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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 5.23 (sec) , antiderivative size = 865, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3782, 6857, 650, 752, 793, 635, 212, 1032, 1050, 1044, 214} \[ \int \frac {\cot ^5(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left (b^2-4 a c\right ) e \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {3 b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 c^{5/2} e}-\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac {\left (3 b^2-2 c \cot (d+e x) b-8 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left (b^2-4 a c\right ) e}+\frac {2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}} \]

[In]

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

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 1032

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((g*c)*((-b
)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a
*b*f))*x), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f
*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*
(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*
(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f
) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}
, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rule 3782

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Dist[-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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^5}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {x}{\left (a+b x+c x^2\right )^{3/2}}+\frac {x^3}{\left (a+b x+c x^2\right )^{3/2}}+\frac {x}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx,x,\cot (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{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 {2 \text {Subst}\left (\int \frac {x (4 a+2 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) e}+\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{2} b \left (b^2-4 a c\right )-\frac {1}{2} (a-c) \left (b^2-4 a c\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) 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}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c^2 e}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )+\frac {1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )+\frac {1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) \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}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{c^2 e}+\frac {\left (b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} b \left (b^2-4 a c\right )^2 \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {\frac {1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )-\frac {1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {\left (b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} b \left (b^2-4 a c\right )^2 \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {\frac {1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )-\frac {1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} e} \\ & = \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+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \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} \\ \end{align*}

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=-\frac {4 \cot (d+e x) (b+2 a \tan (d+e x)) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a e \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right ) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )} \sqrt {1-\left (b^2-4 a c\right ) \left (\frac {b}{b^2-4 a c}+\frac {2 a \tan (d+e x)}{b^2-4 a c}\right )^2}}+\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {4 \cot (d+e x) \left (b^2-2 a c+a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {\frac {3 b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 \left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}}{c \left (b^2-4 a c\right )}\right )}{2 e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}+\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \arctan \left (\frac {i b+2 c-(-2 i a-b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{-4 a+4 i b+4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \arctan \left (\frac {-i b+2 c-(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{-4 a-4 i b+4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b^3+a b (a-3 c)+a \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {4 \left (b^2-4 a c\right ) \left (\frac {a^2}{\left (b^2-4 a c\right ) \left (\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}\right )}\right )^{3/2} \left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right ) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a^2 \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2} \sqrt {1-\frac {\left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right )^2}{\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}}}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \]

[In]

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

[Out]

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

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

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

[In]

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

[Out]

result too large to display

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

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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