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3.1.47 \(\int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx\) [47]

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

[Out]

2*EllipticPi(1/2*(1-csc(f*x+e))^(1/2)*2^(1/2),2*a/(a+b),2^(1/2)*(c/(c+d))^(1/2))*(-cot(f*x+e)^2)^(1/2)*((d+c*c
sc(f*x+e))/(c+d))^(1/2)*(g*sin(f*x+e))^(1/2)*tan(f*x+e)/(a+b)/f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 114, normalized size of antiderivative = 1.00, 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 = {3016} \begin {gather*} \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};\text {ArcSin}\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)}} \end {gather*}

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 3016

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]]/(f*(c + d)*Cot[e + f*x]*Sqrt[a
+ b*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))], 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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3429\) vs. \(2(114)=228\).
time = 47.48, size = 3429, normalized size = 30.08 \begin {gather*} \text {Result too large to show} \end {gather*}

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]))]*(-1/4*(c^2*Sqrt[-c^2 + d^2]*((-(a*c) + (b + S
qrt[-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])])*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)])/(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]))*Ellipti
cPi[(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]]/Sqr
t[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)*S
qrt[-((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 + Sq
rt[-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[(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
[Sin[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) - Sq
rt[-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]]*((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*Sq
rt[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 +
 Sqrt[-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 +
 Sqrt[-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^...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2924\) vs. \(2(107)=214\).
time = 0.39, size = 2925, normalized size = 25.66

method result size
default \(\text {Expression too large to display}\) \(2925\)

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,method=_RETURNVERBOSE)

[Out]

-1/f*(EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^
(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)
/(-c^2+d^2)^(1/2))^(1/2))*a*c*d+EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+
d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(
1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*a*c*(-c^2+d^2)^(1/2)+EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)
^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^
(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*b*c^2-2*EllipticP
i((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^
2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/
2))^(1/2))*b*d^2-2*EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)
/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+
d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*b*d*(-c^2+d^2)^(1/2)-EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*
x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2
+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*c^2*(-a^2+b^2)^(1/2)+2*Ellipti
cPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+
d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(
1/2))^(1/2))*d^2*(-a^2+b^2)^(1/2)+2*EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-
c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2
*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2)-EllipticPi((-(cos(
f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+
d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2)
)*a*c*d-EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e)
)^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+
d)/(-c^2+d^2)^(1/2))^(1/2))*a*c*(-c^2+d^2)^(1/2)-EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(
f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)
+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*b*c^2+2*EllipticPi((-(cos(f*x+e)*c-(-c^2+
d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d
^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*b*d^2+2*Ellip
ticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^
2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)
^(1/2))^(1/2))*b*d*(-c^2+d^2)^(1/2)-EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-
c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2
*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*c^2*(-a^2+b^2)^(1/2)+2*EllipticPi((-(cos(f*x+e)*c-(-c^
2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2
+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*d^2*(-a^2+b
^2)^(1/2)+2*EllipticPi((-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*
x+e))^(1/2),((-c^2+d^2)^(1/2)+d)*a/(a*(-c^2+d^2)^(1/2)-c*(-a^2+b^2)^(1/2)+a*d-b*c),1/2*2^(1/2)*(((-c^2+d^2)^(1
/2)+d)/(-c^2+d^2)^(1/2))^(1/2))*d*(-c^2+d^2)^(1/2)*(-a^2+b^2)^(1/2))*2^(1/2)*(g*sin(f*x+e))^(1/2)*sin(f*x+e)/(
c+d*sin(f*x+e))^(1/2)*(c*(-1+cos(f*x+e))/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2)*((cos(f*x+e)*c+(-c^2+d^2)^(1/2
)*sin(f*x+e)-d*sin(f*x+e)-c)/(-c^2+d^2)^(1/2)/sin(f*x+e))^(1/2)*(-(cos(f*x+e)*c-(-c^2+d^2)^(1/2)*sin(f*x+e)-d*
sin(f*x+e)-c)/((-c^2+d^2)^(1/2)+d)/sin(f*x+e))^(1/2)/(-1+cos(f*x+e))*a/(-a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)
-a*d+b*c)/(a*(-c^2+d^2)^(1/2)+c*(-a^2+b^2)^(1/2)+a*d-b*c)/(-a^2+b^2)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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