\(\int \frac {\cot ^5(d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [11]

Optimal result

Integrand size = 33, antiderivative size = 865 \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {3 b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{5/2} e}-\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e} \] Output:

3/2*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^(5/2)/e-1/2*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2 
-2*a*c+c^2+(a-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*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))*cot(e 
*x+d))*2^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^ 
2+(a-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)^(3/2)/e+1/2*(2*a-2*c+(a^2-2*a*c+b^2+c^2) 
^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*ar 
ctanh(1/2*(b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*c+(a^2-2*a*c 
+b^2+c^2)^(1/2))*cot(e*x+d))*2^(1/2)/(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^( 
1/2)/(a^2-b^2-2*a*c+c^2-(a-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)^(3/2)/e+2*(2*a+b*c 
ot(e*x+d))/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)-2*cot(e*x+ 
d)^2*(2*a+b*cot(e*x+d))/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/ 
2)-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e 
/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)-(3*b^2-8*a*c-2*b*c*cot(e*x+d))*(a+b 
*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^2/(-4*a*c+b^2)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.48 (sec) , antiderivative size = 1674, normalized size of antiderivative = 1.94 \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Cot[d + e*x]^5/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

Output:

(-4*Cot[d + e*x]*(b + 2*a*Tan[d + e*x])*(-((a*(c + b*Tan[d + e*x] + a*Tan[ 
d + e*x]^2))/(b^2 - 4*a*c)))^(3/2))/(a*e*(c + b*Tan[d + e*x] + a*Tan[d + e 
*x]^2)*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]*Sqrt[1 
 - (b^2 - 4*a*c)*(b/(b^2 - 4*a*c) + (2*a*Tan[d + e*x])/(b^2 - 4*a*c))^2]) 
+ (Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((4*Cot[d + e* 
x]*(b^2 - 2*a*c + a*b*Tan[d + e*x]))/(c*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e 
*x] + a*Tan[d + e*x]^2]) + ((3*b*(b^2 - 4*a*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])])/c^(3/2) - (2 
*(3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) 
/c)/(c*(b^2 - 4*a*c))))/(2*e*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*T 
an[d + e*x]^2)]) + (Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^ 
2]*((-2*Tan[d + e*x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x]))/(c*(b^2 - 4*a*c) 
*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (2*(b*Tan[d + e*x]^2*Sqrt[ 
c + b*Tan[d + e*x] + a*Tan[d + e*x]^2] + (((-6*a^2*b^2*c + 24*a^3*c^2)*Arc 
Tanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + 
 e*x]^2])])/(4*a^(5/2)) + ((6*a^2*b*c - 12*a^3*c*Tan[d + e*x])*Sqrt[c + b* 
Tan[d + e*x] + a*Tan[d + e*x]^2])/(2*a^2))/(3*a)))/(c*(b^2 - 4*a*c))))/(e* 
Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]) - (Cot[d + e 
*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((2*((-4*Sqrt[a - I*b - c] 
*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan[(I*b + 2...
 

Rubi [A] (verified)

Time = 3.67 (sec) , antiderivative size = 848, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4184, 7276, 2009}

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.

\(\displaystyle \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (d+e x)^5}{\left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot ^5(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac {3 b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 c^{5/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}+\frac {\left (3 b^2-2 c \cot (d+e x) b-8 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left (b^2-4 a c\right )}-\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}}{e}\)

Input:

Int[Cot[d + e*x]^5/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

Output:

-(((-3*b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] 
 + c*Cot[d + e*x]^2])])/(2*c^(5/2)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2 
*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c 
 + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - 
b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[ 
2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + 
(a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + 
 e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) - (Sqrt[2*a - 2*c + 
Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt 
[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 
- 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e* 
x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^ 
2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d 
+ e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) - 
(2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[ 
d + e*x]^2]) + (2*Cot[d + e*x]^2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*Sq 
rt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b 
*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Cot[ 
d + e*x] + c*Cot[d + e*x]^2]) + ((3*b^2 - 8*a*c - 2*b*c*Cot[d + e*x])*Sqrt 
[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c^2*(b^2 - 4*a*c)))/e)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

Maple [B] (warning: unable to verify)

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

Time = 1.60 (sec) , antiderivative size = 13066867, normalized size of antiderivative = 15106.20

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

Input:

int(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),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. 42456 vs. \(2 (793) = 1586\).

Time = 27.61 (sec) , antiderivative size = 84933, normalized size of antiderivative = 98.19 \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(e*x+d)**5/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"giac")
                                                                                    
                                                                                    
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument 
Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^5}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \] Input:

int(cot(d + e*x)^5/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2),x)
 

Output:

int(cot(d + e*x)^5/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \cot \left (e x +d \right )^{5}}{\cot \left (e x +d \right )^{4} c^{2}+2 \cot \left (e x +d \right )^{3} b c +2 \cot \left (e x +d \right )^{2} a c +\cot \left (e x +d \right )^{2} b^{2}+2 \cot \left (e x +d \right ) a b +a^{2}}d x \] Input:

int(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

Output:

int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*cot(d + e*x)**5)/(cot(d 
+ e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e 
*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)