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

Optimal. Leaf size=829 \[ -\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tan ^{-1}\left (\frac{b \left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tan ^{-1}\left (\frac{b \left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 a^{5/2} e}-\frac{\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left (b^2-4 a c\right ) e}+\frac{2 \cot (d+e x) \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]

[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)

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Rubi [A]  time = 4.93876, antiderivative size = 829, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {3700, 6725, 740, 806, 724, 206, 975, 1036, 1030, 205} \[ -\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tan ^{-1}\left (\frac{b \left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tan ^{-1}\left (\frac{b \left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 a^{5/2} e}-\frac{\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left (b^2-4 a c\right ) e}+\frac{2 \cot (d+e x) \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]

Antiderivative was successfully verified.

[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 3700

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 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 740

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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 724

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 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 975

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 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 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 \frac{\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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{\operatorname{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{\operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 )} \tan ^{-1}\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}} \tan ^{-1}\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 \tanh ^{-1}\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]  time = 6.16579, size = 583, normalized size = 0.7 \[ \frac{-\frac{2 \left (\frac{\left (\frac{1}{2} b \left (8 a c-3 b^2\right )+2 a b c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}-\frac{\left (8 a c-3 b^2\right ) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a}\right )}{a \left (b^2-4 a c\right )}-\frac{2 \left (c \left (2 a c-b^2-2 c^2\right ) \tan (d+e x)+b c (3 a-c)-b^3\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (-\frac{4 \sqrt{a-i b-c} \left (-\frac{1}{4} b \left (b^2-4 a c\right )+\frac{1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac{-2 a-(b-2 i c) \tan (d+e x)+i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}-\frac{4 \sqrt{a+i b-c} \left (-\frac{1}{4} b \left (b^2-4 a c\right )-\frac{1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac{-2 a-(b+2 i c) \tan (d+e x)-i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right )}-\frac{2 \cot (d+e x) \left (2 a c-b^2-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e} \]

Warning: Unable to verify antiderivative.

[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]*(-(b*(b^2 - 4*a*c))/4 + (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]*(-(b*(b^2 - 4*a*c))/4 - (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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( ex+d \right ) \right ) ^{2} \left ( a+b\tan \left ( ex+d \right ) +c \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (e x + d\right )^{2}}{{\left (c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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