3.62 \(\int \frac{x^2}{(a+b \tan (c+d \sqrt [3]{x}))^2} \, dx\)

Optimal. Leaf size=1691 \[ \text{result too large to display} \]

[Out]

((-6*I)*b^2*x^(8/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(8/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^
2) + (24*b^2*x^(7/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(8
/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - ((6*I)*b^2*x^(8/3)*L
og[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((84*I)*b^2*x^2*PolyLog[2, -(((a
- I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (24*b*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (24*b^2*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (252*b^2*x^(5/3)*PolyLog[3, -(((a - I*b)*E^((2*I)*(
c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (84*b*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - ((84*I)*b^2*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a^2 + b^2)^2*d^3) + ((630*I)*b^2*x^(4/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/
(a + I*b))])/((a^2 + b^2)^2*d^5) - (252*b*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)
)])/((I*a - b)*(a - I*b)^2*d^4) + (252*b^2*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b
))])/((a^2 + b^2)^2*d^4) - (1260*b^2*x*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 +
b^2)^2*d^6) - (630*b*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a
+ I*b)*d^5) + ((630*I)*b^2*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^5) - ((1890*I)*b^2*x^(2/3)*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^
2*d^7) + (1260*b*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6)
- (1260*b^2*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^6) + (1890*b^2
*x^(1/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) + (1890*b*x^(2/3)
*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^7) - ((1890*I)*b^2*x
^(2/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^7) + ((945*I)*b^2*Poly
Log[8, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9) - (1890*b*x^(1/3)*PolyLog[8, -
(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^8) + (1890*b^2*x^(1/3)*PolyLog[8,
-(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) - (945*b*PolyLog[9, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^9) + ((945*I)*b^2*PolyLog[9, -(((a - I*b)*E^((2*
I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9)

________________________________________________________________________________________

Rubi [A]  time = 2.88851, antiderivative size = 1691, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3747, 3734, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Tan[c + d*x^(1/3)])^2,x]

[Out]

((-6*I)*b^2*x^(8/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(8/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^
2) + (24*b^2*x^(7/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(8
/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - ((6*I)*b^2*x^(8/3)*L
og[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((84*I)*b^2*x^2*PolyLog[2, -(((a
- I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (24*b*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (24*b^2*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (252*b^2*x^(5/3)*PolyLog[3, -(((a - I*b)*E^((2*I)*(
c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (84*b*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - ((84*I)*b^2*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a^2 + b^2)^2*d^3) + ((630*I)*b^2*x^(4/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/
(a + I*b))])/((a^2 + b^2)^2*d^5) - (252*b*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)
)])/((I*a - b)*(a - I*b)^2*d^4) + (252*b^2*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b
))])/((a^2 + b^2)^2*d^4) - (1260*b^2*x*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 +
b^2)^2*d^6) - (630*b*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a
+ I*b)*d^5) + ((630*I)*b^2*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^5) - ((1890*I)*b^2*x^(2/3)*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^
2*d^7) + (1260*b*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6)
- (1260*b^2*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^6) + (1890*b^2
*x^(1/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) + (1890*b*x^(2/3)
*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^7) - ((1890*I)*b^2*x
^(2/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^7) + ((945*I)*b^2*Poly
Log[8, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9) - (1890*b*x^(1/3)*PolyLog[8, -
(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^8) + (1890*b^2*x^(1/3)*PolyLog[8,
-(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) - (945*b*PolyLog[9, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^9) + ((945*I)*b^2*PolyLog[9, -(((a - I*b)*E^((2*
I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9)

Rule 3747

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 3734

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(
c + d*x)^m, (1/(a - I*b) - (2*I*b)/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x))))^(-n), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]

