\(\int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx\) [390]

Optimal result

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

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

Mathematica [C] (warning: unable to verify)

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

Time = 6.44 (sec) , antiderivative size = 1580, normalized size of antiderivative = 13.39 \[ \int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*c*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])/(a*e*Sqrt[Csc[d + e*x]]) + 
 (a*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*(-((a*AppellF1[-1/2, -1/2, - 
1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + 
 a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Co 
s[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c) 
)*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 
+ c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*S 
qrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcT 
an[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + 
e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 
+ a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - A 
rcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e 
*x - ArcTan[a/c]]]))/(e*Sqrt[Csc[d + e*x]]*Sqrt[b + c*Cos[d + e*x] + a*Sin 
[d + e*x]]) + (c^2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*(-((a*AppellF 
1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a 
/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 
+ a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[ 
1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqr 
t[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[ 
a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*...
 

Rubi [A] (verified)

Time = 0.54 (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, 3598, 3042, 3132}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3647

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3598

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

Input:

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

Output:

(2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*EllipticE[(d + e*x - ArcTan[c 
, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[Csc[d + e*x]] 
*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^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 3598

Int[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]]/Sqrt[(a + 
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]   Int[Sqrt[a/(a + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[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 = 3.42 (sec) , antiderivative size = 1862, normalized size of antiderivative = 15.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(1862\)
default	\(\text {Expression too large to display}\)	\(12657\)

Input:

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

Output:

(c*exp(I*(e*x+d))^2+2*b*exp(I*(e*x+d))-I*a*exp(I*(e*x+d))^2+c+I*a)/e*2^(1/ 
2)*((I*exp(I*(e*x+d))^2*c+2*I*b*exp(I*(e*x+d))+a*exp(I*(e*x+d))^2+I*c-a)/( 
exp(I*(e*x+d))^2-1))^(1/2)/(I*exp(I*(e*x+d))^2*c+2*I*b*exp(I*(e*x+d))+a*ex 
p(I*(e*x+d))^2+I*c-a)/(I*exp(I*(e*x+d))/(exp(I*(e*x+d))^2-1))^(1/2)-I/e*(2 
*I*b*(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)*((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2) 
^(1/2))/(I*a-c))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*a-c))^(1/2)*((exp(I*(e*x+d)) 
-(b+(-a^2+b^2-c^2)^(1/2))/(I*a-c))/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)-(b+ 
(-a^2+b^2-c^2)^(1/2))/(I*a-c)))^(1/2)*(-1/(-b+(-a^2+b^2-c^2)^(1/2))*(I*a-c 
)*exp(I*(e*x+d)))^(1/2)/(-exp(I*(e*x+d))^3*c-2*b*exp(I*(e*x+d))^2+I*exp(I* 
(e*x+d))^3*a-c*exp(I*(e*x+d))-I*a*exp(I*(e*x+d)))^(1/2)*EllipticF(((exp(I* 
(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*a 
-c))^(1/2),(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)/(-(-b+(-a^2+b^2-c^2)^(1/2)) 
/(I*a-c)-(b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)))^(1/2))+(I*c-a)*(2*(-c*exp(I*(e 
*x+d))^2-2*b*exp(I*(e*x+d))+I*a*exp(I*(e*x+d))^2-c-I*a)/(I*a+c)/(exp(I*(e* 
x+d))*(-c*exp(I*(e*x+d))^2-2*b*exp(I*(e*x+d))+I*a*exp(I*(e*x+d))^2-c-I*a)) 
^(1/2)+2*(1/(I*a+c)*(I*a-c)-(2*I*a-2*c)/(I*a+c))*(-b+(-a^2+b^2-c^2)^(1/2)) 
/(I*a-c)*((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c))/(-b+(-a^2+b^2 
-c^2)^(1/2))*(I*a-c))^(1/2)*((exp(I*(e*x+d))-(b+(-a^2+b^2-c^2)^(1/2))/(I*a 
-c))/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)-(b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)) 
)^(1/2)*(-1/(-b+(-a^2+b^2-c^2)^(1/2))*(I*a-c)*exp(I*(e*x+d)))^(1/2)/(-e...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1361, normalized size of antiderivative = 11.53 \[ \int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((I*a*b - b*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)) + 
(-I*a*b - b*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)) + 3*(a^2 
 + c^2)*sqrt(-2*I*a - 2*c)*weierstrassZeta(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), 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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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