\(\int \frac {\sqrt {c+d \sin (e+f x)}}{3+b \sin (e+f x)} \, dx\) [747]

Optimal result

Integrand size = 27, antiderivative size = 152 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+b \sin (e+f x)} \, dx=\frac {2 d \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b f \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-3 d) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b (3+b) f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2*d*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)
*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/b/f/(c+d*sin(f*x+e))^(1/2)-2*(-a*d+b*c)*(sin(1/2*e+1/4*Pi+1/2
*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2
))*((c+d*sin(f*x+e))/(c+d))^(1/2)/b/(a+b)/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2882, 2742, 2740, 2886, 2884} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+b \sin (e+f x)} \, dx=\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}+\frac {2 d \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2882

Int[Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[d/b
, Int[1/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e +
f*x]]), x], 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]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

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

Mathematica [A] (verified)

Time = 5.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+b \sin (e+f x)} \, dx=-\frac {2 \left ((3+b) d \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+(b c-3 d) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b (3+b) f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(-2*((3 + b)*d*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (b*c - 3*d)*EllipticPi[(2*b)/(3 + b), (-2*e +
Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(3 + b)*f*Sqrt[c + d*Sin[e + f*x]])

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {2 \left (F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )-\Pi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, -\frac {\left (c -d \right ) b}{d a -c b}, \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \left (c -d \right )}{b \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(181\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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