3.1.11 \(\int \frac {\cot ^7(d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [11]

3.1.11.1 Optimal result

Integrand size = 33, antiderivative size = 1189 \[ \int \frac {\cot ^7(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 {5 b \left (7 b^2-12 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^{9/2} 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 \cot ^4(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 (7 b^2-16 a c\right ) \cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c^2 \left (b^2-4 a c\right ) e}+\frac {2 b \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\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}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2-2 b c \left (35 b^2-116 a c\right ) \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^4 \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+5/16*b*(-12*a*c+7*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^(9/2)/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*c 
ot(e*x+d)^4*(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)-1/3*(-16*a*c+7*b^2)*cot(e*x+ 
d)^2*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^2/(-4*a*c+b^2)/e+2*b*cot(e*x+ 
d)^3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c/(-4*a*c+b^2)/e+(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- 
1/24*(105*b^4-460*a*b^2*c+256*a^2*c^2-2*b*c*(-116*a*c+35*b^2)*cot(e*x+d))* 
(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^4/(-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...
 

3.1.11.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.24 (sec) , antiderivative size = 2097, normalized size of antiderivative = 1.76 \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Result too large to show} \]

input

Integrate[Cot[d + e*x]^7/(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*Ta 
n[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)*ArcT 
anh[(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*T 
an[d + e*x] + a*Tan[d + e*x]^2])/(2*a^2))/(3*a)))/(c*(b^2 - 4*a*c))))/(e*S 
qrt[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*...
 

3.1.11.3 Rubi [A] (verified)

Time = 5.63 (sec) , antiderivative size = 1159, normalized size of antiderivative = 0.97, 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.

\(\displaystyle \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (d+e x)^7}{\left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot ^7(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {2 (2 a+b \cot (d+e x)) \cot ^4(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 b \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)}{c \left (b^2-4 a c\right )}+\frac {\left (7 b^2-16 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^2(d+e x)}{3 c^2 \left (b^2-4 a c\right )}-\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 {5 b \left (7 b^2-12 a c\right ) \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 )}{16 c^{9/2}}+\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 {\left (105 b^4-460 a c b^2-2 c \left (35 b^2-116 a c\right ) \cot (d+e x) b+256 a^2 c^2\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{24 c^4 \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]^7/(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)) - (5*b*(7*b^2 - 12*a*c)*ArcTanh[(b + 2* 
c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/ 
(16*c^(9/2)) - (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)*Sqrt[a + b*Cot[d + e*x] + 
 c*Cot[d + e*x]^2]) + (2*Cot[d + e*x]^4*(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*(a*(b^2 - 2*(a -...
 

3.1.11.3.1 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]
 

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

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

Time = 1.57 (sec) , antiderivative size = 13067599, normalized size of antiderivative = 10990.41

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

input

int(cot(e*x+d)^7/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

output

result too large to display
 

3.1.11.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44563 vs. \(2 (1097) = 2194\).

Time = 35.38 (sec) , antiderivative size = 89177, normalized size of antiderivative = 75.00 \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

input

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"fricas")
 

output

Too large to include
 

3.1.11.6 Sympy [F]

\[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{7}{\left (d + e x \right )}}{\left (a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(cot(e*x+d)**7/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(3/2),x)
 

output

Integral(cot(d + e*x)**7/(a + b*cot(d + e*x) + c*cot(d + e*x)**2)**(3/2), 
x)
 

3.1.11.7 Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"maxima")
 

output

Timed out
 

3.1.11.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(cot(e*x+d)^7/(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
 

3.1.11.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Hanged} \]

input

int(cot(d + e*x)^7/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2),x)
 

output

\text{Hanged}