\(\int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) [1]

Optimal result

Integrand size = 33, antiderivative size = 547 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}+\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e} \]

[Out]

-1/2*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^(3/2)/e+1/16*b*(-12*a*c
+5*b^2)*arctanh(1/2*(b+2*c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/c^(7/2)/e+(a+b*cot(e*x+d
)+c*cot(e*x+d)^2)^(1/2)/c/e-1/3*cot(e*x+d)^2*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c/e-1/24*(15*b^2-16*a*c-10*
b*c*cot(e*x+d))*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^3/e-1/2*arctanh(1/2*(a-c+b*cot(e*x+d)-(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)/(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2))*(a-c-(a^2-2*a*
c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/2)+1/2*arctanh(1/2*(a-c+b*cot(e*x+d)+(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)/(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2))*(a-c+(a^2-2*a*
c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/2)

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3782, 6857, 654, 635, 212, 756, 793, 1050, 1044, 214} \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac {\left (-16 a c+15 b^2-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]

[In]

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

[Out]

-((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x]
)/(Sqrt[2]*Sqrt[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]
*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 +
b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[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]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sq
rt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) + (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[
d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(7/2)*e) + Sqrt[a + b*Cot[d + e*x] +
c*Cot[d + e*x]^2]/(c*e) - (Cot[d + e*x]^2*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(3*c*e) - ((15*b^2 - 1
6*a*c - 10*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^3*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 654

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 756

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 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 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 ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {x}{\sqrt {a+b x+c x^2}}+\frac {x^3}{\sqrt {a+b x+c x^2}}+\frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\cot (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\text {Subst}\left (\int \frac {x \left (-2 a-\frac {5 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{3 c e}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c e}+\frac {\text {Subst}\left (\int \frac {-b+\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\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 {\text {Subst}\left (\int \frac {-b+\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\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 {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}-\frac {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 e}+\frac {\left (b \left (5 b^2-12 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{16 c^3 e}+\frac {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\left (b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {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} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {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} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {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^{3/2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}+\frac {\left (b \left (5 b^2-12 a c\right )\right ) \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 )}{8 c^3 e} \\ & = -\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {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} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {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} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {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^{3/2} e}+\frac {b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac {\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.77 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{c e}-\frac {i \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 \sqrt {a+i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {i \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 \sqrt {a-i b-c} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {b \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 c^{3/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {16 \cot ^3(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}-\frac {\frac {20 b \cot ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}+\frac {\frac {3 b \left (5 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 \left (15 b^2-16 a c\right ) \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}}{c}}{c}\right )}{48 e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \]

[In]

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

[Out]

Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]/(c*e) - ((I/2)*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])]*Tan[d + e*x]*Sqrt[Cot[d + e*x
]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(Sqrt[a + I*b - c]*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]
) - ((I/2)*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*Ta
n[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(Sqrt[a - I*b - c]*
e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (b*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d
+ e*x] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(2*c^
(3/2)*e*Sqrt[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]*((16*Cot[d + e*x]^3*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])/c - ((20*b*Cot[d + e*x]^2*Sqrt[c + b*Tan[d
+ e*x] + a*Tan[d + e*x]^2])/c + ((3*b*(5*b^2 - 12*a*c)*ArcTanh[(2*c + b*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*T
an[d + e*x] + a*Tan[d + e*x]^2])])/c^(3/2) - (2*(15*b^2 - 16*a*c)*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan
[d + e*x]^2])/c)/c)/c))/(48*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 = 3.55 (sec) , antiderivative size = 9581948, normalized size of antiderivative = 17517.27

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

[In]

int(cot(e*x+d)^5/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),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. 12095 vs. \(2 (484) = 968\).

Time = 6.09 (sec) , antiderivative size = 24241, normalized size of antiderivative = 44.32 \[ \int \frac {\cot ^5(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

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

Giac [F(-2)]

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

[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]

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

[In]

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

[Out]

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