\(\int \frac {\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx\) [41]

Optimal result

Integrand size = 33, antiderivative size = 146 \[ \int \frac {\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}}-\frac {2 b \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (a+b) f \sqrt {c+d \sin (e+f x)}} \] Output:

-2*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2,2^(1/2)*(d/(c+d))^(1/2))*((c+d*s 
in(f*x+e))/(c+d))^(1/2)/a/f/(c+d*sin(f*x+e))^(1/2)+2*b*EllipticPi(cos(1/2* 
e+1/4*Pi+1/2*f*x),2*b/(a+b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+ 
d))^(1/2)/a/(a+b)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.03 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.39 \[ \int \frac {\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 i \left ((-b c+a d) \operatorname {EllipticPi}\left (\frac {c+d}{c},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+b c \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-a d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}}{a c \sqrt {-\frac {1}{c+d}} (b c-a d) f} \] Input:

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

Output:

((-2*I)*((-(b*c) + a*d)*EllipticPi[(c + d)/c, I*ArcSinh[Sqrt[-(c + d)^(-1) 
]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + b*c*EllipticPi[(b*(c + d)) 
/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c 
+ d)/(c - d)])*Sec[e + f*x]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[ 
-((d*(1 + Sin[e + f*x]))/(c - d))])/(a*c*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*f 
)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 3420, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3420

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

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

rule 3420

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

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {2 \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\operatorname {EllipticPi}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {c -d}{c}, \sqrt {\frac {c -d}{c +d}}\right ) a d -\operatorname {EllipticPi}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {c -d}{c}, \sqrt {\frac {c -d}{c +d}}\right ) b c +b \operatorname {EllipticPi}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, -\frac {\left (c -d \right ) b}{a d -b c}, \sqrt {\frac {c -d}{c +d}}\right ) c \right )}{a c \left (a d -b c \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(254\)

Input:

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

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\csc (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \csc \left (f x +e \right )}{\sin \left (f x +e \right )^{2} b d +\sin \left (f x +e \right ) a d +\sin \left (f x +e \right ) b c +a c}d x \] Input:

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

Output:

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