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

Optimal. Leaf size=749 \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left (\frac{-b \left (-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left (\frac{-b \left (\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2} e}-\frac{2 \left (-2 a c+b^2+b c \cot (d+e x)\right )}{a e \left (b^2-4 a c\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{e \left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 4.47623, antiderivative size = 749, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {3701, 6725, 740, 12, 724, 206, 1018, 1036, 1030, 208} \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left (\frac{-b \left (-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tanh ^{-1}\left (\frac{-b \left (\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2} e}-\frac{2 \left (-2 a c+b^2+b c \cot (d+e x)\right )}{a e \left (b^2-4 a c\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{e \left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1018

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)*((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))/((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[(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 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]

Rubi steps

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

Mathematica [C]  time = 48.8399, size = 558961, normalized size = 746.28 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 50.648, size = 21243685, normalized size = 28362.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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