\(\int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx\) [391]

Optimal result

Integrand size = 33, antiderivative size = 118 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {2 \sqrt {\csc (d+e x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \] Output:

2*csc(e*x+d)^(1/2)*InverseJacobiAM(1/2*d+1/2*e*x-1/2*arctan(a,c),2^(1/2)*( 
(a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2)))^(1/2))*((b+c*cos(e*x+d)+a*sin(e*x+d)) 
/(b+(a^2+c^2)^(1/2)))^(1/2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 0.85 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{b-a \sqrt {1+\frac {c^2}{a^2}}},\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{b+a \sqrt {1+\frac {c^2}{a^2}}}\right ) \sqrt {\csc (d+e x)} \sec \left (d+e x+\arctan \left (\frac {c}{a}\right )\right ) \sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \sqrt {-\frac {a \sqrt {1+\frac {c^2}{a^2}} \left (-1+\sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )\right )}{b+a \sqrt {1+\frac {c^2}{a^2}}}} \sqrt {\frac {a \sqrt {1+\frac {c^2}{a^2}} \left (1+\sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )\right )}{-b+a \sqrt {1+\frac {c^2}{a^2}}}} \sqrt {b+a \sqrt {1+\frac {c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}}{a \sqrt {1+\frac {c^2}{a^2}} e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \] Input:

Integrate[Sqrt[Csc[d + e*x]]/Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]],x]
 

Output:

(2*AppellF1[1/2, 1/2, 1/2, 3/2, (b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + Arc 
Tan[c/a]])/(b - a*Sqrt[1 + c^2/a^2]), (b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x 
 + ArcTan[c/a]])/(b + a*Sqrt[1 + c^2/a^2])]*Sqrt[Csc[d + e*x]]*Sec[d + e*x 
 + ArcTan[c/a]]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]*Sqrt[-((a*Sqrt[1 
 + c^2/a^2]*(-1 + Sin[d + e*x + ArcTan[c/a]]))/(b + a*Sqrt[1 + c^2/a^2]))] 
*Sqrt[(a*Sqrt[1 + c^2/a^2]*(1 + Sin[d + e*x + ArcTan[c/a]]))/(-b + a*Sqrt[ 
1 + c^2/a^2])]*Sqrt[b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]]])/( 
a*Sqrt[1 + c^2/a^2]*e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 118, 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, 3647, 3042, 3606, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3647

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3606

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\csc (d+e x)} \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a+b \csc (d+e x)+c \cot (d+e x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \sqrt {\csc (d+e x)} \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}}\)

Input:

Int[Sqrt[Csc[d + e*x]]/Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]],x]
 

Output:

(2*Sqrt[Csc[d + e*x]]*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + 
c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b 
 + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*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 3606

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq 
rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]   Int[1/Sqrt[a/(a 
 + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - 
 ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 
, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3647

Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + 
cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_), x_Symbol] :> Simp[Csc[d + e*x]^n*((b + 
 a*Sin[d + e*x] + c*Cos[d + e*x])^n/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n 
)   Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[m + n, 0] &&  !IntegerQ[n]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.68 (sec) , antiderivative size = 639, normalized size of antiderivative = 5.42

method	result	size
default	\(\frac {2 i \sqrt {\frac {\left (-i \cos \left (e x +d \right )-i+\sin \left (e x +d \right )\right )^{2} \left (b -c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}{\left (\cos \left (e x +d \right )+1\right ) \left (-b +c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}}\, \sqrt {\frac {\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )-a \sin \left (e x +d \right )+b \cos \left (e x +d \right )-c \cos \left (e x +d \right )-b +c}{\left (1+i \sin \left (e x +d \right )-\cos \left (e x +d \right )\right ) \left (-b +c -i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}}\, \sqrt {\frac {-\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )-a \sin \left (e x +d \right )+b \cos \left (e x +d \right )-c \cos \left (e x +d \right )-b +c}{\left (1+i \sin \left (e x +d \right )-\cos \left (e x +d \right )\right ) \left (-b +c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (-i \cos \left (e x +d \right )-i+\sin \left (e x +d \right )\right )^{2} \left (b -c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}{\left (\cos \left (e x +d \right )+1\right ) \left (-b +c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}}}{2}, \sqrt {\frac {\left (-b +c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right ) \left (-i \sqrt {a^{2}-b^{2}+c^{2}}+i a +b -c \right )}{\left (b -c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right ) \left (-b +c -i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}}\right ) \left (i \cos \left (e x +d \right ) \sin \left (e x +d \right )+\sin \left (e x +d \right )^{2}\right ) \sqrt {\csc \left (e x +d \right )}\, \sqrt {2 a +2 c \cot \left (e x +d \right )+2 b \csc \left (e x +d \right )}\, \left (-b +c +i \sqrt {a^{2}-b^{2}+c^{2}}+i a \right )}{e \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}\)	\(639\)

