\(\int \frac {\tan ^5(d+e x)}{(a+b \tan (d+e x)+c \tan ^2(d+e x))^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 33, antiderivative size = 864 \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {3 b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 c^{5/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \tan ^2(d+e x) (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {\left (3 b^2-8 a c-2 b c \tan (d+e x)\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e} \]

[Out]

-3/2*b*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^(5/2)/e-1/2*arctanh(1/2
*(b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))*tan(e*x+d))*2^(1/2)/(2*a-2*c
+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-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))*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1
/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)+1/2*arctanh(1/2*(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2
*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))*tan(e*x+d))*2^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c
+c^2+(a-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))*(2*a-2*c-(a^2-2*a*c+b^2+c^2
)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-2
*(2*a+b*tan(e*x+d))/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)+2*tan(e*x+d)^2*(2*a+b*tan(e*x+d))/(-4
*a*c+b^2)/e/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)+(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*(3*b^2-8*a*c-2*b*c*tan
(e*x+d))/c^2/(-4*a*c+b^2)/e+2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*tan(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*tan(e
*x+d)+c*tan(e*x+d)^2)^(1/2)

Rubi [A] (verified)

Time = 5.38 (sec) , antiderivative size = 864, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3781, 6857, 650, 752, 793, 635, 212, 1032, 1050, 1044, 214} \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {2 (2 a+b \tan (d+e x)) \tan ^2(d+e x)}{\left (b^2-4 a c\right ) e \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}-\frac {3 b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 c^{5/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac {\left (3 b^2-2 c \tan (d+e x) b-8 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{c^2 \left (b^2-4 a c\right ) e}-\frac {2 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]

[In]

Int[Tan[d + e*x]^5/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]

[Out]

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

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 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 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 752

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 + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(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] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 793

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 1032

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)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 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), 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 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^5}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {x}{\left (a+b x+c x^2\right )^{3/2}}+\frac {x^3}{\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,\tan (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {2 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \tan ^2(d+e x) (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \text {Subst}\left (\int \frac {x (4 a+2 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{\left (b^2-4 a c\right ) e}-\frac {2 \text {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,\tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e} \\ & = -\frac {2 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \tan ^2(d+e x) (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {\left (3 b^2-8 a c-2 b c \tan (d+e x)\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 c^2 e}+\frac {\text {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,\tan (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 {\text {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,\tan (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 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \tan ^2(d+e x) (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {\left (3 b^2-8 a c-2 b c \tan (d+e x)\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac {(3 b) \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 )}{c^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 ) \text {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 ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^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 ) \text {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 ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} e} \\ & = -\frac {3 b \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 )}{2 c^{5/2} 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}} \text {arctanh}\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 ) \tan (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 \tan (d+e x)+c \tan ^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 )} \text {arctanh}\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 ) \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \tan ^2(d+e x) (2 a+b \tan (d+e x))}{\left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {\left (3 b^2-8 a c-2 b c \tan (d+e x)\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 7.70 (sec) , antiderivative size = 2272, normalized size of antiderivative = 2.63 \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[Tan[d + e*x]^5/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]

[Out]

