\(\int \frac {a+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx\) [733]

Optimal result

Integrand size = 25, antiderivative size = 140 \[ \int \frac {a+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 b E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-a d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \] Output:

-2*b*EllipticE(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))^(1/2)/d/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2*(-a*d+b*c)*InverseJaco 
biAM(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)/d/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 3.41 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \left (b (c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+(-b c+a d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

(2*b*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]] 
)/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*EllipticF[(e - 
 Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sq 
rt[c + d*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 = 2.90 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {2 \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) a d +\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b d -\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b c -\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(243\)
parts	\(\frac {2 a \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 b \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c +\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(330\)
risch	\(\text {Expression too large to display}\)	\(1429\)

Input:

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

Output:

2*(c-d)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d 
*(1+sin(f*x+e))/(c-d))^(1/2)*(EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c 
-d)/(c+d))^(1/2))*a*d+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d 
))^(1/2))*b*d-EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2) 
)*b*c-EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*b*d)/d 
^2/cos(f*x+e)/(c+d*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 = 369, normalized size of antiderivative = 2.64 \[ \int \frac {a+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (3 i \, b \sqrt {\frac {1}{2} i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 i \, b \sqrt {-\frac {1}{2} i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {\frac {1}{2} i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-\frac {1}{2} i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}}{3 \, d^{2} f} \] Input:

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

Output:

-2/3*(3*I*b*sqrt(1/2*I*d)*d*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 
7*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, 
 -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + 
e) - 2*I*c)/d)) - 3*I*b*sqrt(-1/2*I*d)*d*weierstrassZeta(-4/3*(4*c^2 - 3*d 
^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 
 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3 
*I*d*sin(f*x + e) + 2*I*c)/d)) + (2*b*c - 3*a*d)*sqrt(1/2*I*d)*weierstrass 
PInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3 
*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (2*b*c - 3*a*d)*sqrt(-1 
/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9* 
I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d))/(d^2 
*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 18.06 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {b\,\left (2\,c\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )-2\,\left (c+d\right )\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{d\,f\,\cos \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}-\frac {2\,a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |\frac {2\,d}{c+d}\right )\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{f\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \] Input:

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

Output:

(b*(2*c*ellipticF(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*d)/(c + d 
)) - 2*(c + d)*ellipticE(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*d) 
/(c + d)))*(cos(e + f*x)^2)^(1/2)*((c + d*sin(e + f*x))/(c + d))^(1/2))/(d 
*f*cos(e + f*x)*(c + d*sin(e + f*x))^(1/2)) - (2*a*ellipticF(pi/4 - e/2 - 
(f*x)/2, (2*d)/(c + d))*((c + d*sin(e + f*x))/(c + d))^(1/2))/(f*(c + d*si 
n(e + f*x))^(1/2))
 

Reduce [F]

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

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

Output:

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