\(\int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\) [6]

Optimal result

Integrand size = 33, antiderivative size = 747 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{5/2} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e} \] Output:

1/2*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1 
/2)))^(1/2)*arctan(1/2*(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(a^2-2 
*a*c+b^2+c^2)^(1/2)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/(a^2+b^2 
+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/ 
(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e 
+1/2*b*arctanh(1/2*(b+2*c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d) 
^2)^(1/2))/c^(1/2)/e-1/16*b*(-4*a*c+b^2)*arctanh(1/2*(b+2*c*cot(e*x+d))/c^ 
(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/c^(5/2)/e-1/2*(a^2+b^2+c*(c-( 
a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)*arctanh 
(1/2*(b^2+(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))+b*(a^2-2*a*c+b^2+c^2)^(1/2 
)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/(a^2+b^2+c*(c-(a^2-2*a*c+b 
^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a+b*cot(e*x+d)+c* 
cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e+(a+b*cot(e*x+d)+c 
*cot(e*x+d)^2)^(1/2)/e+1/8*b*(b+2*c*cot(e*x+d))*(a+b*cot(e*x+d)+c*cot(e*x+ 
d)^2)^(1/2)/c^2/e-1/3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2)/c/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.02 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.51 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \left (-3 b \left (b^2-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \sqrt {c} \left (12 i \sqrt {a+i b-c} c^2 \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+12 i \sqrt {a-i b-c} c^2 \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\cot (d+e x) \left (3 b^2-8 a c+24 c^2-2 b c \cot (d+e x)-8 c^2 \cot ^2(d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )}{48 c^{5/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \] Input:

Integrate[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
 

Output:

(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(-3*b*(b^2 - 4*c 
*(a + 2*c))*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e 
*x] + a*Tan[d + e*x]^2])] + 2*Sqrt[c]*((12*I)*Sqrt[a + I*b - c]*c^2*ArcTan 
[(I*b - 2*c + ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b* 
Tan[d + e*x] + a*Tan[d + e*x]^2])] + (12*I)*Sqrt[a - I*b - c]*c^2*ArcTan[( 
I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Ta 
n[d + e*x] + a*Tan[d + e*x]^2])] + Cot[d + e*x]*(3*b^2 - 8*a*c + 24*c^2 - 
2*b*c*Cot[d + e*x] - 8*c^2*Cot[d + e*x]^2)*Sqrt[c + b*Tan[d + e*x] + a*Tan 
[d + e*x]^2])))/(48*c^(5/2)*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])
 

Rubi [F]

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.

\(\displaystyle \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (d+e x)^3 \sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot ^3(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {\int \left (\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}-\frac {\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}\right )d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {\int \left (\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}-\frac {\cot (d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}\right )d\cot (d+e x)}{e}\)

Input:

Int[Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 4.00 (sec) , antiderivative size = 21948621, normalized size of antiderivative = 29382.36

\[\text {output too large to display}\]

Input:

int(cot(e*x+d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5953 vs. \(2 (670) = 1340\).

Time = 2.00 (sec) , antiderivative size = 11927, normalized size of antiderivative = 15.97 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm= 
"fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot ^{3}{\left (d + e x \right )}\, dx \] Input:

integrate(cot(e*x+d)**3*(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2),x)
 

Output:

Integral(sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2)*cot(d + e*x)**3, x)
 

Maxima [F]

\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \] Input:

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm= 
"maxima")
 

Output:

integrate(sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a)*cot(e*x + d)^3, x)
 

Giac [F]

\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \] Input:

integrate(cot(e*x+d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm= 
"giac")
                                                                                    
                                                                                    
 

Output:

integrate(sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a)*cot(e*x + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a} \,d x \] Input:

int(cot(d + e*x)^3*(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2),x)
 

Output:

int(cot(d + e*x)^3*(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2), x)
 

Reduce [F]

\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int \sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \cot \left (e x +d \right )^{3}d x \] Input:

int(cot(e*x+d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
 

Output:

int(sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x)**3,x)