(Sqrt[(a + c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])/(1 + Cos[2*(d + e*x)])]*((-3*a^3*
b^2 - 3*a*b^4 + 8*a^4*c + 15*a^2*b^2*c + b^4*c - 16*a^3*c^2 - 7*a*b^2*c^2 + 12*a^2*c^3 + b^2*c^3 - 4*a*c^4)/((
a - c)*(a - I*b - c)*(a + I*b - c)*c^2*(-b^2 + 4*a*c)) - (2*(-2*a^3*b^2 - 2*a*b^4 + 4*a^4*c + 8*a^2*b^2*c - 4*
a^3*c^2 - a^4*b*Sin[2*(d + e*x)] - 2*a^2*b^3*Sin[2*(d + e*x)] - b^5*Sin[2*(d + e*x)] + 6*a^3*b*c*Sin[2*(d + e*
x)] + 5*a*b^3*c*Sin[2*(d + e*x)] - 5*a^2*b*c^2*Sin[2*(d + e*x)]))/((a - c)*(a - I*b - c)*(a + I*b - c)*c*(-b^2
+ 4*a*c)*(a + c + a*Cos[2*(d + e*x)] - c*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]))))/e - (((3*a^2*b + 3*b^3 - 6
*a*b*c + 2*b*c^2)*ArcTanh[(2*Sqrt[c]*Tan[(d + e*x)/2])/(Sqrt[a]*(-1 + Tan[(d + e*x)/2]^2) - Sqrt[a*(-1 + Tan[(
d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2)])]*(1 + Cos[d + e*x])*
Sqrt[(1 + Cos[2*(d + e*x)])/(1 + Cos[d + e*x])^2]*Sqrt[(a + c + (a - c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)])
/(1 + Cos[2*(d + e*x)])]*(-1 + Tan[(d + e*x)/2]^2)*(1 + Tan[(d + e*x)/2]^2)*Sqrt[(a*(-1 + Tan[(d + e*x)/2]^2)^
2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2))/(1 + Tan[(d + e*x)/2]^2)^2])/(Sqrt[c
]*Sqrt[a + c + (a - c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]]*Sqrt[(-1 + Tan[(d + e*x)/2]^2)^2]*Sqrt[a*(-1 + T
an[(d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2)]) + ((-(a*c^2) + c
^3)*(1 + Cos[d + e*x])*Sqrt[(1 + Cos[2*(d + e*x)])/(1 + Cos[d + e*x])^2]*RootSum[a^2 + b^2 + 4*b*Sqrt[c]*#1 -
2*a*#1^2 + 4*c*#1^2 + #1^4 & , (-(a*Log[-1 + Tan[(d + e*x)/2]^2]) + a*Log[#1 - 2*Sqrt[c]*Tan[(d + e*x)/2] - #1
*Tan[(d + e*x)/2]^2 + Sqrt[a + 2*b*Tan[(d + e*x)/2] + (-2*a + 4*c)*Tan[(d + e*x)/2]^2 - 2*b*Tan[(d + e*x)/2]^3
+ a*Tan[(d + e*x)/2]^4]] + Log[-1 + Tan[(d + e*x)/2]^2]*#1^2 - Log[#1 - 2*Sqrt[c]*Tan[(d + e*x)/2] - #1*Tan[(
d + e*x)/2]^2 + Sqrt[a + 2*b*Tan[(d + e*x)/2] + (-2*a + 4*c)*Tan[(d + e*x)/2]^2 - 2*b*Tan[(d + e*x)/2]^3 + a*T
an[(d + e*x)/2]^4]]*#1^2)/(-(b*Sqrt[c]) + a*#1 - 2*c*#1 - #1^3) & ]*Sqrt[(a + c + (a - c)*Cos[2*(d + e*x)] + b
*Sin[2*(d + e*x)])/(1 + Cos[2*(d + e*x)])]*(-1 + Tan[(d + e*x)/2]^2)*(1 + Tan[(d + e*x)/2]^2)*Sqrt[(a*(-1 + Ta
n[(d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2))/(1 + Tan[(d + e*x)
/2]^2)^2])/(2*Sqrt[a + c + (a - c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]]*Sqrt[(-1 + Tan[(d + e*x)/2]^2)^2]*Sq
rt[a*(-1 + Tan[(d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2)]) - (b
*c^(3/2)*(1 + Cos[d + e*x])*Sqrt[(1 + Cos[2*(d + e*x)])/(1 + Cos[d + e*x])^2]*(Log[-1 + Tan[(d + e*x)/2]^2] -
Log[-4*c*Tan[(d + e*x)/2] + b*(-1 + Tan[(d + e*x)/2]^2) + 2*Sqrt[c]*Sqrt[a*(-1 + Tan[(d + e*x)/2]^2)^2 + 2*Tan
[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2)]] + Sqrt[c]*RootSum[a^2 + b^2 + 4*b*Sqrt[c]*#1
- 2*a*#1^2 + 4*c*#1^2 + #1^4 & , (-(b*Log[-1 + Tan[(d + e*x)/2]^2]) + b*Log[#1 - 2*Sqrt[c]*Tan[(d + e*x)/2] -
#1*Tan[(d + e*x)/2]^2 + Sqrt[a + 2*b*Tan[(d + e*x)/2] + (-2*a + 4*c)*Tan[(d + e*x)/2]^2 - 2*b*Tan[(d + e*x)/2
]^3 + a*Tan[(d + e*x)/2]^4]] - 2*Sqrt[c]*Log[-1 + Tan[(d + e*x)/2]^2]*#1 + 2*Sqrt[c]*Log[#1 - 2*Sqrt[c]*Tan[(d
+ e*x)/2] - #1*Tan[(d + e*x)/2]^2 + Sqrt[a + 2*b*Tan[(d + e*x)/2] + (-2*a + 4*c)*Tan[(d + e*x)/2]^2 - 2*b*Tan
[(d + e*x)/2]^3 + a*Tan[(d + e*x)/2]^4]]*#1)/(b*Sqrt[c] - a*#1 + 2*c*#1 + #1^3) & ])*Sqrt[(a + c + (a - c)*Cos
[2*(d + e*x)] + b*Sin[2*(d + e*x)])/(1 + Cos[2*(d + e*x)])]*(-1 + Tan[(d + e*x)/2]^2)*(1 + Tan[(d + e*x)/2]^2)
*Sqrt[(a*(-1 + Tan[(d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d + e*x)/2]^2))/(
1 + Tan[(d + e*x)/2]^2)^2])/(2*Sqrt[a + c + (a - c)*Cos[2*(d + e*x)] + b*Sin[2*(d + e*x)]]*Sqrt[(-1 + Tan[(d +
e*x)/2]^2)^2]*Sqrt[a*(-1 + Tan[(d + e*x)/2]^2)^2 + 2*Tan[(d + e*x)/2]*(b + 2*c*Tan[(d + e*x)/2] - b*Tan[(d +
e*x)/2]^2)]))/((I*a + b - I*c)*((-I)*a + b + I*c)*c^2*e)

Maple [B] (warning: unable to verify)

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

Time = 2.37 (sec) , antiderivative size = 13066870, normalized size of antiderivative = 15123.69

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

[In]

int(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),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. 19865 vs. \(2 (794) = 1588\).

Time = 4.70 (sec) , antiderivative size = 39731, normalized size of antiderivative = 45.98 \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan ^{5}{\left (d + e x \right )}}{\left (a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(e*x+d)**5/(a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(3/2),x)

[Out]

Integral(tan(d + e*x)**5/(a + b*tan(d + e*x) + c*tan(d + e*x)**2)**(3/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^5(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^5}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]

[In]

int(tan(d + e*x)^5/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(3/2),x)

[Out]

int(tan(d + e*x)^5/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(3/2), x)