\(\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [45]
Optimal result
Integrand size = 39, antiderivative size = 246 \[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{a c f \sqrt {g}}-\frac {2 d \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt {a+b \sin (e+f x)}}
\]
[Out]
-2*EllipticF(g^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(g*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)
*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a/c/f/g^(1/2)-2*d*EllipticPi(1/2*(1-
csc(f*x+e))^(1/2)*2^(1/2),2*c/(c+d),2^(1/2)*(a/(a+b))^(1/2))*(-cot(f*x+e)^2)^(1/2)*((b+a*csc(f*x+e))/(a+b))^(1
/2)*(g*sin(f*x+e))^(1/2)*tan(f*x+e)/c/(c+d)/f/g/(a+b*sin(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.38 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3018, 2895, 3016}
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=-\frac {2 d \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {a \csc (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{c f g (c+d) \sqrt {a+b \sin (e+f x)}}-\frac {2 \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a c f \sqrt {g}}
\]
[In]
Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(-2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sq
rt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*
Sqrt[g]) - (2*d*Sqrt[-Cot[e + f*x]^2]*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)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e
+ f*x]])
Rule 2895
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
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]
Rule 3018
Int[1/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[1/c, Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x] - Dist[d
/(c*g), Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*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] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{c}-\frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{c g} \\ & = -\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{a c f \sqrt {g}}-\frac {2 d \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{c (c+d) f g \sqrt {a+b \sin (e+f x)}} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(5612\) vs. \(2(246)=492\).
Time = 31.09 (sec) , antiderivative size = 5612, normalized size of antiderivative = 22.81
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\text {Result too large to show}
\]
[In]
Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
Result too large to show
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(3342\) vs. \(2(227)=454\).
Time = 2.86 (sec) , antiderivative size = 3343, normalized size of antiderivative =
13.59
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| method | result | size |
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\(\text {Expression too large to display}\) |
\(3343\) |
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[In]
int(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/f*(2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(
1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1
/2))*a*b*d*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)-EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-
a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+
(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*c*d*(-a^2+b^2)^(1/2)+2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-
(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-
a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d^2*(-a^2+b^2)^(1/2)+2*EllipticPi((-(a
*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^
(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d*(-a^2+b^2)^
(1/2)*(-c^2+d^2)^(1/2)+EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2)
,(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^
2+b^2)^(1/2))^(1/2))*a^2*c*d*(-a^2+b^2)^(1/2)-2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b
+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*(
(b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b*d^2*(-a^2+b^2)^(1/2)-4*EllipticF((-(a*cot(f*x+e)-a*csc(f*x+e
)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a
*b*d*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)+4*EllipticF((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b
^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^2*c*(-a^2+b^2)^(1/2)*(-c^2+d^2)
^(1/2)-EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(
1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1
/2))*a^3*d*(-c^2+d^2)^(1/2)+2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2))
)^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2
))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c^2+d^2)^(1/2)-EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-
b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1
/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d^2-EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^
(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c),1/
2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b*c*d+2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(
-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a
*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d^2-EllipticPi((-(a*cot(f*x+e)-a*csc(
f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)
^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)+2*EllipticPi
((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+
b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c^
2+d^2)^(1/2)+EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+(-a^2+
b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/
2))^(1/2))*a^3*d^2+EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),(b+
(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^
2)^(1/2))^(1/2))*a^2*b*c*d-2*EllipticPi((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))
^(1/2),(b+(-a^2+b^2)^(1/2))*c/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1/2)-a*d+b*c),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2)
)/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d^2+2*EllipticF((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2
)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^3*d*(-c^2+d^2)^(1/2)-2*EllipticF(
(-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))
/(-a^2+b^2)^(1/2))^(1/2))*a^2*b*c*(-c^2+d^2)^(1/2)-4*EllipticF((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b
)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a*b^2*d*(-c^2+d^2)^(1
/2)+4*EllipticF((-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(
-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*b^3*c*(-c^2+d^2)^(1/2))*(-(a*cot(f*x+e)-a*csc(f*x+e)-(-a^2+b^2)^(1/2
)-b)/(b+(-a^2+b^2)^(1/2)))^(1/2)*((a*cot(f*x+e)-a*csc(f*x+e)+(-a^2+b^2)^(1/2)-b)/(-a^2+b^2)^(1/2))^(1/2)*((-cs
c(f*x+e)+cot(f*x+e))*a/(b+(-a^2+b^2)^(1/2)))^(1/2)/(a+b*sin(f*x+e))^(1/2)*2^(1/2)*(1+cos(f*x+e))/(g*sin(f*x+e)
)^(1/2)/a/(c*(-a^2+b^2)^(1/2)-a*(-c^2+d^2)^(1/2)-a*d+b*c)/(-c^2+d^2)^(1/2)/(c*(-a^2+b^2)^(1/2)+a*(-c^2+d^2)^(1
/2)-a*d+b*c)
Fricas [F]
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x }
\]
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
integral(-sqrt(b*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/((b*c + a*d)*g*cos(f*x + e)^2 - (b*c + a*d)*g + (b*d*g
*cos(f*x + e)^2 - (a*c + b*d)*g)*sin(f*x + e)), x)
Sympy [F]
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int \frac {1}{\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}} \left (c + d \sin {\left (e + f x \right )}\right )}\, dx
\]
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(1/2),x)
[Out]
Integral(1/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x)
Maxima [F]
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x }
\]
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)
Giac [F]
\[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \sqrt {g \sin \left (f x + e\right )}} \,d x }
\]
[In]
integrate(1/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\int \frac {1}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x
\]
[In]
int(1/((g*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))),x)
[Out]
int(1/((g*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))), x)