Rule 2185

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2191

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p +
1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^8}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{x^8}{(a-i b)^2}-\frac{4 b^2 x^8}{(i a+b)^2 \left (i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}\right )^2}+\frac{4 b x^8}{(a-i b)^2 \left (i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{x^3}{3 (a-i b)^2}+\frac{(12 b) \operatorname{Subst}\left (\int \frac{x^8}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{x^8}{\left (i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{(i a+b)^2}\\ &=\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}+\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{x^8}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2}-\frac{(12 b) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^8}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^8}{\left (i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}\\ &=-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^8}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a+i b)^2 (i a+b)}-\frac{(48 b) \operatorname{Subst}\left (\int x^7 \log \left (1+\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}+\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int \frac{x^7}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{(168 b) \operatorname{Subst}\left (\int x^6 \text{Li}_2\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{\left (48 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^7}{i a \left (1+\frac{i b}{a}\right )+i a \left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b) (a+i b)^2 d}+\frac{\left (48 i b^2\right ) \operatorname{Subst}\left (\int x^7 \log \left (1+\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{(504 b) \operatorname{Subst}\left (\int x^5 \text{Li}_3\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{\left (168 b^2\right ) \operatorname{Subst}\left (\int x^6 \log \left (1+\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{\left (168 b^2\right ) \operatorname{Subst}\left (\int x^6 \text{Li}_2\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{(1260 b) \operatorname{Subst}\left (\int x^4 \text{Li}_4\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{\left (504 i b^2\right ) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{\left (504 i b^2\right ) \operatorname{Subst}\left (\int x^5 \text{Li}_3\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^3}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{(2520 b) \operatorname{Subst}\left (\int x^3 \text{Li}_5\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d^5}-\frac{\left (1260 b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{\left (1260 b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_4\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^4}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{(3780 b) \operatorname{Subst}\left (\int x^2 \text{Li}_6\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{\left (2520 i b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{\left (2520 i b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_5\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^5}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{1260 b^2 x \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{1260 b^2 x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{1890 b x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac{(3780 b) \operatorname{Subst}\left (\int x \text{Li}_7\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d^7}+\frac{\left (3780 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{\left (3780 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_6\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^6}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{1260 b^2 x \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{1890 i b^2 x^{2/3} \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{1260 b^2 x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{1890 b x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac{1890 i b^2 x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}-\frac{1890 b \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^8}+\frac{(1890 b) \operatorname{Subst}\left (\int \text{Li}_8\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^8}+\frac{\left (3780 i b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_6\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{\left (3780 i b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_7\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^7}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{1260 b^2 x \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{1890 i b^2 x^{2/3} \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{1260 b^2 x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}+\frac{1890 b x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac{1890 i b^2 x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}-\frac{1890 b \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^8}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}-\frac{(945 b) \operatorname{Subst}\left (\int \frac{\text{Li}_8\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{(a-i b)^2 (a+i b) d^9}-\frac{\left (1890 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_7\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^8}-\frac{\left (1890 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_8\left (-\frac{\left (1-\frac{i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac{i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^8}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{1260 b^2 x \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{1890 i b^2 x^{2/3} \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{1260 b^2 x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}+\frac{1890 b x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac{1890 i b^2 x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}-\frac{1890 b \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^8}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}-\frac{945 b \text{Li}_9\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^9}+\frac{\left (945 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_7\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^9}+\frac{\left (945 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_8\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{\left (a^2+b^2\right )^2 d^9}\\ &=-\frac{6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac{x^3}{3 (a-i b)^2}+\frac{4 b x^3}{3 (i a-b) (a-i b)^2}-\frac{4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac{24 b^2 x^{7/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{6 b x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac{6 i b^2 x^{8/3} \log \left (1+\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac{84 i b^2 x^2 \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{24 b x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac{24 b^2 x^{7/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac{84 b x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac{84 i b^2 x^2 \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{252 b x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac{252 b^2 x^{5/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac{1260 b^2 x \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac{630 b x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac{630 i b^2 x^{4/3} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac{1890 i b^2 x^{2/3} \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{1260 b x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac{1260 b^2 x \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}+\frac{1890 b x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac{1890 i b^2 x^{2/3} \text{Li}_7\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac{945 i b^2 \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9}-\frac{1890 b \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^8}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_8\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}-\frac{945 b \text{Li}_9\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^9}+\frac{945 i b^2 \text{Li}_9\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9}\\ \end{align*}

