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

Optimal. Leaf size=976 \[ -\frac{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2} \cot ^2(d+e x)}{5 c e}-\frac{\left (35 b^2-42 c \cot (d+e x) b-32 a c\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}{240 c^3 e}+\frac{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}{3 c e}-\frac{\sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac{b^2-\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{256 c^{9/2} e}+\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{16 c^{5/2} e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 \sqrt{c} e}+\frac{\sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac{b^2+\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac{b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{128 c^4 e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{8 c^2 e}-\frac{\sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{e} \]

[Out]

-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b
^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*
(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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)^(1/4)*e)
) - (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*Sqrt[c]*e)
+ (b*(b^2 - 4*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])])/(1
6*c^(5/2)*e) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*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])])/(256*c^(9/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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*Sqrt[a^2
+ b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2
+ b^2 - 2*a*c + c^2]) - a*(2*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)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e - (b*(b + 2*c*Co
t[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) + (b*(7*b^2 - 12*a*c)*(b + 2*c*Cot[d + e*x]
)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(128*c^4*e) + (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*
c*e) - (Cot[d + e*x]^2*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(5*c*e) - ((35*b^2 - 32*a*c - 42*b*c*Cot
[d + e*x])*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(240*c^3*e)

________________________________________________________________________________________

Rubi [A]  time = 24.3123, antiderivative size = 976, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424, Rules used = {3701, 6725, 640, 612, 621, 206, 742, 779, 1021, 1078, 1036, 1030, 208, 205} \[ -\frac{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2} \cot ^2(d+e x)}{5 c e}-\frac{\left (35 b^2-42 c \cot (d+e x) b-32 a c\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}{240 c^3 e}+\frac{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}{3 c e}-\frac{\sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac{b^2-\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{256 c^{9/2} e}+\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{16 c^{5/2} e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 \sqrt{c} e}+\frac{\sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac{b^2+\sqrt{a^2-2 c a+b^2+c^2} \cot (d+e x) b+(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac{b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{128 c^4 e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{8 c^2 e}-\frac{\sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b
^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*
(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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)^(1/4)*e)
) - (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*Sqrt[c]*e)
+ (b*(b^2 - 4*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])])/(1
6*c^(5/2)*e) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*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])])/(256*c^(9/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*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*Sqrt[a^2
+ b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2
+ b^2 - 2*a*c + c^2]) - a*(2*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)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e - (b*(b + 2*c*Co
t[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) + (b*(7*b^2 - 12*a*c)*(b + 2*c*Cot[d + e*x]
)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(128*c^4*e) + (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*
c*e) - (Cot[d + e*x]^2*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(5*c*e) - ((35*b^2 - 32*a*c - 42*b*c*Cot
[d + e*x])*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(240*c^3*e)

Rule 3701

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 6725

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]

Rule 640

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

Rule 612

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

Rule 621

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 206

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

Rule 742

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

Rule 779

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 1021

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

Rule 1078

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d
+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]

