\(\int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx\) [42]

Optimal result

Integrand size = 39, antiderivative size = 254 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\frac {2 \sqrt {a+b} \sqrt {g} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{d f}-\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {b+a \csc (e+f x)}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{d (c+d) f \sqrt {a+b \sin (e+f x)}} \]

[Out]

2*EllipticPi(g^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(g*sin(f*x+e))^(1/2),(a+b)/b,((-a-b)/(a-b))^(1/2))*(a+
b)^(1/2)*g^(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)/d/f-2*(-a*d+b*c)*Ell
ipticPi(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)/d/(c+d)/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 254, 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 = {3008, 2888, 3016} \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\frac {2 \sqrt {g} \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{d f}-\frac {2 (b c-a d) \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {a \csc (e+f x)+b}{a+b}} \operatorname {EllipticPi}\left (\frac {2 c}{c+d},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{d f (c+d) \sqrt {a+b \sin (e+f x)}} \]

[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[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticPi[(a
+ b)/b, ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Ta
n[e + f*x])/(d*f) - (2*(b*c - a*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])/(d*(c + d)*f
*Sqrt[a + b*Sin[e + f*x]])

Rule 2888

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*b*(Tan
[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*El
lipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)],
x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]

Rule 3008

Int[(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]])/((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)]), x_Symbol] :> Dist[b/d, Int[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[(b*c -
a*d)/d, 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]

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 {b \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}} \, dx}{d}-\frac {(b c-a d) \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{d} \\ & = \frac {2 \sqrt {a+b} \sqrt {g} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\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)}{d f}-\frac {2 (b c-a 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)}{d (c+d) f \sqrt {a+b \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 31.90 (sec) , antiderivative size = 23019, normalized size of antiderivative = 90.63 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\text {Result too large to show} \]

[In]

Integrate[(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 [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.66 (sec) , antiderivative size = 5478, normalized size of antiderivative = 21.57

method	result	size
default	\(\text {Expression too large to display}\)	\(5478\)

[In]

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

[Out]

result too large to display

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

[Out]

Timed out

Sympy [F]

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

[In]

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

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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