\(\int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx\) [785]

Optimal result

Integrand size = 29, antiderivative size = 196 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {3+b} \operatorname {EllipticPi}\left (\frac {(3+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b \sqrt {c+d} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.01, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2890} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt {c+d}} \]

[In]

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

[Out]

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

Rule 2890

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b \sqrt {c+d} f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {3+b} \operatorname {EllipticPi}\left (\frac {(3+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),-\frac {(3+b) (c-d)}{(-3+b) (c+d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (-1+\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(-3+b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b \sqrt {c+d} f} \]

[In]

Integrate[Sqrt[c + d*Sin[e + f*x]]/Sqrt[3 + b*Sin[e + f*x]],x]

[Out]

(2*Sqrt[3 + b]*EllipticPi[((3 + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[3 + b*Sin[e + f*x]])/(Sqrt[3 + b]*
Sqrt[c + d*Sin[e + f*x]])], -(((3 + b)*(c - d))/((-3 + b)*(c + d)))]*Sec[e + f*x]*Sqrt[-(((b*c - 3*d)*(-1 + Si
n[e + f*x]))/((3 + b)*(c + d*Sin[e + f*x])))]*Sqrt[((b*c - 3*d)*(1 + Sin[e + f*x]))/((-3 + b)*(c + d*Sin[e + f
*x]))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.88 (sec) , antiderivative size = 223623, normalized size of antiderivative = 1140.93

method	result	size
default	\(\text {Expression too large to display}\)	\(223623\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {b \sin \left (f x + e\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+b \sin (e+f x)}} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

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

[Out]

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