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

Optimal. Leaf size=691 \[ -\frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{3/2} e}+\frac{\tan ^2(d+e x) (2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e} \]

[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)
) - (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - ((b^
2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(3/2)
*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])]*ArcTa
nh[(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])/(Sqr
t[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) + ((2*a + b*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(4*a*e)

________________________________________________________________________________________

Rubi [A]  time = 23.712, antiderivative size = 691, 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, 720, 724, 206, 734, 843, 621, 1021, 1078, 1036, 1030, 208, 205} \[ -\frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{3/2} e}+\frac{\tan ^2(d+e x) (2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^3,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)
) - (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - ((b^
2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(3/2)
*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])]*ArcTa
nh[(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])/(Sqr
t[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) + ((2*a + b*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(4*a*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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, 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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

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 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 \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3 \left (1+x^2\right )} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sqrt{a+b x+c x^2}}{x^3}-\frac{\sqrt{a+b x+c x^2}}{x}+\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 \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,\cot (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x} \, 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{(2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a e}-\frac{\operatorname{Subst}\left (\int \frac{-2 a-b x}{x \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 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{\left (b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{8 a e}\\ &=\frac{(2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a 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{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\left (b^2-4 a c\right ) \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 )}{4 a e}\\ &=-\frac{\left (b^2-4 a c\right ) \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 )}{8 a^{3/2} e}+\frac{(2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a e}-\frac{(2 a) \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 )}{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{\sqrt{a} \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 )}{e}-\frac{\left (b^2-4 a c\right ) \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 )}{8 a^{3/2} e}+\frac{(2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a 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{\sqrt{a} \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 )}{e}-\frac{\left (b^2-4 a c\right ) \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 )}{8 a^{3/2} 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{(2 a+b \cot (d+e x)) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a e}\\ \end{align*}

Mathematica [C]  time = 47.529, size = 465721, normalized size = 673.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 75.419, size = 4237674, normalized size = 6132.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,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} \tan \left (e x + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a)*tan(e*x + d)^3, 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((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,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 )}} \tan ^{3}{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2)*tan(e*x+d)**3,x)

[Out]

Integral(sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2)*tan(d + e*x)**3, 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} \tan \left (e x + d\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm="giac")

[Out]

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