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

Optimal. Leaf size=570 \[ \frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e} \]

[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 - (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])]*ArcTanh[(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)

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Rubi [A]  time = 23.6221, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {3701, 6725, 734, 843, 621, 206, 724, 1021, 1078, 1036, 1030, 208, 205} \[ \frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac{-b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \cot (d+e x)}{2 \sqrt{a} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

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 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 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 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 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 (d+e x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x \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}-\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} \, 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{\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{\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{(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 \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{\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}\\ \end{align*}

Mathematica [C]  time = 14.2964, size = 278, normalized size = 0.49 \[ \frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left (i \sqrt{a-i b-c} \tan ^{-1}\left (\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )-\sqrt{a+i b-c} \tanh ^{-1}\left (\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}}\right )+2 \sqrt{a} \tanh ^{-1}\left (\frac{2 a \tan (d+e x)+b}{2 \sqrt{a} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )\right )}{2 e \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I*Sqrt[a - I*b - c]*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + Tan[d + e*
x]*(b + a*Tan[d + e*x])])] - Sqrt[a + I*b - c]*ArcTanh[(b + (2*I)*c + (2*a + I*b)*Tan[d + e*x])/(2*Sqrt[a + I*
b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + 2*Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqr
t[c + Tan[d + e*x]*(b + a*Tan[d + e*x])])])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(2*e*Sqr
t[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])

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Maple [C]  time = 11.046, size = 249428, normalized size = 437.6 \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),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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),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{\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),x)

[Out]

Integral(sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2)*tan(d + e*x), 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 )\,{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),x, algorithm="giac")

[Out]

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