3.1.12 \(\int \frac {\cot ^5(d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [12]

3.1.12.1 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+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*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+1/2*arctanh(1/2*(b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*cot( 
e*x+d)*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))*2^(1/2)/(a+b*cot(e*x+d)+c*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))*(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^2- 
2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-1/2*arctanh(1/2*(b^2-b*cot(e*x+d)*(2*a-2*c- 
(a^2-2*a*c+b^2+c^2)^(1/2))-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))*2^(1/2)/ 
(a+b*cot(e*x+d)+c*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))*(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^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)
 

3.1.12.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.91 (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...
 

3.1.12.3 Rubi [A] (verified)

Time = 4.44 (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)
 

3.1.12.3.1 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]
 

3.1.12.4 Maple [B] (warning: unable to verify)

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

Time = 2.91 (sec) , antiderivative size = 13067692, normalized size of antiderivative = 15107.16

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

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
 

3.1.12.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42426 vs. \(2 (793) = 1586\).

Time = 29.83 (sec) , antiderivative size = 84903, normalized size of antiderivative = 98.15 \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

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
 

3.1.12.6 Sympy [F]

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

input

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

output

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

3.1.12.7 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
 

3.1.12.8 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
 

3.1.12.9 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)