Input:

int(csc(e*x+d)^(1/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

2*I/e*((-I*cos(e*x+d)-I+sin(e*x+d))^2/(cos(e*x+d)+1)*(b-c+I*(a^2-b^2+c^2)^ 
(1/2)+I*a)/(-b+c+I*(a^2-b^2+c^2)^(1/2)+I*a))^(1/2)*(((a^2-b^2+c^2)^(1/2)*s 
in(e*x+d)-a*sin(e*x+d)+b*cos(e*x+d)-c*cos(e*x+d)-b+c)/(1+I*sin(e*x+d)-cos( 
e*x+d))/(-b+c-I*(a^2-b^2+c^2)^(1/2)+I*a))^(1/2)*((-(a^2-b^2+c^2)^(1/2)*sin 
(e*x+d)-a*sin(e*x+d)+b*cos(e*x+d)-c*cos(e*x+d)-b+c)/(1+I*sin(e*x+d)-cos(e* 
x+d))/(-b+c+I*(a^2-b^2+c^2)^(1/2)+I*a))^(1/2)*EllipticF(1/2*2^(1/2)*((-I*c 
os(e*x+d)-I+sin(e*x+d))^2/(cos(e*x+d)+1)*(b-c+I*(a^2-b^2+c^2)^(1/2)+I*a)/( 
-b+c+I*(a^2-b^2+c^2)^(1/2)+I*a))^(1/2),((-b+c+I*(a^2-b^2+c^2)^(1/2)+I*a)*( 
-I*(a^2-b^2+c^2)^(1/2)+I*a+b-c)/(b-c+I*(a^2-b^2+c^2)^(1/2)+I*a)/(-b+c-I*(a 
^2-b^2+c^2)^(1/2)+I*a))^(1/2))*(I*cos(e*x+d)*sin(e*x+d)+sin(e*x+d)^2)*csc( 
e*x+d)^(1/2)*(2*a+2*c*cot(e*x+d)+2*b*csc(e*x+d))^(1/2)*(-b+c+I*(a^2-b^2+c^ 
2)^(1/2)+I*a)/(b+c*cos(e*x+d)+a*sin(e*x+d))/(I*b-I*c-(a^2-b^2+c^2)^(1/2)-a 
)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.31 \[ \int \frac {\sqrt {\csc (d+e x)}}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}} \, dx=\frac {{\left (i \, a - c\right )} \sqrt {-2 i \, a - 2 \, c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {-2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (i \, a^{2} + i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right ) + \sqrt {2 i \, a - 2 \, c} {\left (-i \, a - c\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (-i \, a^{2} - i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right )}{{\left (a^{2} + c^{2}\right )} e} \] Input:

integrate(csc(e*x+d)^(1/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2),x, algorith 
m="fricas")
 

Output:

((I*a - c)*sqrt(-2*I*a - 2*c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 
 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 
 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a 
^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)* 
c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(-2*I*a*b + 2*b*c + 3*(a^2 + c 
^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + sqrt(2*I 
*a - 2*c)*(-I*a - c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^ 
2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 
-8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^2*b + 4*b 
^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 
3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x 
 + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)))/((a^2 + c^2)*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(csc(e*x+d)^(1/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate(csc(e*x+d)^(1/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(cot(d + e*x)*c + csc(d + e*x)*b + a)*sqrt(csc(d + e*x)))/(cot(d 
+ e*x)*c + csc(d + e*x)*b + a),x)