Mathematica [A]  time = 4.98868, size = 1136, normalized size = 0.67 \[ \frac{\frac{(a-i b)^2 (a+i b) (a \cos (c)-b \sin (c)) x^3}{a \cos (c)+b \sin (c)}+\frac{9 (a-i b)^2 (a+i b) b^2 \sin \left (d \sqrt [3]{x}\right ) x^{8/3}}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt [3]{x}\right )+b \sin \left (c+d \sqrt [3]{x}\right )\right )}-\frac{i b \left (4 a (a+i b) (i a+b) x^3 d^9+18 (a+i b) b (i a+b) x^{8/3} d^8+18 a (a-i b) \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) x^{8/3} \log \left (\frac{e^{-2 i \left (c+d \sqrt [3]{x}\right )} (a+i b)}{a-i b}+1\right ) d^8+72 (a-i b) b \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) x^{7/3} \log \left (\frac{e^{-2 i \left (c+d \sqrt [3]{x}\right )} (a+i b)}{a-i b}+1\right ) d^7+63 b (i a+b) \left (b \left (-1+e^{2 i c}\right )+i a \left (1+e^{2 i c}\right )\right ) \left (-4 i x^2 \text{PolyLog}\left (2,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^6-12 x^{5/3} \text{PolyLog}\left (3,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^5+15 i \left (2 x^{4/3} \text{PolyLog}\left (4,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^4-4 i x \text{PolyLog}\left (5,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^3-6 x^{2/3} \text{PolyLog}\left (6,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^2+6 i \sqrt [3]{x} \text{PolyLog}\left (7,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text{PolyLog}\left (8,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )\right )+9 a (a-i b) \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (8 i x^{7/3} \text{PolyLog}\left (2,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^7+28 x^2 \text{PolyLog}\left (3,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^6-84 i x^{5/3} \text{PolyLog}\left (4,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^5-105 \left (2 x^{4/3} \text{PolyLog}\left (5,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^4-4 i x \text{PolyLog}\left (6,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^3-6 x^{2/3} \text{PolyLog}\left (7,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^2+6 i \sqrt [3]{x} \text{PolyLog}\left (8,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text{PolyLog}\left (9,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )\right )\right )}{d^9 \left (-e^{2 i c} b+b-i a \left (1+e^{2 i c}\right )\right )}}{3 (a-i b)^3 (a+i b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Tan[c + d*x^(1/3)])^2,x]

[Out]

(((-I)*b*(18*(a + I*b)*b*(I*a + b)*d^8*x^(8/3) + 4*a*(a + I*b)*(I*a + b)*d^9*x^3 + 72*(a - I*b)*b*d^7*((-I)*b*
(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(7/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 1
8*a*(a - I*b)*d^8*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(8/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2
*I)*(c + d*x^(1/3))))] + 63*b*(I*a + b)*(b*(-1 + E^((2*I)*c)) + I*a*(1 + E^((2*I)*c)))*((-4*I)*d^6*x^2*PolyLog
[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 12*d^5*x^(5/3)*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*
I)*(c + d*x^(1/3))))] + (15*I)*(2*d^4*x^(4/3)*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (
4*I)*d^3*x*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*PolyLog[6, (-a - I*b)/
((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (6*I)*d*x^(1/3)*PolyLog[7, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1
/3))))] + 3*PolyLog[8, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))])) + 9*a*(a - I*b)*((-I)*b*(-1 + E^((2
*I)*c)) + a*(1 + E^((2*I)*c)))*((8*I)*d^7*x^(7/3)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]
+ 28*d^6*x^2*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (84*I)*d^5*x^(5/3)*PolyLog[4, (-a
- I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 105*(2*d^4*x^(4/3)*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*
(c + d*x^(1/3))))] - (4*I)*d^3*x*PolyLog[6, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*
PolyLog[7, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (6*I)*d*x^(1/3)*PolyLog[8, (-a - I*b)/((a - I*b
)*E^((2*I)*(c + d*x^(1/3))))] + 3*PolyLog[9, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]))))/(d^9*(b - b
*E^((2*I)*c) - I*a*(1 + E^((2*I)*c)))) + ((a - I*b)^2*(a + I*b)*x^3*(a*Cos[c] - b*Sin[c]))/(a*Cos[c] + b*Sin[c
]) + (9*(a - I*b)^2*(a + I*b)*b^2*x^(8/3)*Sin[d*x^(1/3)])/(d*(a*Cos[c] + b*Sin[c])*(a*Cos[c + d*x^(1/3)] + b*S
in[c + d*x^(1/3)])))/(3*(a - I*b)^3*(a + I*b)^2)