Rule 1036

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 1030

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 208

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

Rule 205

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

Rubi steps

\begin{align*} \int \cot ^5(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^5 \sqrt{a+b x+c x^2}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-x \sqrt{a+b x+c x^2}+x^3 \sqrt{a+b x+c x^2}+\frac{x \sqrt{a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\cot (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int x \sqrt{a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int x^3 \sqrt{a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{x \sqrt{a+b x+c x^2}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac{\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}+\frac{\operatorname{Subst}\left (\int \frac{\frac{b}{2}-(a-c) x-\frac{b x^2}{2}}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int x \left (-2 a-\frac{7 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{5 c e}-\frac{b \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{2 c e}\\ &=-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}+\frac{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac{\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}-\frac{\left (35 b^2-32 a c-42 b c \cot (d+e x)\right ) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{240 c^3 e}+\frac{\operatorname{Subst}\left (\int \frac{b+(-a+c) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 e}+\frac{\left (b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{32 c^3 e}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{16 c^2 e}\\ &=-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}+\frac{b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{128 c^4 e}+\frac{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac{\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}-\frac{\left (35 b^2-32 a c-42 b c \cot (d+e x)\right ) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{240 c^3 e}-\frac{b \operatorname{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 )}{e}+\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{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 )}{8 c^2 e}-\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{256 c^4 e}-\frac{\operatorname{Subst}\left (\int \frac{-b \sqrt{a^2+b^2-2 a c+c^2}+\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 )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\operatorname{Subst}\left (\int \frac{b \sqrt{a^2+b^2-2 a c+c^2}+\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 )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=-\frac{b \tanh ^{-1}\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 \sqrt{c} e}+\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\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^{5/2} e}-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}+\frac{b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{128 c^4 e}+\frac{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac{\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}-\frac{\left (35 b^2-32 a c-42 b c \cot (d+e x)\right ) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{240 c^3 e}-\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \operatorname{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 )}{128 c^4 e}+\frac{\left (b \left (b^2+(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 b \sqrt{a^2+b^2-2 a c+c^2} \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{-b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )+b \sqrt{a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}+\frac{\left (b \left (b^2+(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \sqrt{a^2+b^2-2 a c+c^2} \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{-b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )-b \sqrt{a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}\\ &=-\frac{\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 )} \tan ^{-1}\left (\frac{b^2+(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )-b \sqrt{a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt{2} \sqrt [4]{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 )} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac{b \tanh ^{-1}\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 \sqrt{c} e}+\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\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^{5/2} e}-\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\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 )}{256 c^{9/2} e}+\frac{\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 )} \tanh ^{-1}\left (\frac{b^2+(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )+b \sqrt{a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt{2} \sqrt [4]{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 )} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac{b (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}+\frac{b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{128 c^4 e}+\frac{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac{\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}-\frac{\left (35 b^2-32 a c-42 b c \cot (d+e x)\right ) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{240 c^3 e}\\ \end{align*}

