\(\int \frac {\cot ^2(d+e x)}{(a+b \tan (d+e x)+c \tan ^2(d+e x))^{3/2}} \, dx\) [25]

Optimal result

Integrand size = 33, antiderivative size = 829 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2} e}+\frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e} \]

[Out]

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

Rubi [A] (verified)

Time = 5.49 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3781, 6857, 754, 820, 738, 212, 989, 1050, 1044, 211} \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 a^{5/2} e}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left (b^2-4 a c\right ) e}+\frac {2 \cot (d+e x) \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) e \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]

[In]

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

[Out]

-((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]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*
a*c + c^2]))*Tan[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*Tan[d + e*x] + c*Tan[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]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a
- c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[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*Tan[d + e*x] + c*Tan[d + e*x]
^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (3*b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b
*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(5/2)*e) + (2*Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2
- 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(b*(b^2 - (3*a - c)*c) + c*(b^2 - 2*(a - c)*c)*T
an[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - ((3*b^2 - 8*a*
c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a^2*(b^2 - 4*a*c)*e)

Rule 211

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

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 754

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

Rule 820

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

Rule 989

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b^3*f + b*c*(c*d
- 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(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))), 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[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*
f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(
p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, 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]) &
&  !IGtQ[q, 0]

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 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(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*Tan[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 {1}{x^2 \left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{\left (-1-x^2\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} (a-c) \left (b^2-4 a c\right )-\frac {1}{2} b \left (b^2-4 a c\right ) x}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e} \\ & = \frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 a^2 e}+\frac {\text {Subst}\left (\int \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 ) x}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (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} \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 ) x}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (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 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a^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} 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 ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^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} 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 ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{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 )} \arctan \left (\frac {b \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 ) \tan (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 \tan (d+e x)+c \tan ^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}} \arctan \left (\frac {b \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 ) \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2} e}+\frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.24 (sec) , antiderivative size = 583, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a+i b-(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a-i b-(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \cot (d+e x) \left (-b^2+2 a c-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (-b^3+b (3 a-c) c+c \left (-b^2+2 a c-2 c^2\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (\frac {\left (2 a b c+\frac {1}{2} b \left (-3 b^2+8 a c\right )\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}+\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a}\right )}{a \left (b^2-4 a c\right )}}{e} \]

[In]

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

[Out]

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

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39249 vs. \(2 (764) = 1528\).

Time = 12.46 (sec) , antiderivative size = 78535, normalized size of antiderivative = 94.73 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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