\(\int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx\) [514]

Optimal result

Integrand size = 27, antiderivative size = 170 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}} \] Output:

-cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))+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)/a/f/((c+d* 
sin(f*x+e))/(c+d))^(1/2)+(c+d)*InverseJacobiAM(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)/a/f/(c+d*sin(f*x+e))^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (c+d \sin (e+f x)-(c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+(c+d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )}{a f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*Sin[(e + f*x)/2]*(c + d*Sin[e + 
f*x]) - (Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x] - (c + d 
)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x] 
)/(c + d)] + (c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[ 
(c + d*Sin[e + f*x])/(c + d)])))/(a*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e 
+ f*x]])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3248, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3248

\(\displaystyle \frac {d \int \frac {a-a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \int \frac {a-a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {d \left (\frac {a (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {a (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {d \left (\frac {a (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d \left (\frac {a (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )}{2 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}\)

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

-((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) + (d*( 
(-2*a*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*a*(c + d)*EllipticF[(e - 
Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqr 
t[c + d*Sin[e + f*x]])))/(2*a^2)
 

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]
 

rule 3248

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(381\) vs. \(2(164)=328\).

Time = 0.88 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.25

method	result	size
default	\(\frac {\sqrt {\cos \left (f x +e \right )^{2} \sin \left (f x +e \right ) d +\cos \left (f x +e \right )^{2} c}\, \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-d^{2} \cos \left (f x +e \right )^{2}+d \sin \left (f x +e \right ) c -d^{2} \sin \left (f x +e \right )-c d +d^{2}\right )}{d \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (-1+\sin \left (f x +e \right )\right ) \left (1+\sin \left (f x +e \right )\right )}\, a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(382\)

Input:

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

Output:

(cos(f*x+e)^2*sin(f*x+e)*d+cos(f*x+e)^2*c)^(1/2)*((d/(c-d)*sin(f*x+e)+1/(c 
-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c 
-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1 
/2))*c^2-(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d) 
)^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+ 
1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^2-d^2*cos(f*x+e)^2+d*sin(f*x+e)*c- 
d^2*sin(f*x+e)-c*d+d^2)/d/(-(c+d*sin(f*x+e))*(-1+sin(f*x+e))*(1+sin(f*x+e) 
))^(1/2)/a/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 = 515, normalized size of antiderivative = 3.03 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/3*(((2*c + 3*d)*cos(f*x + e) + (2*c + 3*d)*sin(f*x + e) + 2*c + 3*d)*sqr 
t(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) + ( 
(2*c + 3*d)*cos(f*x + e) + (2*c + 3*d)*sin(f*x + e) + 2*c + 3*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) - 3*(-I 
*d*cos(f*x + e) - I*d*sin(f*x + e) - I*d)*sqrt(1/2*I*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, weierstrassPInver 
se(-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*d*cos(f*x + e) + I*d*si 
n(f*x + e) + I*d)*sqrt(-1/2*I*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)) - 3*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(d*si 
n(f*x + e) + c))/(a*d*f*cos(f*x + e) + a*d*f*sin(f*x + e) + a*d*f)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \] Input:

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

Output:

Integral(sqrt(c + d*sin(e + f*x))/(sin(e + f*x) + 1), x)/a
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )+1}d x}{a} \] Input:

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

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x) + 1),x)/a