\(\int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx\) [210]

Optimal result

Integrand size = 23, antiderivative size = 222 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{a f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}-\frac {b \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{a f \sqrt {a+b \sin (e+f x)}} \] Output:

-cot(f*x+e)*(a+b*sin(f*x+e))^(1/2)/a/f+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)/a/f/((a+b*sin(f*x+e))/(a+ 
b))^(1/2)+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)/f/(a+b*sin(f*x+e))^(1/2)+b*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)/a/f/(a+b*sin(f*x+e))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.98 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\frac {\frac {2 i \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {b (-1+\sin (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sin (e+f x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-4 \cot (e+f x) \sqrt {a+b \sin (e+f x)}+\frac {6 b \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}}{4 a f} \] Input:

Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]],x]
 

Output:

(((2*I)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*S 
in[e + f*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b 
)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b) 
/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - 
b)]))*Sec[e + f*x]*Sqrt[-((b*(-1 + Sin[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + 
 Sin[e + f*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - 4*Cot[e + f*x]*Sqrt 
[a + b*Sin[e + f*x]] + (6*b*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a 
+ b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]])/(4*a*f 
)
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3281, 27, 3042, 3539, 25, 3042, 3134, 3042, 3132, 3481, 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.

\(\displaystyle \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (e+f x)^2 \sqrt {a+b \sin (e+f x)}}dx\)

\(\Big \downarrow \) 3281

\(\displaystyle \frac {\int -\frac {\csc (e+f x) \left (b \sin ^2(e+f x)+b\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\csc (e+f x) \left (b \sin ^2(e+f x)+b\right )}{\sqrt {a+b \sin (e+f x)}}dx}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {b \sin (e+f x)^2+b}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3539

\(\displaystyle -\frac {\int \sqrt {a+b \sin (e+f x)}dx-\frac {\int -\frac {\csc (e+f x) \left (b^2-a b \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{b}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {\csc (e+f x) \left (b^2-a b \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{b}+\int \sqrt {a+b \sin (e+f x)}dx}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {b^2-a b \sin (e+f x)}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{b}+\int \sqrt {a+b \sin (e+f x)}dx}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {\int \frac {b^2-a b \sin (e+f x)}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{b}+\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {b^2-a b \sin (e+f x)}{\sin (e+f x) \sqrt {a+b \sin (e+f x)}}dx}{b}+\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}\)

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

Int[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]],x]
 

Output:

-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(a*f)) - ((2*EllipticE[(e - Pi/2 
 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + 
 f*x])/(a + b)]) + ((-2*a*b*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*S 
qrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (2*b^2*E 
llipticPi[2, (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]]))/b)/(2*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3281

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 
))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n 
 + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2* 
n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 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]
 

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3539

Int[((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) 
 + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp 
[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Simp[1/(b*d)   Int[Simp[a 
*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*( 
c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b 
*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.85

method	result	size
default	\(\frac {\sqrt {-\left (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (-\frac {\sqrt {-\left (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{a \sin \left (f x +e \right )}-\frac {b \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {b \left (1-\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{a \sqrt {-\left (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {b^{2} \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {b \left (1-\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (f x +e \right )\right )}{a -b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, -\frac {\left (-\frac {a}{b}+1\right ) b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{a^{2} \sqrt {-\left (-a -b \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(410\)

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(csc(e + f*x)**2/sqrt(a + b*sin(e + f*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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