________________________________________________________________________________________

Maple [F]  time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\tan \left ( c+d\sqrt [3]{x} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

[Out]

int(x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

________________________________________________________________________________________

Maxima [B]  time = 18.3881, size = 11051, normalized size = 6.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*((2*a*b*log(b*tan(d*x^(1/3) + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*x^(1/3) + c)^2 + 1)/(a^4 + 2*a
^2*b^2 + b^4) + (a^2 - b^2)*(d*x^(1/3) + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x^(
1/3) + c)))*c^8 + ((35*a^3 - 35*I*a^2*b + 35*a*b^2 - 35*I*b^3)*(d*x^(1/3) + c)^9 - (315*a^3 - 315*I*a^2*b + 31
5*a*b^2 - 315*I*b^3)*(d*x^(1/3) + c)^8*c + (1260*a^3 - 1260*I*a^2*b + 1260*a*b^2 - 1260*I*b^3)*(d*x^(1/3) + c)
^7*c^2 - (2940*a^3 - 2940*I*a^2*b + 2940*a*b^2 - 2940*I*b^3)*(d*x^(1/3) + c)^6*c^3 + (4410*a^3 - 4410*I*a^2*b
+ 4410*a*b^2 - 4410*I*b^3)*(d*x^(1/3) + c)^5*c^4 - (4410*a^3 - 4410*I*a^2*b + 4410*a*b^2 - 4410*I*b^3)*(d*x^(1
/3) + c)^4*c^5 + (2940*a^3 - 2940*I*a^2*b + 2940*a*b^2 - 2940*I*b^3)*(d*x^(1/3) + c)^3*c^6 - (1260*a^3 - 1260*
I*a^2*b + 1260*a*b^2 - 1260*I*b^3)*(d*x^(1/3) + c)^2*c^7 - (2520*(I*a*b^2 + b^3)*c^7*cos(2*d*x^(1/3) + 2*c) -
(2520*a*b^2 - 2520*I*b^3)*c^7*sin(2*d*x^(1/3) + 2*c) + 2520*(I*a*b^2 - b^3)*c^7)*arctan2(-b*cos(2*d*x^(1/3) +
2*c) + a*sin(2*d*x^(1/3) + 2*c) + b, a*cos(2*d*x^(1/3) + 2*c) + b*sin(2*d*x^(1/3) + 2*c) + a) + ((-10080*I*a^2
*b + 10080*a*b^2)*(d*x^(1/3) + c)^8 + (-23040*I*a*b^2 + 23040*b^3 + (46080*I*a^2*b - 46080*a*b^2)*c)*(d*x^(1/3
) + c)^7 + ((-94080*I*a^2*b + 94080*a*b^2)*c^2 - 94080*(-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^6 + ((112896*I*a^2*
b - 112896*a*b^2)*c^3 - 169344*(I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^5 + ((-88200*I*a^2*b + 88200*a*b^2)*c^4 -
176400*(-I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^4 + ((47040*I*a^2*b - 47040*a*b^2)*c^5 - 117600*(I*a*b^2 - b^3)*c
^4)*(d*x^(1/3) + c)^3 + ((-17640*I*a^2*b + 17640*a*b^2)*c^6 - 52920*(-I*a*b^2 + b^3)*c^5)*(d*x^(1/3) + c)^2 +
