\(\int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx\) [463]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3247, 3198, 2732} \[ \int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx=\frac {2 \sqrt {a+b \csc (d+e x)+c \cot (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\csc (d+e x)} \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}} \]

[In]

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

[Out]

(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 + Sq
rt[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])])

Rule 2732

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 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[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
+ 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 3247

Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_)
, x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{\sqrt {\csc (d+e x)} \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \\ & = \frac {\sqrt {a+c \cot (d+e x)+b \csc (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 {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}} \\ & = \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}}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 6.68 (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=\frac {2 c \sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{a e \sqrt {\csc (d+e x)}}+\frac {a \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \left (-\frac {a \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (-1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c}\right ) \sin \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} c \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}-c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{b+c \sqrt {\frac {a^2+c^2}{c^2}}}} \sqrt {b+c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}+c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{-b+c \sqrt {\frac {a^2+c^2}{c^2}}}}}-\frac {\frac {2 c \left (b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )\right )}{a^2+c^2}-\frac {a \sin \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} c}}{\sqrt {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}}\right )}{e \sqrt {\csc (d+e x)} \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}+\frac {c^2 \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \left (-\frac {a \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (-1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c}\right ) \sin \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} c \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}-c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{b+c \sqrt {\frac {a^2+c^2}{c^2}}}} \sqrt {b+c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}+c \sqrt {\frac {a^2+c^2}{c^2}} \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{-b+c \sqrt {\frac {a^2+c^2}{c^2}}}}}-\frac {\frac {2 c \left (b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )\right )}{a^2+c^2}-\frac {a \sin \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} c}}{\sqrt {b+\sqrt {1+\frac {a^2}{c^2}} c \cos \left (d+e x-\arctan \left (\frac {a}{c}\right )\right )}}\right )}{a e \sqrt {\csc (d+e x)} \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}+\frac {2 b \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 )}{a \sqrt {1+\frac {c^2}{a^2}} \left (1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )},-\frac {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}} \left (-1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )}\right ) \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sec \left (d+e x+\arctan \left (\frac {c}{a}\right )\right ) \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}-a \sqrt {\frac {a^2+c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{b+a \sqrt {\frac {a^2+c^2}{a^2}}}} \sqrt {b+a \sqrt {\frac {a^2+c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )} \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}+a \sqrt {\frac {a^2+c^2}{a^2}} \sin \left (d+e x+\arctan \left (\frac {c}{a}\right )\right )}{-b+a \sqrt {\frac {a^2+c^2}{a^2}}}}}{a \sqrt {1+\frac {c^2}{a^2}} e \sqrt {\csc (d+e x)} \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 28.20 (sec) , antiderivative size = 1862, normalized size of antiderivative = 15.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(1862\)
default	\(\text {Expression too large to display}\)	\(12826\)

[In]

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

[Out]

(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*
exp(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)*(-exp(I*(e*x+d))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*a-c))^(1/2)/(-exp(I*(e*x+d))^3*c-2*b*exp(I*(e*x+d))^2+I*ex
p(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*(I*a*exp(I*(e*x+d))^2-c*exp(I*(e*x+d))^2-I
*a-2*b*exp(I*(e*x+d))-c)/(I*a+c)/(exp(I*(e*x+d))*(I*a*exp(I*(e*x+d))^2-c*exp(I*(e*x+d))^2-I*a-2*b*exp(I*(e*x+d
))-c))^(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)*(-exp(I*(e*x+d))/(-b
+(-a^2+b^2-c^2)^(1/2))*(I*a-c))^(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)*((-(-b+(-a^2+b^2-c^2)^(1/2))/(I*a-c)-(b+(-a^2+b^2-c^2)^(1/2))/(I*a-c))*Ellip
ticE(((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))+(b+(-a^2
+b^2-c^2)^(1/2))/(I*a-c)*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)))))*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*exp(I*(e*x+d))^2+I*c-a)/(I*exp(I*(e*x+d))/(e
xp(I*(e*x+d))^2-1))^(1/2)*(I*(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
)))^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 1361, normalized size of antiderivative = 11.53 \[ \int \frac {\sqrt {a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt {\csc (d+e x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*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*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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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