3.28 \(\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx\)
Optimal. Leaf size=168 \[ \frac{2 d \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} \sqrt{c} f \sqrt{g} (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f \sqrt{g} (c-d)} \]
[Out]
-((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sq
rt[a]*(c - d)*f*Sqrt[g])) + (2*d*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]*Sqrt[c]*(c - d)*Sqrt[c + d]*f*Sqrt[g])
________________________________________________________________________________________
Rubi [A] time = 0.536077, antiderivative size = 168, 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 =
{2938, 2782, 205, 2930} \[ \frac{2 d \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} \sqrt{c} f \sqrt{g} (c-d) \sqrt{c+d}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{a} f \sqrt{g} (c-d)} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
-((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sq
rt[a]*(c - d)*f*Sqrt[g])) + (2*d*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]*Sqrt[c]*(c - d)*Sqrt[c + d]*f*Sqrt[g])
Rule 2938
Int[1/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x
] - Dist[d/(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] /; Fr
eeQ[{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{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=\frac{\int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}} \, dx}{c-d}-\frac{d \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) \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 d) \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} \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 \sqrt{g}}+\frac{2 d \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} \sqrt{c} (c-d) \sqrt{c+d} f \sqrt{g}}\\ \end{align*}
Mathematica [C] time = 40.2357, size = 99997, normalized size = 595.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(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 = 0.379, size = 621, 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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(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)*(-(-1+co
s(f*x+e))/sin(f*x+e))^(1/2)*(-1+cos(f*x+e)-sin(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)+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))^(1/2)*d+(((-(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)^(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))*d^2-arctan((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*c/(((-(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)*d+((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*arct
an((-(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c*d-arctan((-(-1+cos(f*x+e))/sin(
f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*((-d+(-(c-d)*(c+d))^(1/2))*c)^(1/2)*d^2)/(g*sin(f*x+e))^(1
/2)/(a*(1+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{1}{\sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)
________________________________________________________________________________________
Fricas [B] time = 18.0833, size = 7646, normalized size = 45.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[-1/4*(sqrt(2)*(a*c^2 + a*c*d)*g*sqrt(-1/(a*g))*log((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(-1/(a*g)) + 17*cos(f*x + e)^3 +
3*cos(f*x + e)^2 + (17*cos(f*x + e)^2 + 14*cos(f*x + e) - 4)*sin(f*x + e) - 18*cos(f*x + e) - 4)/(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(-(a*c
^2 + a*c*d)*g)*d*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a
*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c
^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g
*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16*c^3 +
24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^4 - (24*c^3 + 28*c^2*d + 7*c*d^2)*cos(f*x + e)^3 + 51*c^3 + 59*c^2*d
+ 17*c*d^2 + d^3 - (66*c^3 + 83*c^2*d + 27*c*d^2 + 2*d^3)*cos(f*x + e)^2 + (25*c^3 + 28*c^2*d + 7*c*d^2)*cos(f
*x + e) + ((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^3 - 51*c^3 - 59*c^2*d - 17*c*d^2 - d^3 + (40*c^3
+ 52*c^2*d + 17*c*d^2 + d^3)*cos(f*x + e)^2 - (26*c^3 + 31*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))
*sqrt(-(a*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4
*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64
*a*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 - 2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2*d^2 +
18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) +
(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*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))))/((a*c^3 - a*c*d^2)*f*g), 1/4*(2*sqrt(2)*(a*c^2 + a*c*d)*g*sq
rt(1/(a*g))*arctan(1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(1/(a*g))*(3*sin(f*x + e) - 1
)/(cos(f*x + e)*sin(f*x + e))) + sqrt(-(a*c^2 + a*c*d)*g)*d*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32
*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x +
e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*
c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c
*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^4 - (24*c^3 + 28*c^2*d + 7
*c*d^2)*cos(f*x + e)^3 + 51*c^3 + 59*c^2*d + 17*c*d^2 + d^3 - (66*c^3 + 83*c^2*d + 27*c*d^2 + 2*d^3)*cos(f*x +
e)^2 + (25*c^3 + 28*c^2*d + 7*c*d^2)*cos(f*x + e) + ((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^3 - 51
*c^3 - 59*c^2*d - 17*c*d^2 - d^3 + (40*c^3 + 52*c^2*d + 17*c*d^2 + d^3)*cos(f*x + e)^2 - (26*c^3 + 31*c^2*d +
10*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-(a*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x +
e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 3
2*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64*a*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 -
2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2*d^2 + 18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d +
56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*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))))/((a*c^3 - a*c*d^2
)*f*g), -1/4*(sqrt(2)*(a*c^2 + a*c*d)*g*sqrt(-1/(a*g))*log((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(-1/(a*g)) + 17*cos(f*x +
e)^3 + 3*cos(f*x + e)^2 + (17*cos(f*x + e)^2 + 14*cos(f*x + e) - 4)*sin(f*x + e) - 18*cos(f*x + e) - 4)/(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)) + 2*s
qrt((a*c^2 + a*c*d)*g)*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*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/((2*a*c^3 + 3*a*c^2*
d + a*c*d^2)*g*cos(f*x + e)^3 + (a*c^3 + a*c^2*d)*g*cos(f*x + e)*sin(f*x + e) - (2*a*c^3 + 3*a*c^2*d + a*c*d^2
)*g*cos(f*x + e))))/((a*c^3 - a*c*d^2)*f*g), 1/2*(sqrt(2)*(a*c^2 + a*c*d)*g*sqrt(1/(a*g))*arctan(1/4*sqrt(2)*s
qrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(1/(a*g))*(3*sin(f*x + e) - 1)/(cos(f*x + e)*sin(f*x + e))) -
sqrt((a*c^2 + a*c*d)*g)*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*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/((2*a*c^3 + 3*a*c^
2*d + a*c*d^2)*g*cos(f*x + e)^3 + (a*c^3 + a*c^2*d)*g*cos(f*x + e)*sin(f*x + e) - (2*a*c^3 + 3*a*c^2*d + a*c*d
^2)*g*cos(f*x + e))))/((a*c^3 - a*c*d^2)*f*g)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \sqrt{g \sin{\left (e + f x \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(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**(1/2),x)
[Out]
Integral(1/(sqrt(a*(sin(e + f*x) + 1))*sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)