((5040*I*a^2*b - 5040*a*b^2)*c^7 - 17640*(I*a*b^2 - b^3)*c^6)*(d*x^(1/3) + c) + ((-10080*I*a^2*b - 10080*a*b^2
)*(d*x^(1/3) + c)^8 + (-23040*I*a*b^2 - 23040*b^3 + (46080*I*a^2*b + 46080*a*b^2)*c)*(d*x^(1/3) + c)^7 + ((-94
080*I*a^2*b - 94080*a*b^2)*c^2 - 94080*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^6 + ((112896*I*a^2*b + 112896*a*b^2
)*c^3 - 169344*(I*a*b^2 + b^3)*c^2)*(d*x^(1/3) + c)^5 + ((-88200*I*a^2*b - 88200*a*b^2)*c^4 - 176400*(-I*a*b^2
- b^3)*c^3)*(d*x^(1/3) + c)^4 + ((47040*I*a^2*b + 47040*a*b^2)*c^5 - 117600*(I*a*b^2 + b^3)*c^4)*(d*x^(1/3) +
c)^3 + ((-17640*I*a^2*b - 17640*a*b^2)*c^6 - 52920*(-I*a*b^2 - b^3)*c^5)*(d*x^(1/3) + c)^2 + ((5040*I*a^2*b +
5040*a*b^2)*c^7 - 17640*(I*a*b^2 + b^3)*c^6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (10080*(a^2*b - I*a*b^
2)*(d*x^(1/3) + c)^8 + (23040*a*b^2 - 23040*I*b^3 - 46080*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^7 + (94080*(a^2
*b - I*a*b^2)*c^2 - (94080*a*b^2 - 94080*I*b^3)*c)*(d*x^(1/3) + c)^6 - (112896*(a^2*b - I*a*b^2)*c^3 - (169344
*a*b^2 - 169344*I*b^3)*c^2)*(d*x^(1/3) + c)^5 + (88200*(a^2*b - I*a*b^2)*c^4 - (176400*a*b^2 - 176400*I*b^3)*c
^3)*(d*x^(1/3) + c)^4 - (47040*(a^2*b - I*a*b^2)*c^5 - (117600*a*b^2 - 117600*I*b^3)*c^4)*(d*x^(1/3) + c)^3 +
(17640*(a^2*b - I*a*b^2)*c^6 - (52920*a*b^2 - 52920*I*b^3)*c^5)*(d*x^(1/3) + c)^2 - (5040*(a^2*b - I*a*b^2)*c^
7 - (17640*a*b^2 - 17640*I*b^3)*c^6)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*arctan2((2*a*b*cos(2*d*x^(1/3) +
2*c) - (a^2 - b^2)*sin(2*d*x^(1/3) + 2*c))/(a^2 + b^2), (2*a*b*sin(2*d*x^(1/3) + 2*c) + a^2 + b^2 + (a^2 - b^
2)*cos(2*d*x^(1/3) + 2*c))/(a^2 + b^2)) + ((35*a^3 - 105*I*a^2*b - 105*a*b^2 + 35*I*b^3)*(d*x^(1/3) + c)^9 + (
-630*I*a*b^2 - 630*b^3 - (315*a^3 - 945*I*a^2*b - 945*a*b^2 + 315*I*b^3)*c)*(d*x^(1/3) + c)^8 - 5040*(-I*a*b^2
- b^3)*(d*x^(1/3) + c)*c^7 + ((1260*a^3 - 3780*I*a^2*b - 3780*a*b^2 + 1260*I*b^3)*c^2 - 5040*(-I*a*b^2 - b^3)
*c)*(d*x^(1/3) + c)^7 - ((2940*a^3 - 8820*I*a^2*b - 8820*a*b^2 + 2940*I*b^3)*c^3 + 17640*(I*a*b^2 + b^3)*c^2)*
(d*x^(1/3) + c)^6 + ((4410*a^3 - 13230*I*a^2*b - 13230*a*b^2 + 4410*I*b^3)*c^4 - 35280*(-I*a*b^2 - b^3)*c^3)*(
d*x^(1/3) + c)^5 - ((4410*a^3 - 13230*I*a^2*b - 13230*a*b^2 + 4410*I*b^3)*c^5 + 44100*(I*a*b^2 + b^3)*c^4)*(d*
