3.27 \(\int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx\)
Optimal. Leaf size=166 \[ \frac{\sqrt{2} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f (c-d)}-\frac{2 \sqrt{c} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f (c-d) \sqrt{c+d}} \]
[Out]
(Sqrt[2]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])
])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*
Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)
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Rubi [A] time = 0.515423, antiderivative size = 166, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used =
{2936, 2782, 205, 2930} \[ \frac{\sqrt{2} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f (c-d)}-\frac{2 \sqrt{c} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f (c-d) \sqrt{c+d}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(Sqrt[2]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])
])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*
Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)
Rule 2936
Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> -Dist[(a*g)/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x]
, x] + Dist[(c*g)/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x
] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])
Rule 2782
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c
+ d*Sin[e + f*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]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 2930
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> Dist[(-2*b)/f, Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[g*
Sin[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] && EqQ[a^
2 - b^2, 0]
Rubi steps
\begin{align*} \int \frac{\sqrt{g \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=-\frac{g \int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}} \, dx}{c-d}+\frac{(c g) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{a (c-d)}\\ &=\frac{(2 a g) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a g x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{(c-d) f}-\frac{(2 c g) \operatorname{Subst}\left (\int \frac{1}{a c+a d+c g x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{(c-d) f}\\ &=\frac{\sqrt{2} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{\sqrt{a} (c-d) f}-\frac{2 \sqrt{c} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{\sqrt{a} (c-d) \sqrt{c+d} f}\\ \end{align*}
Mathematica [C] time = 39.4428, size = 61316, normalized size = 369.37 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
Result too large to show
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Maple [B] time = 0.329, size = 614, normalized size = 3.7 \begin{align*} \text{result 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)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x)
[Out]
-1/f/(c-d)/(-(c-d)*(c+d))^(1/2)/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)/((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*(g*sin(
f*x+e))^(1/2)*(1-cos(f*x+e)+sin(f*x+e))*(2*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2))*(-(c-d)*(c+d))^(1/2)*((
(-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)+(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/
2)+d)*c)^(1/2)*arctanh((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*c/((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2))*c+(((-(c-d)*
(c+d))^(1/2)+d)*c)^(1/2)*arctanh((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*c/((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2))*c^
2-(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*arctanh((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*c/((-d+(-(c-d)*(c+d))^(1/2))*
c)^(1/2))*c*d-(-(c-d)*(c+d))^(1/2)*((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1
/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c+((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*arctan((-(-1+cos(f*x+e))/sin(f
*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c^2-((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*arctan((-(-1+cos(f*
x+e))/sin(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c*d)/(a*(1+sin(f*x+e)))^(1/2)/sin(f*x+e)/(-(-1+c
os(f*x+e))/sin(f*x+e))^(1/2)
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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 )}}{\sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(g*sin(f*x + e))/(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), x)
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Fricas [A] time = 14.2722, size = 7360, normalized size = 44.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[-1/4*(sqrt(2)*sqrt(-g/a)*log((17*g*cos(f*x + e)^3 - 4*sqrt(2)*(3*cos(f*x + e)^2 + (3*cos(f*x + e) + 4)*sin(f*
x + e) - cos(f*x + e) - 4)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-g/a) + 3*g*cos(f*x + e)^2 - 18*
g*cos(f*x + e) + (17*g*cos(f*x + e)^2 + 14*g*cos(f*x + e) - 4*g)*sin(f*x + e) - 4*g)/(cos(f*x + e)^3 + 3*cos(f
*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + sqrt(-c*g/(a*c + a*d))
*log(((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^5 - (128*c^4 + 192*c^3*d + 64*c^2*d^
