3.1 \(\int \tan ^5(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\)

Optimal. Leaf size=975 \[ \frac{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2} \tan ^2(d+e x)}{5 c e}+\frac{\left (35 b^2-42 c \tan (d+e x) b-32 a c\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{240 c^3 e}-\frac{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{3 c e}+\frac{\sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac{b^2-\sqrt{a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{256 c^{9/2} e}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{16 c^{5/2} e}+\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 \sqrt{c} e}-\frac{\sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac{b^2+\sqrt{a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac{b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac{\sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e} \]

[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]*Tan[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*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) +
(b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) - (
b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c
^(5/2)*e) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x
] + c*Tan[d + e*x]^2])])/(256*c^(9/2)*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]*Tan[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*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(
Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e + (b*(b + 2*c*Tan[d
+ e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) - (b*(7*b^2 - 12*a*c)*(b + 2*c*Tan[d + e*x])*S
qrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(128*c^4*e) - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e
) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c*e) + ((35*b^2 - 32*a*c - 42*b*c*Tan[d
+ e*x])*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(240*c^3*e)

________________________________________________________________________________________

Rubi [A]  time = 24.4627, antiderivative size = 975, 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 = {3700, 6725, 640, 612, 621, 206, 742, 779, 1021, 1078, 1036, 1030, 208, 205} \[ \frac{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2} \tan ^2(d+e x)}{5 c e}+\frac{\left (35 b^2-42 c \tan (d+e x) b-32 a c\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{240 c^3 e}-\frac{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{3 c e}+\frac{\sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \tan ^{-1}\left (\frac{b^2-\sqrt{a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{256 c^{9/2} e}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{16 c^{5/2} e}+\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 \sqrt{c} e}-\frac{\sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac{b^2+\sqrt{a^2-2 c a+b^2+c^2} \tan (d+e x) b+(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt{a^2-\left (2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt{a^2-2 c a+b^2+c^2}\right )} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}-\frac{b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{128 c^4 e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{8 c^2 e}+\frac{\sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}{e} \]

Antiderivative was successfully verified.

[In]

Int[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],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]*Tan[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*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) +
(b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) - (
b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c
^(5/2)*e) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x
] + c*Tan[d + e*x]^2])])/(256*c^(9/2)*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]*Tan[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*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(
Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e + (b*(b + 2*c*Tan[d
+ e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) - (b*(7*b^2 - 12*a*c)*(b + 2*c*Tan[d + e*x])*S
qrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(128*c^4*e) - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e
) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c*e) + ((35*b^2 - 32*a*c - 42*b*c*Tan[d
+ e*x])*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(240*c^3*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 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 742

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

Rule 779

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

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 \tan ^5(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5 \sqrt{a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \left (-x \sqrt{a+b x+c x^2}+x^3 \sqrt{a+b x+c x^2}+\frac{x \sqrt{a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\tan (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int x \sqrt{a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int x^3 \sqrt{a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int \frac{x \sqrt{a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c 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,\tan (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int x \left (-2 a-\frac{7 b x}{2}\right ) \sqrt{a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{5 c e}+\frac{b \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{2 c e}\\ &=\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac{\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 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,\tan (d+e x)\right )}{e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 e}-\frac{\left (b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{32 c^3 e}-\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{16 c^2 e}\\ &=\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac{b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac{\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac{\left (b \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^2 e}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{256 c^4 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,\tan (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,\tan (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt{c} e}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac{b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac{\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}+\frac{\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^4 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} \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^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} \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^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} \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt{c} e}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{256 c^{9/2} 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} \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e}+\frac{b (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{8 c^2 e}-\frac{b \left (7 b^2-12 a c\right ) (b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{128 c^4 e}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c e}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c e}+\frac{\left (35 b^2-32 a c-42 b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{240 c^3 e}\\ \end{align*}

Mathematica [C]  time = 6.22143, size = 690, normalized size = 0.71 \[ \frac{\frac{b \left (\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}\right )}{2 c}+\frac{\frac{\left (15 a b c-\frac{35 b^3}{4}\right ) \left (\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}\right )}{8 c^2}-\frac{\left (8 a c-\frac{35 b^2}{4}+\frac{21}{2} b c \tan (d+e x)\right ) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{12 c^2}}{5 c}+\frac{\tan ^2(d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{5 c}-\frac{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c}+\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}+\frac{1}{4} \left (\frac{(b-2 i c) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{c}}-2 \sqrt{a-i b-c} \tanh ^{-1}\left (\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )+\frac{1}{4} \left (\frac{(b+2 i c) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{c}}-2 \sqrt{a+i b-c} \tanh ^{-1}\left (\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )}{e} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]

[Out]

((-2*Sqrt[a - I*b - 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])] + ((b - (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x
] + c*Tan[d + e*x]^2])])/Sqrt[c])/4 + (-2*Sqrt[a + I*b - 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])] + ((b + (2*I)*c)*ArcTanh[(b + 2*c*Tan[d + e*
x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[c])/4 + Sqrt[a + b*Tan[d + e*x] + c*Tan[d +
e*x]^2] - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d
+ e*x]^2)^(3/2))/(5*c) + (b*(-((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x
] + c*Tan[d + e*x]^2])])/(8*c^(3/2)) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4
*c)))/(2*c) + (-(((-35*b^2)/4 + 8*a*c + (21*b*c*Tan[d + e*x])/2)*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)
)/(12*c^2) + (((-35*b^3)/4 + 15*a*b*c)*(-((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*T
an[d + e*x] + c*Tan[d + e*x]^2])])/(8*c^(3/2)) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e
*x]^2])/(4*c)))/(8*c^2))/(5*c))/e

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Maple [B]  time = 1.056, size = 17767396, normalized size = 18223. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^5,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 \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^5,x, algorithm="maxima")

[Out]

integrate(sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*tan(e*x + d)^5, 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*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^5,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 \tan{\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{5}{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2)*tan(e*x+d)**5,x)

[Out]

Integral(sqrt(a + b*tan(d + e*x) + c*tan(d + e*x)**2)*tan(d + e*x)**5, x)

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Giac [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*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^5,x, algorithm="giac")

[Out]

Timed out