\(\int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [21]

Optimal result

Integrand size = 35, antiderivative size = 270 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {\left (b^3+2 b^2 c-4 b (a-2 c) c-8 c^2 (a+2 c)\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{32 c^{5/2} e}+\frac {\left ((b-2 c) (b+4 c)+2 c (b+2 c) \cot ^2(d+e x)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{16 c^2 e}-\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{6 c e} \] Output:

1/2*(a-b+c)^(1/2)*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)/( 
a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e-1/32*(b^3+2*b^2*c-4*b*(a-2*c)*c- 
8*c^2*(a+2*c))*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+ 
c*cot(e*x+d)^4)^(1/2))/c^(5/2)/e+1/16*((b-2*c)*(b+4*c)+2*c*(b+2*c)*cot(e*x 
+d)^2)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/c^2/e-1/6*(a+b*cot(e*x+d)^2 
+c*cot(e*x+d)^4)^(3/2)/c/e
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1017\) vs. \(2(270)=540\).

Time = 6.45 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.77 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx =\text {Too large to display} \] Input:

Integrate[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

-1/2*(((b*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e 
*x]^2 + a*Tan[d + e*x]^4])])/(2*Sqrt[a]) - Sqrt[c]*ArcTanh[(2*c + b*Tan[d 
+ e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])] + ((( 
2*a - b)*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e* 
x]^2 + a*Tan[d + e*x]^4])])/Sqrt[a] - (4*Sqrt[a - b + c]*(2*a - 2*b + 2*c) 
*ArcTanh[(b - 2*c - (-2*a + b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + 
 b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/(4*a - 4*b + 4*c))/2)*Tan[d + e*x 
]^2*Sqrt[Cot[d + e*x]^4*(c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4)])/(e*Sqr 
t[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]) + (Tan[d + e*x]^2*Sqrt[Cot[d + 
 e*x]^4*(c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4)]*(2*Sqrt[a]*ArcTanh[(b + 
 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x] 
^4])] - (b*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + 
e*x]^2 + a*Tan[d + e*x]^4])])/Sqrt[c] - 2*Cot[d + e*x]^2*Sqrt[c + b*Tan[d 
+ e*x]^2 + a*Tan[d + e*x]^4]))/(4*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + 
e*x]^4]) - (Tan[d + e*x]^2*Sqrt[Cot[d + e*x]^4*(c + b*Tan[d + e*x]^2 + a*T 
an[d + e*x]^4)]*(((b^2 - 4*a*c)*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c 
]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/c^(3/2) - (2*Cot[d + e* 
x]^4*(2*c + b*Tan[d + e*x]^2)*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4 
])/c))/(16*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]) - (Tan[d + e*x 
]^2*Sqrt[Cot[d + e*x]^4*(c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4)]*((16...
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {3042, 4184, 1578, 1267, 27, 1231, 27, 1269, 1092, 219, 1154, 219}

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.

\(\displaystyle \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (d+e x)^5 \sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {\int \frac {\cot ^4(d+e x) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot ^2(d+e x)}{2 e}\)

\(\Big \downarrow \) 1267

\(\displaystyle -\frac {\frac {\int -\frac {3 \left ((b+2 c) \cot ^2(d+e x)+b\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{2 \left (\cot ^2(d+e x)+1\right )}d\cot ^2(d+e x)}{3 c}+\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\int \frac {\left ((b+2 c) \cot ^2(d+e x)+b\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}{\cot ^2(d+e x)+1}d\cot ^2(d+e x)}{2 c}}{2 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\int \frac {\left (b^3+2 c b^2-4 (a-2 c) c b-8 c^2 (a+2 c)\right ) \cot ^2(d+e x)+(b-2 c) \left (b^2+4 c b-4 a c\right )}{2 \left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{4 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\int \frac {\left (b^3+2 c b^2-4 (a-2 c) c b-8 c^2 (a+2 c)\right ) \cot ^2(d+e x)+(b-2 c) \left (b^2+4 c b-4 a c\right )}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \int \frac {1}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+16 c^2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {2 \left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \int \frac {1}{4 c-\cot ^4(d+e x)}d\frac {2 c \cot ^2(d+e x)+b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}+16 c^2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{8 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {16 c^2 (a-b+c) \int \frac {1}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)+\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}}{8 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-32 c^2 (a-b+c) \int \frac {1}{4 (a-b+c)-\cot ^4(d+e x)}d\frac {(b-2 c) \cot ^2(d+e x)+2 a-b}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}}{8 c}}{2 c}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}}{3 c}-\frac {\frac {\left (2 c (b+2 c) \cot ^2(d+e x)+(b-2 c) (b+4 c)\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{4 c}-\frac {\frac {\left (-4 b c (a-2 c)-8 c^2 (a+2 c)+b^3+2 b^2 c\right ) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{\sqrt {c}}-16 c^2 \sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{8 c}}{2 c}}{2 e}\)

