\(\int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx\) [205]

Optimal result

Integrand size = 21, antiderivative size = 128 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=\frac {2 b \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \] Output:

2*b*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(b/(a+b))^(1/2))*((a+b*si 
n(f*x+e))/(a+b))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)-2*a*EllipticPi(cos(1/2*e+1 
/4*Pi+1/2*f*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/f 
/(a+b*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 17.00 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=-\frac {2 \left (b \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )+a \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right )\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 128, 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, 3282, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3282

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

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

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 3282

Int[Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/((a_.) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[d/b   Int[1/Sqrt[c + d*Sin[e + f*x]], x], x 
] + Simp[(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 3284

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 3286

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[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] && NeQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {b \left (-1+\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{\cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(169\)

Input:

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

Fricas [F(-1)]

Timed out. \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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