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

Optimal result

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

-2*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*InverseJacobiAM(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)
 

Mathematica [A] (verified)

Time = 2.88 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 \left ((a+b) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{b f \sqrt {a+b \sin (e+f x)}} \] Input:

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

Output:

(-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]])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}}dx\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {a+b \sin (e+f x)}dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {a \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {a \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {a \int \frac {1}{\sqrt {a+b \sin (e+f x)}}dx}{b}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}}dx}{b \sqrt {a+b \sin (e+f x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b f \sqrt {a+b \sin (e+f x)}}\)

Input:

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

Output:

(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*Sin 
[e + f*x]])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3231

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

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.53

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

Input:

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

Output:

-2*(a-b)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(-b*(-1+sin(f*x+e))/(a+b))^(1/2)*( 
-b*(1+sin(f*x+e))/(a-b))^(1/2)*(EllipticE(((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 complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.67 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {2 \, {\left (2 \, a \sqrt {\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, a \sqrt {-\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right ) + 3 i \, \sqrt {\frac {1}{2} i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 i \, \sqrt {-\frac {1}{2} i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )\right )\right )}}{3 \, b^{2} f} \] Input:

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

Output:

-2/3*(2*a*sqrt(1/2*I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/2 
7*(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*a*sqrt(-1/2*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*sqrt(1/2*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)) - 3*I*sqrt(-1/2*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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 16.79 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {\left (2\,a\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,b}{a+b}\right )-2\,\left (a+b\right )\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,b}{a+b}\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}}{b\,f\,\cos \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \] Input:

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

Output:

((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))
 

Reduce [F]

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

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

Output:

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