Input:

Int[Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

-1/2*((a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2)/(3*c) - (-1/8*(-16*c 
^2*Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a 
- b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])] + ((b^3 + 2*b^2*c 
 - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2* 
Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/Sqrt[c])/c + (((b 
 - 2*c)*(b + 4*c) + 2*c*(b + 2*c)*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^ 
2 + c*Cot[d + e*x]^4])/(4*c))/(2*c))/e
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(\frac {-\frac {\left (a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}{6 c}+\frac {b \left (\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}-\frac {\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}+\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}}{e}\)	\(455\)
default	\(\frac {-\frac {\left (a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}{6 c}+\frac {b \left (\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}-\frac {\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}+\frac {\left (b +2 c \cot \left (e x +d \right )^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{16 c^{\frac {3}{2}}}}{e}\)	\(455\)

Input:

int(cot(e*x+d)^5*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

1/e*(-1/6*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2)/c+1/4*b/c*(1/4*(b+2*c*co 
t(e*x+d)^2)/c*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/8*(4*a*c-b^2)/c^(3 
/2)*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1 
/2)))-1/2*(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2)-1/4* 
(b-2*c)*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1/2)+(c*(cot(e*x+d)^2+1)^2+(b-2 
*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/c^(1/2)+1/2*(a-b+c)^(1/2)*ln((2*a-2*b+2 
*c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*(c*(cot(e*x+d)^2+1)^2+(b-2*c)* 
(cot(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e*x+d)^2+1))+1/8*(b+2*c*cot(e*x+d)^2)/ 
c*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/16*(4*a*c-b^2)/c^(3/2)*ln((1/2 
*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (242) = 484\).

Time = 2.63 (sec) , antiderivative size = 3019, normalized size of antiderivative = 11.18 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)^5*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorith 
m="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(cot(e*x+d)**5*(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cot(e*x+d)^5*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorith 
m="maxima")
 

Output:

integrate(sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a)*cot(e*x + d)^5, x)
 

Giac [F(-1)]

Timed out. \[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Timed out} \] Input:

integrate(cot(e*x+d)^5*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorith 
m="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Hanged} \] Input:

int(cot(d + e*x)^5*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \cot ^5(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {too large to display} \] Input:

int(cot(e*x+d)^5*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)
 

Output:

( - 4*sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x)**4*b*c 
- 8*sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x)**4*c**2 - 
 sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x)**2*b**2 + 4* 
sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x)**2*b*c + 12*s 
qrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x)**2*c**2 + 2*sq 
rt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*a*b + 4*sqrt(cot(d + e*x)**4 
*c + cot(d + e*x)**2*b + a)*a*c + 3*sqrt(cot(d + e*x)**4*c + cot(d + e*x)* 
*2*b + a)*b**2 - 18*sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*b*c + 
12*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot 
(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d 
 + e*x)**2*b*c + a*b + 2*a*c),x)*a*b**2*c*e + 48*int((sqrt(cot(d + e*x)**4 
*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot(d + e*x)**4*b*c + 2*cot(d + 
 e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c 
),x)*a*b*c**2*e + 48*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)* 
cot(d + e*x))/(cot(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x) 
**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c),x)*a*c**3*e + 3*int((sqrt( 
cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot(d + e*x)**4* 
b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2*b* 
c + a*b + 2*a*c),x)*b**4*e - 12*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x) 
**2*b + a)*cot(d + e*x))/(cot(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 ...