3.1.27 \(\int \frac {\cot ^7(d+e x)}{(a+b \cot ^2(d+e x)+c \cot ^4(d+e x))^{3/2}} \, dx\) [27]

3.1.27.1 Optimal result

Integrand size = 35, antiderivative size = 236 \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {\text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 c^{3/2} e}-\frac {a \left (b^2-a (b+2 c)\right )+\left (b^3+2 a^2 c-a b (b+3 c)\right ) \cot ^2(d+e x)}{c (a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]

output

-1/2*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^(3/2)/e-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))/(a-b+c)^(3/2)/e+(-a*(b^2- 
a*(b+2*c))-(b^3+2*a^2*c-a*b*(b+3*c))*cot(e*x+d)^2)/c/(a-b+c)/(-4*a*c+b^2)/ 
e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)
 

3.1.27.2 Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)} \left (-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2}}+\frac {\left (-\frac {b^2}{2}+2 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{c^{3/2}}+\frac {b^2-2 a c+a b \tan ^2(d+e x)}{c \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}-\frac {b^2-a (b+2 c)+a (-2 a+b) \tan ^2(d+e x)}{(a-b+c) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]

input

Integrate[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x 
]
 

output

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

3.1.27.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {3042, 4184, 1578, 1264, 27, 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.

\(\displaystyle \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (d+e x)^7}{\left (a+b \cot (d+e x)^2+c \cot (d+e x)^4\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot ^7(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {\int \frac {\cot ^6(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^4(d+e x)+b \cot ^2(d+e x)+a\right )^{3/2}}d\cot ^2(d+e x)}{2 e}\)

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

input

Int[Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]
 

output

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

3.1.27.3.1 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 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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]
 

3.1.27.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs. \(2(215)=430\).

Time = 0.59 (sec) , antiderivative size = 686, normalized size of antiderivative = 2.91

method	result	size
derivativedivides	\(\frac {-\frac {b +2 c \cot \left (e x +d \right )^{2}}{\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}+\frac {\cot \left (e x +d \right )^{2}}{2 c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}+\frac {b \left (-\frac {1}{c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}-\frac {b \left (b +2 c \cot \left (e x +d \right )^{2}\right )}{c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}\right )}{4 c}-\frac {\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 )}{2 c^{\frac {3}{2}}}-\frac {2 a +b \cot \left (e x +d \right )^{2}}{\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}+\frac {2 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 {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\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 )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(686\)
default	\(\frac {-\frac {b +2 c \cot \left (e x +d \right )^{2}}{\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}+\frac {\cot \left (e x +d \right )^{2}}{2 c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}+\frac {b \left (-\frac {1}{c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}-\frac {b \left (b +2 c \cot \left (e x +d \right )^{2}\right )}{c \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}\right )}{4 c}-\frac {\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 )}{2 c^{\frac {3}{2}}}-\frac {2 a +b \cot \left (e x +d \right )^{2}}{\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \left (4 a c -b^{2}\right )}+\frac {2 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 {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\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 )}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \sqrt {a -b +c}}+\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c -\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}+b -2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}-\frac {2 c \sqrt {\left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2} c +\sqrt {-4 a c +b^{2}}\, \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{\left (\sqrt {-4 a c +b^{2}}-b +2 c \right ) \left (-4 a c +b^{2}\right ) \left (\cot \left (e x +d \right )^{2}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}{e}\)	\(686\)

input

int(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x,method=_RETURNV 
ERBOSE)
 

output

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

3.1.27.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1484 vs. \(2 (215) = 430\).

Time = 2.60 (sec) , antiderivative size = 6011, normalized size of antiderivative = 25.47 \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

input

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorith 
m="fricas")
 

output

Too large to include
 

3.1.27.6 Sympy [F]

\[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{7}{\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(cot(e*x+d)**7/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(3/2),x)
 

output

Integral(cot(d + e*x)**7/(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)**(3/2 
), x)
 

3.1.27.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorith 
m="maxima")
 

output

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

3.1.27.8 Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(cot(e*x+d)^7/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorith 
m="giac")
 

output

Timed out
 

3.1.27.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^7(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^7}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a\right )}^{3/2}} \,d x \]

input

int(cot(d + e*x)^7/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(3/2),x)
 

output

int(cot(d + e*x)^7/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(3/2), x)