\(\int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) [2]

Optimal result

Integrand size = 33, antiderivative size = 889 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^{7/2} e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e} \]

[Out]

1/8*(-4*a*c+b^2)*arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))/c^(3/2)/e-1/128
*(-4*a*c+b^2)*(-4*a*c+5*b^2)*arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))/c^(
7/2)/e+arctanh(1/2*(b+2*c*tan(e*x+d))/c^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*c^(1/2)/e-1/2*arctan(1/2*
(b*(a^2-2*a*c+b^2+c^2)^(1/2)-(b^2+(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4)
*2^(1/2)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a+b*tan(e*x+d)+c*t
an(e*x+d)^2)^(1/2))*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a
*c+b^2+c^2)^(1/4)/e*2^(1/2)-1/2*arctanh(1/2*(b*(a^2-2*a*c+b^2+c^2)^(1/2)+(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(
1/2)))*tan(e*x+d))/(a^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*
c+b^2+c^2)^(1/2)))^(1/2)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2))*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*
c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)-1/4*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1
/2)*(b+2*c*tan(e*x+d))/c/e+1/64*(-4*a*c+5*b^2)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*(b+2*c*tan(e*x+d))/c^3/e-
5/24*b*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c^2/e+1/4*tan(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2)/c/e

Rubi [A] (verified)

Time = 24.64 (sec) , antiderivative size = 889, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3781, 6857, 626, 635, 212, 756, 654, 1004, 1050, 1044, 214, 211} \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\tan (d+e x) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{4 c e}-\frac {5 b \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{24 c^2 e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{64 c^3 e}-\frac {(b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{4 c e}-\frac {\sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \arctan \left (\frac {b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{128 c^{7/2} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{e}-\frac {\sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \text {arctanh}\left (\frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e} \]

[In]

Int[Tan[d + e*x]^4*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
*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
]*(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[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])])/e + ((
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)*e) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*T
an[d + e*x]^2])])/(128*c^(7/2)*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*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]*(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 + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(
4*c*e) + ((5*b^2 - 4*a*c)*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(64*c^3*e) - (5*
b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(24*c^2*e) + (Tan[d + e*x]*(a + b*Tan[d + e*x] + c*Tan[d + e*
x]^2)^(3/2))/(4*c*e)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 626

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 635

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 654

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 756

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 1004

Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] :> Dist[c/f, Int[1/Sqrt[a + b*x +
c*x^2], x], x] - Dist[1/f, Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b,
c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1044

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 1050

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 3781

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 6857

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (-\sqrt {a+b x+c x^2}+x^2 \sqrt {a+b x+c x^2}+\frac {\sqrt {a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int x^2 \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\text {Subst}\left (\int \frac {-a+c-b x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \left (-a-\frac {5 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{4 c e}+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{8 c e} \\ & = -\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}+\frac {(2 c) \text {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^2-4 a c\right ) \text {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 )}{4 c e}+\frac {\left (5 b^2-4 a c\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{16 c^2 e}-\frac {\text {Subst}\left (\int \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} 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 {\text {Subst}\left (\int \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} 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 {\sqrt {c} \text {arctanh}\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 )}{e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\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} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{128 c^3 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 ) \text {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 \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 ) \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 ) \text {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 \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 ) \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 )} \arctan \left (\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 ) \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 {c} \text {arctanh}\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 )}{e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\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} 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 )} \text {arctanh}\left (\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 ) \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+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {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 )}{64 c^3 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 )} \arctan \left (\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 ) \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 {c} \text {arctanh}\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 )}{e}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\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} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \text {arctanh}\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 )}{128 c^{7/2} 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 )} \text {arctanh}\left (\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 ) \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+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.62 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.57 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-4 i \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+4 i \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+8 \sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c^2}+\frac {2 \tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{c}-\frac {\left (-5 b^2+4 a c\right ) \left (-\left (\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )\right )+2 \sqrt {c} (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}\right )}{16 c^{7/2}}}{8 e} \]

[In]

Integrate[Tan[d + e*x]^4*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2],x]

[Out]

((-4*I)*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])] + (4*I)*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])] + 8*Sqrt[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])] + ((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])])/c^(3/2) - (2*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x]
+ c*Tan[d + e*x]^2])/c - (5*b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(3*c^2) + (2*Tan[d + e*x]*(a + b*
Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/c - ((-5*b^2 + 4*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])]) + 2*Sqrt[c]*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d
+ e*x] + c*Tan[d + e*x]^2]))/(16*c^(7/2)))/(8*e)

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 2.22 (sec) , antiderivative size = 17248526, normalized size of antiderivative = 19402.17

\[\text {output too large to display}\]

[In]

int((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^4,x)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2427 vs. \(2 (795) = 1590\).

