\(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [44]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.15 (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} \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx=\frac {2 \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {a \csc (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{f (c+d) \sqrt {a+b \sin (e+f x)}} \]

[In]

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

[Out]

(2*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 + d)*f*Sqrt[a + b*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 {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+d) f \sqrt {a+b \sin (e+f x)}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(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. \(2582\) vs. \(2(107)=214\).

Time = 2.80 (sec) , antiderivative size = 2583, normalized size of antiderivative = 22.66

method	result	size
default	\(\text {Expression too large to display}\)	\(2583\)

[In]

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

[Out]

-1/f*(EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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*d-(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))
^(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+EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+
b))^(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*c+(-a^2+b^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*
x+e)+(-a^2+b^2)^(1/2)+b))^(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*c-2*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*c
ot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b^2*d+2*(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^
(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b^2-2*(-a^2+b^2)^(1/2)*Ellipt
icPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b*d+2*(
-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2
)+b))^(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))*b-EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/
2)+b))^(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*d-(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot
(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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-EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*
cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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*c-(-a^2+b^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(
1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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*c+2*EllipticPi((1/(b+(-a^2+b
^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b^2*d+2*(-c^2+d^2)^(1/2)*
EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b
^2+2*(-a^2+b^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b*d+2*(-a^2+b^2)^(1/2)*(-c^2+d^2)^(1/2)*EllipticPi((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-
a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(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))*b)*(g*sin(f*x+e))^(1/2)*2^(1/2)*((-csc(f*x+e)+cot(
f*x+e))*a/(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)*
(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)/(a+b*sin(f*x+e))^(1/2)*(cot(f*x+
e)+csc(f*x+e))*c/(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(-1)]

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

[In]

integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x)

Maxima [F]

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

[In]

integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(g*sin(f*x + e))/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), x)

Giac [F]

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

[In]

integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+b*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(g*sin(f*x + e))/(sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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