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

Optimal result

Integrand size = 39, antiderivative size = 114 \[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{(a+b) f \sqrt {c+d \sin (e+f x)}} \]

[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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {3016} \[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {c \csc (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}} \]

[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*} \text {integral}& = \frac {2 \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \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 [F]

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

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2589\) vs. \(2(107)=214\).

Time = 2.92 (sec) , antiderivative size = 2590, normalized size of antiderivative = 22.72

method	result	size
default	\(\text {Expression too large to display}\)	\(2590\)

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \]

[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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

[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)