Time = 1.13 (sec) , antiderivative size = 4855, normalized size of antiderivative = 5.46 \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^4,x, algorithm="fricas")

[Out]

[1/768*(192*c^4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 -
(4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^
3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c
)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*
b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a
^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x +
d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b +
a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1))
- 192*c^4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^
4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^
3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)
*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*
a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3
+ b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2
+ 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3
+ (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) + 192*
c^4*e*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b +
3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c -
2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt
(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) + ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3
)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^
5 - 2*(4*a^3*b + a*b^3)*c)*e + ((4*a^3*b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(
4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*
a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - 192*c^4*
e*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a
^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a
*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^
2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) - ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)
*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 -
2*(4*a^3*b + a*b^3)*c)*e + ((4*a^3*b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^
4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*
b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - 3*(5*b^4 - 2
4*a*b^2*c + 64*a*c^3 - 128*c^4 + 16*(a^2 - b^2)*c^2)*sqrt(c)*log(8*c^2*tan(e*x + d)^2 + 8*b*c*tan(e*x + d) + b
^2 + 4*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*(2*c*tan(e*x + d) + b)*sqrt(c) + 4*a*c) + 4*(48*c^4*tan(e*x
+ d)^3 + 8*b*c^3*tan(e*x + d)^2 + 15*b^3*c - 52*a*b*c^2 - 48*b*c^3 - 2*(5*b^2*c^2 - 12*a*c^3 + 48*c^4)*tan(e*
x + d))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a))/(c^4*e), 1/384*(96*c^4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c
)/e^2)*log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2
- 2*(4*a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 +
3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*t
an(e*x + d) + a) + ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 +
a*b^4 + b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*
b + 3*a*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)
*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sq
rt((e^2*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) - 96*c^4*e*sqrt((e^2*sqrt(-b^2/e^4) - a + c)/e^2)*
log((2*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*
a^3*b + 3*a*b^3)*c)*tan(e*x + d) + (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*
b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x
+ d) + a) - ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 +
b^4*c - 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e - ((4*a^3*b + 3*a
*b^3 - 8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 -
2*(4*a^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt((e^2
*sqrt(-b^2/e^4) - a + c)/e^2))/(tan(e*x + d)^2 + 1)) + 96*c^4*e*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2)*log((2
*(4*a^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b
+ 3*a*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 +
b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) +
a) + ((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c
- 2*a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e + ((4*a^3*b + 3*a*b^3 -
8*a*b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a
^3 + 3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt
(-b^2/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) - 96*c^4*e*sqrt(-(e^2*sqrt(-b^2/e^4) + a - c)/e^2)*log((2*(4*a
^3*b^2 + 2*a*b^4 + 2*b^4*c - 4*a*b^2*c^2 - (4*a^4*b + 3*a^2*b^3 + b^5 + (4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 3*a
*b^3)*c)*tan(e*x + d) - (2*(2*a^3*b + a*b^3 + b^3*c - 2*a*b*c^2)*e^2*tan(e*x + d) + (4*a^4 + 3*a^2*b^2 + b^4 +
(4*a^2 - b^2)*c^2 - 2*(4*a^3 + 3*a*b^2)*c)*e^2)*sqrt(-b^2/e^4))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a) -
((b^5 + 2*(4*a^2*b - b^3)*c^2 - 2*(4*a^3*b + 5*a*b^3)*c)*e*tan(e*x + d)^2 - 4*(2*a^3*b^2 + a*b^4 + b^4*c - 2*
a*b^2*c^2)*e*tan(e*x + d) - (8*a^4*b + 6*a^2*b^3 + b^5 - 2*(4*a^3*b + a*b^3)*c)*e + ((4*a^3*b + 3*a*b^3 - 8*a*
b*c^2 - (4*a^2*b - 3*b^3)*c)*e^3*tan(e*x + d)^2 + 2*(4*a^4 + 3*a^2*b^2 + b^4 + (4*a^2 - b^2)*c^2 - 2*(4*a^3 +
3*a*b^2)*c)*e^3*tan(e*x + d) - (4*a^3*b + a*b^3 + (4*a^2*b + b^3)*c)*e^3)*sqrt(-b^2/e^4))*sqrt(-(e^2*sqrt(-b^2
/e^4) + a - c)/e^2))/(tan(e*x + d)^2 + 1)) + 3*(5*b^4 - 24*a*b^2*c + 64*a*c^3 - 128*c^4 + 16*(a^2 - b^2)*c^2)*
sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*(2*c*tan(e*x + d) + b)*sqrt(-c)/(c^2*tan(e*x +
d)^2 + b*c*tan(e*x + d) + a*c)) + 2*(48*c^4*tan(e*x + d)^3 + 8*b*c^3*tan(e*x + d)^2 + 15*b^3*c - 52*a*b*c^2 -
48*b*c^3 - 2*(5*b^2*c^2 - 12*a*c^3 + 48*c^4)*tan(e*x + d))*sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a))/(c^4*
e)]

Sympy [F]

\[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{4}{\left (d + e x \right )}\, dx \]

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2)*tan(e*x+d)**4,x)

[Out]

Integral(sqrt(a + b*tan(d + e*x) + c*tan(d + e*x)**2)*tan(d + e*x)**4, x)

Maxima [F]

\[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{4} \,d x } \]

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)*tan(e*x + d)^4, x)

Giac [F(-1)]

Timed out. \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^4,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Hanged} \]

[In]

int(tan(d + e*x)^4*(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)

[Out]

\text{Hanged}