2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^4 - 2*(208*c^4 + 368*c^3*d + 195*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^3
+ 2*(64*c^4 + 94*c^3*d + 29*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^2 + (289*c^4 + 480*c^3*d + 230*c^2*d^2 + 3
2*c*d^3 + d^4)*g*cos(f*x + e) + 8*((16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 +
110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4
+ 149*c^3*d + 110*c^2*d^2 + 29*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos
(f*x + e) - (51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d
^4)*cos(f*x + e)^3 - (40*c^4 + 92*c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 4
1*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-c*
g/(a*c + a*d)) + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*g + ((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^
3 + d^4)*g*cos(f*x + e)^4 + 4*(64*c^4 + 112*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e)^3 - 2*(80*c^4 + 144*c
^3*d + 83*c^2*d^2 + 18*c*d^3 + d^4)*g*cos(f*x + e)^2 - 4*(72*c^4 + 119*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x
+ e) + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*g)*sin(f*x + e))/(d^4*cos(f*x + e)^5 + (4*c*d^3 + d^4)*cos
(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^3 - 2*(2*c^3*d + 3*
c^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4)*cos(f*x + e) + (d^4*cos(f*x + e)^4 - 4*c*d^3
*cos(f*x + e)^3 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + 2*c*d^3 + d^4)*cos(f*x + e)^2 + 4
*(c^3*d + c*d^3)*cos(f*x + e))*sin(f*x + e))))/((c - d)*f), -1/4*(sqrt(2)*sqrt(-g/a)*log((17*g*cos(f*x + e)^3
- 4*sqrt(2)*(3*cos(f*x + e)^2 + (3*cos(f*x + e) + 4)*sin(f*x + e) - cos(f*x + e) - 4)*sqrt(a*sin(f*x + e) + a)
*sqrt(g*sin(f*x + e))*sqrt(-g/a) + 3*g*cos(f*x + e)^2 - 18*g*cos(f*x + e) + (17*g*cos(f*x + e)^2 + 14*g*cos(f*
x + e) - 4*g)*sin(f*x + e) - 4*g)/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*s
in(f*x + e) - 2*cos(f*x + e) - 4)) - 2*sqrt(c*g/(a*c + a*d))*arctan(1/4*((8*c^2 + 8*c*d + d^2)*cos(f*x + e)^2
- 9*c^2 - 8*c*d - d^2 + 2*(4*c^2 + 3*c*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(c*g
/(a*c + a*d))/((2*c^2 + c*d)*g*cos(f*x + e)^3 + c^2*g*cos(f*x + e)*sin(f*x + e) - (2*c^2 + c*d)*g*cos(f*x + e)
)))/((c - d)*f), -1/4*(2*sqrt(2)*sqrt(g/a)*arctan(1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sq
rt(g/a)*(3*sin(f*x + e) - 1)/(g*cos(f*x + e)*sin(f*x + e))) + sqrt(-c*g/(a*c + a*d))*log(((128*c^4 + 256*c^3*d
+ 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^5 - (128*c^4 + 192*c^3*d + 64*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f
*x + e)^4 - 2*(208*c^4 + 368*c^3*d + 195*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^3 + 2*(64*c^4 + 94*c^3*d + 2
9*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^2 + (289*c^4 + 480*c^3*d + 230*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x +
e) + 8*((16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 + 110*c^3*d + 76*c^2*d^2 +
18*c*d^3 + d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4 + 149*c^3*d + 110*c^2*d^2
+ 29*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e) - (51*c^4 + 110*
c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^3 - (40*c
^4 + 92*c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 41*c^2*d^2 + 11*c*d^3 + d^4
)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-c*g/(a*c + a*d)) + (c^4 + 4*
c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*g + ((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^4
+ 4*(64*c^4 + 112*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e)^3 - 2*(80*c^4 + 144*c^3*d + 83*c^2*d^2 + 18*c*d
^3 + d^4)*g*cos(f*x + e)^2 - 4*(72*c^4 + 119*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e) + (c^4 + 4*c^3*d + 6
*c^2*d^2 + 4*c*d^3 + d^4)*g)*sin(f*x + e))/(d^4*cos(f*x + e)^5 + (4*c*d^3 + d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*
d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^3 - 2*(2*c^3*d + 3*c^2*d^2 + 4*c*d^3 + d^4)*c
os(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4)*cos(f*x + e) + (d^4*cos(f*x + e)^4 - 4*c*d^3*cos(f*x + e)^3 + c^4 + 4*
c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + 2*c*d^3 + d^4)*cos(f*x + e)^2 + 4*(c^3*d + c*d^3)*cos(f*x +
e))*sin(f*x + e))))/((c - d)*f), -1/2*(sqrt(2)*sqrt(g/a)*arctan(1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(g*s
in(f*x + e))*sqrt(g/a)*(3*sin(f*x + e) - 1)/(g*cos(f*x + e)*sin(f*x + e))) - sqrt(c*g/(a*c + a*d))*arctan(1/4*
((8*c^2 + 8*c*d + d^2)*cos(f*x + e)^2 - 9*c^2 - 8*c*d - d^2 + 2*(4*c^2 + 3*c*d)*sin(f*x + e))*sqrt(a*sin(f*x +
e) + a)*sqrt(g*sin(f*x + e))*sqrt(c*g/(a*c + a*d))/((2*c^2 + c*d)*g*cos(f*x + e)^3 + c^2*g*cos(f*x + e)*sin(f
*x + e) - (2*c^2 + c*d)*g*cos(f*x + e))))/((c - d)*f)]
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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 )}}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (c + d \sin{\left (e + f x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))**(1/2),x)
[Out]
Integral(sqrt(g*sin(e + f*x))/(sqrt(a*(sin(e + f*x) + 1))*(c + d*sin(e + f*x))), x)
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError