrt[-c^2 + d^2])/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sqrt[-c^2 + d^2])), ArcSin[S
qrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]/Sqrt[2]], (2*Sqrt[-c^2 + d^2])/(d + Sqrt[-c^
2 + d^2])])*Sqrt[Sin[e + f*x]]*((c*d*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(c^2 - d^2) + (c*Sec[(e + f*x)/2]^2*(c +
d*Sin[e + f*x])*Tan[(e + f*x)/2])/(c^2 - d^2)))/(2*Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt
[c + d*Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e + f*x]))/(c^2 - d^2)]*Sqrt[-((c*Tan[(e + f*x)/2])
/(d + Sqrt[-c^2 + d^2]))]) - (c*Sqrt[-c^2 + d^2]*Sqrt[Sin[e + f*x]]*Sqrt[(c*Sec[(e + f*x)/2]^2*(c + d*Sin[e +
f*x]))/(c^2 - d^2)]*((c*(-(a*c) + (b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2
]*Sqrt[-c^2 + d^2]*Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]*Sqrt[1 - (d + Sqrt[-c^2
+ d^2] + c*Tan[(e + f*x)/2])/(2*Sqrt[-c^2 + d^2])]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(d + S
qrt[-c^2 + d^2])]*(1 - (a*(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2]))/(-(b*c) - Sqrt[-a^2 + b^2]*c + a*(d + S
qrt[-c^2 + d^2])))) + (c*(a*c + (-b + Sqrt[-a^2 + b^2])*(d + Sqrt[-c^2 + d^2]))*Sec[(e + f*x)/2]^2)/(4*Sqrt[2]
*Sqrt[-c^2 + d^2]*Sqrt[(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/Sqrt[-c^2 + d^2]]*Sqrt[1 - (d + Sqrt[-c^2 +
d^2] + c*Tan[(e + f*x)/2])/(2*Sqrt[-c^2 + d^2])]*Sqrt[1 - (d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2])/(d + Sq
rt[-c^2 + d^2])]*(1 - (a*(d + Sqrt[-c^2 + d^2] + c*Tan[(e + f*x)/2]))/(-(b*c) + Sqrt[-a^2 + b^2]*c + a*(d + Sq
rt[-c^2 + d^2]))))))/(Sqrt[-a^2 + b^2]*(b*c - a*d)*(d + Sqrt[-c^2 + d^2])^2*Sqrt[c + d*Sin[e + f*x]]*Sqrt[-((c
*Tan[(e + f*x)/2])/(d + Sqrt[-c^2 + d^2]))]))))

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Maple [B]  time = 0.426, size = 2932, normalized size = 25.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*a/(-c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b)/(-a^2+b
^2)^(1/2)*(2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(
f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^2+d^2
)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)+2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*
x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)
^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*d*(-c^2+d^2)^(1/
2)*(-a^2+b^2)^(1/2)+EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2
))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-
c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*a*c*(-c^2+d^2)^(1/2)-2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*s
in(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a*
(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*d*(-c^2+d^2)^(1/2)-Elli
pticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d
+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d
^2)^(1/2))^(1/2))*a*c*(-c^2+d^2)^(1/2)+2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-
c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c
*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*d*(-c^2+d^2)^(1/2)-EllipticPi((-(-(-c^2+d^2)^
(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c
*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*c^2*(
-a^2+b^2)^(1/2)+2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))
/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^
2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*d^2*(-a^2+b^2)^(1/2)-EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f
*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c
^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*c^2*(-a^2+b^2)^(1/2)+2*Ellip
ticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+
(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^
2)^(1/2))^(1/2))*d^2*(-a^2+b^2)^(1/2)+EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/
(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),
1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*a*c*d+EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*
sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)+a
*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*c^2-2*EllipticPi((-(-(
-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),(d+(-c^2+d^2)^(
1/2))*a/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)+d*a-c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1
/2))*b*d^2-EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*
x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)
^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*a*c*d-EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)
/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b
),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*c^2+2*EllipticPi((-(-(-c^2+d^2)^(1/2)*sin(f*x+e
)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2),-(d+(-c^2+d^2)^(1/2))*a/(c*(-a^2+b^2)^(1
/2)-a*(-c^2+d^2)^(1/2)-d*a+c*b),1/2*2^(1/2)*((d+(-c^2+d^2)^(1/2))/(-c^2+d^2)^(1/2))^(1/2))*b*d^2)*(g*sin(f*x+e
))^(1/2)*sin(f*x+e)/(c+d*sin(f*x+e))^(1/2)*2^(1/2)*((-1+cos(f*x+e))*c/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2)*(
((-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(-c^2+d^2)^(1/2)/sin(f*x+e))^(1/2)*(-(-(-c^2+d^2)^(1
/2)*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-c)/(d+(-c^2+d^2)^(1/2))/sin(f*x+e))^(1/2)/(-1+cos(f*x+e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(g*sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(g*sin(f*x + e))/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)), x)