\(\int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx\) [207]

Optimal result

Integrand size = 21, antiderivative size = 132 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {2 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{b f \sqrt {a+b \sin (e+f x)}} \]

[Out]

-2*(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)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(
b/(a+b))^(1/2))*(a+b*sin(f*x+e))^(1/2)/b/f/((a+b*sin(f*x+e))/(a+b))^(1/2)+2*a*(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)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(
a+b))^(1/2)/b/f/(a+b*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2831, 2742, 2740, 2734, 2732} \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}} \]

[In]

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

[Out]

(2*EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(b*f*Sqrt[(a + b*Sin[e + f*x])/(a +
b)]) - (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*S
in[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[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 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 2831

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

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

Mathematica [A] (verified)

Time = 2.57 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 \left ((a+b) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{b f \sqrt {a+b \sin (e+f x)}} \]

[In]

Integrate[Sin[e + f*x]/Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(-2*((a + b)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*b)/(a + b
)])*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*Sin[e + f*x]])

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )+1\right ) b}{a -b}}\, \left (E\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a +E\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b -F\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b \right )}{b^{2} \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(202\)
risch	\(\text {Expression too large to display}\)	\(1072\)

[In]

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

[Out]

-2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-(sin(f*x+e)-1)*b/(a+b))^(1/2)*(-(sin(f*x+e)+1)*b/(a-b))^(1/2)*(Ellip
ticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a+EllipticE(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+
b))^(1/2))*b-EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b)/b^2/cos(f*x+e)/(a+b*sin(f*x+e))^
(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.77 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 \, \sqrt {2} a \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} a \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right ) + 3 i \, \sqrt {2} \sqrt {i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 i \, \sqrt {2} \sqrt {-i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )\right )}{3 \, b^{2} f} \]

[In]

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

[Out]

-1/3*(2*sqrt(2)*a*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3
*(3*b*cos(f*x + e) - 3*I*b*sin(f*x + e) - 2*I*a)/b) + 2*sqrt(2)*a*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 -
3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) + 3*I*b*sin(f*x + e) + 2*I*a)/b) + 3*I*sq
rt(2)*sqrt(I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPInver
se(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) - 3*I*b*sin(f*x + e) - 2*I
*a)/b)) - 3*I*sqrt(2)*sqrt(-I*b)*b*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3,
weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(f*x + e) + 3*I*b
*sin(f*x + e) + 2*I*a)/b)))/(b^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.43 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {\left (2\,a\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,b}{a+b}\right )-2\,\left (a+b\right )\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,b}{a+b}\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}}{b\,f\,\cos \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \]

[In]

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

[Out]

((2*a*ellipticF(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*b)/(a + b)) - 2*(a + b)*ellipticE(asin((2^(1/2)
*(1 - sin(e + f*x))^(1/2))/2), (2*b)/(a + b)))*(cos(e + f*x)^2)^(1/2)*((a + b*sin(e + f*x))/(a + b))^(1/2))/(b
*f*cos(e + f*x)*(a + b*sin(e + f*x))^(1/2))