Mathematica [C]  time = 37.5363, size = 4224, normalized size = 4.33 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-(-105*b^4 + 460*a*b^2*c - 256*a^2*c^2 + 296*b^2*c^2 - 768*a*c^3 + 2944*c^4)/(1920*c^4) + ((-35*b^3*Cos[d +
e*x] + 116*a*b*c*Cos[d + e*x] + 104*b*c^2*Cos[d + e*x])*Csc[d + e*x])/(960*c^3) + ((7*b^2 - 16*a*c + 176*c^2)*
Csc[d + e*x]^2)/(240*c^2) - (b*Cot[d + e*x]*Csc[d + e*x]^2)/(40*c) - Csc[d + e*x]^4/5)*Sqrt[(-a - c + a*Cos[2*
(d + e*x)] - c*Cos[2*(d + e*x)] - b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/e + (Sqrt[a + b*Cot[d + e*x] +
c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*S
qrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqr
t[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^
2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(
b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x]
+ Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e
*x]))])*((7*b^5*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos
[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])
/(128*c^4*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) - (5*a*b^3*Sqrt[-(a/(-1 + Co
s[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*
x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(16*c^3*(a + c - a*Cos[2*(d + e*
x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (3*a^2*b*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(
d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*
Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(8*c^2*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*
(d + e*x)])) - (b^3*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 +
Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)
])])/(8*c^2*(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (a*b*Sqrt[-(a/(-1 + Cos[
2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)
])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(2*c*(a + c - a*Cos[2*(d + e*x)] +
c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])) + (b*Cos[2*(d + e*x)]*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + C
os[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])
- (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(
d + e*x)]) + (a*Sin[2*(d + e*x)]*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d +
e*x)])/(-1 + Cos[2*(d + e*x)]) - (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Co
s[2*(d + e*x)])])/(a + c - a*Cos[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]) - (c*Sin[2*(d + e*x)]
*Sqrt[-(a/(-1 + Cos[2*(d + e*x)])) - c/(-1 + Cos[2*(d + e*x)]) + (a*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])
- (c*Cos[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)]) - (b*Sin[2*(d + e*x)])/(-1 + Cos[2*(d + e*x)])])/(a + c - a*Cos
[2*(d + e*x)] + c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]))*Tan[d + e*x])/(256*c^(9/2)*e*Sqrt[c + Tan[d + e*x]*(
b + a*Tan[d + e*x])]*(-(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(
3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b
*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^
(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b
*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*
c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x
])]))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x]))])*Tan[d + e*x]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d +
e*x]^2*(b + a*Tan[d + e*x])))/(512*c^(9/2)*(c + Tan[d + e*x]*(b + a*Tan[d + e*x]))^(3/2)) + (Sqrt[a + b*Cot[d
+ e*x] + c*Cot[d + e*x]^2]*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[Tan[d + e*x]
] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b
- c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]))] - b*(7*b
^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[c + Tan[d
+ e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan
[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/2)*c^4*(I +
Tan[d + e*x]))])*Sec[d + e*x]^2)/(256*c^(9/2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) + ((-(b*Csc[d + e*x
]^2) - 2*c*Cot[d + e*x]*Csc[d + e*x]^2)*(b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[
Tan[d + e*x]] - 128*Sqrt[a - I*b - c]*c^(9/2)*Log[(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*
Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x]
))] - b*(7*b^4 - 8*b^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Log[2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sq
rt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]] + 128*Sqrt[a + I*b - c]*c^(9/2)*Log[(2*c + b*(-I + Tan[d + e*x]) -
(2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(128*(a + I*b - c)^(3/
2)*c^4*(I + Tan[d + e*x]))])*Tan[d + e*x])/(512*c^(9/2)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Sqrt[c + T
an[d + e*x]*(b + a*Tan[d + e*x])]) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]*(b*(7*b^4 - 8*b
^2*c*(5*a + 2*c) + 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*Csc[d + e*x]*Sec[d + e*x] - (b*(7*b^4 - 8*b^2*c*(5*a + 2*c)
+ 16*c^2*(3*a^2 + 4*a*c + 8*c^2))*(b*Sec[d + e*x]^2 + (Sqrt[c]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^
2*(b + a*Tan[d + e*x])))/Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]))/(2*c + b*Tan[d + e*x] + 2*Sqrt[c]*Sqrt[
c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) - (16384*(a - I*b - c)^2*c^(17/2)*(-I + Tan[d + e*x])*(((-2*I)*a*Sec[d
+ e*x]^2 - b*Sec[d + e*x]^2 + (I*Sqrt[a - I*b - c]*(a*Sec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan
[d + e*x])))/Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x])) -
(Sec[d + e*x]^2*(-2*c - (2*I)*a*Tan[d + e*x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d +
e*x]*(b + a*Tan[d + e*x])]))/(128*(a - I*b - c)^(3/2)*c^4*(-I + Tan[d + e*x])^2)))/(-2*c - (2*I)*a*Tan[d + e*
x] - b*(I + Tan[d + e*x]) + (2*I)*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])]) + (16384*(a +
I*b - c)^2*c^(17/2)*(I + Tan[d + e*x])*((b*Sec[d + e*x]^2 - (2*I)*(a*Sec[d + e*x]^2 + (Sqrt[a + I*b - c]*(a*S
ec[d + e*x]^2*Tan[d + e*x] + Sec[d + e*x]^2*(b + a*Tan[d + e*x])))/(2*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x
])])))/(128*(a + I*b - c)^(3/2)*c^4*(I + Tan[d + e*x])) - (Sec[d + e*x]^2*(2*c + b*(-I + Tan[d + e*x]) - (2*I)
*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])))/(128*(a + I*b - c)^(3/2)*c
^4*(I + Tan[d + e*x])^2)))/(2*c + b*(-I + Tan[d + e*x]) - (2*I)*(a*Tan[d + e*x] + Sqrt[a + I*b - c]*Sqrt[c + T
an[d + e*x]*(b + a*Tan[d + e*x])]))))/(256*c^(9/2)*Sqrt[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])))

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Maple [B]  time = 0.157, size = 17768513, normalized size = 18205.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cot{\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot ^{5}{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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