x^(1/3) + c)^4 + ((2940*a^3 - 8820*I*a^2*b - 8820*a*b^2 + 2940*I*b^3)*c^6 - 35280*(-I*a*b^2 - b^3)*c^5)*(d*x^(
1/3) + c)^3 - ((1260*a^3 - 3780*I*a^2*b - 3780*a*b^2 + 1260*I*b^3)*c^7 + 17640*(I*a*b^2 + b^3)*c^6)*(d*x^(1/3)
+ c)^2)*cos(2*d*x^(1/3) + 2*c) + ((-40320*I*a^2*b + 40320*a*b^2)*(d*x^(1/3) + c)^7 + (2520*I*a^2*b - 2520*a*b
^2)*c^7 + (-80640*I*a*b^2 + 80640*b^3 + (161280*I*a^2*b - 161280*a*b^2)*c)*(d*x^(1/3) + c)^6 - 8820*(I*a*b^2 -
b^3)*c^6 + ((-282240*I*a^2*b + 282240*a*b^2)*c^2 - 282240*(-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^5 + ((282240*I*
a^2*b - 282240*a*b^2)*c^3 - 423360*(I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^4 + ((-176400*I*a^2*b + 176400*a*b^2)*
c^4 - 352800*(-I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^3 + ((70560*I*a^2*b - 70560*a*b^2)*c^5 - 176400*(I*a*b^2 -
b^3)*c^4)*(d*x^(1/3) + c)^2 + ((-17640*I*a^2*b + 17640*a*b^2)*c^6 - 52920*(-I*a*b^2 + b^3)*c^5)*(d*x^(1/3) + c
) + ((-40320*I*a^2*b - 40320*a*b^2)*(d*x^(1/3) + c)^7 + (2520*I*a^2*b + 2520*a*b^2)*c^7 + (-80640*I*a*b^2 - 80
640*b^3 + (161280*I*a^2*b + 161280*a*b^2)*c)*(d*x^(1/3) + c)^6 - 8820*(I*a*b^2 + b^3)*c^6 + ((-282240*I*a^2*b
- 282240*a*b^2)*c^2 - 282240*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^5 + ((282240*I*a^2*b + 282240*a*b^2)*c^3 - 42
3360*(I*a*b^2 + b^3)*c^2)*(d*x^(1/3) + c)^4 + ((-176400*I*a^2*b - 176400*a*b^2)*c^4 - 352800*(-I*a*b^2 - b^3)*
c^3)*(d*x^(1/3) + c)^3 + ((70560*I*a^2*b + 70560*a*b^2)*c^5 - 176400*(I*a*b^2 + b^3)*c^4)*(d*x^(1/3) + c)^2 +
((-17640*I*a^2*b - 17640*a*b^2)*c^6 - 52920*(-I*a*b^2 - b^3)*c^5)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (4
0320*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^7 - 2520*(a^2*b - I*a*b^2)*c^7 + (80640*a*b^2 - 80640*I*b^3 - 161280*(a
^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^6 + (8820*a*b^2 - 8820*I*b^3)*c^6 + (282240*(a^2*b - I*a*b^2)*c^2 - (282240
*a*b^2 - 282240*I*b^3)*c)*(d*x^(1/3) + c)^5 - (282240*(a^2*b - I*a*b^2)*c^3 - (423360*a*b^2 - 423360*I*b^3)*c^
2)*(d*x^(1/3) + c)^4 + (176400*(a^2*b - I*a*b^2)*c^4 - (352800*a*b^2 - 352800*I*b^3)*c^3)*(d*x^(1/3) + c)^3 -
(70560*(a^2*b - I*a*b^2)*c^5 - (176400*a*b^2 - 176400*I*b^3)*c^4)*(d*x^(1/3) + c)^2 + (17640*(a^2*b - I*a*b^2)
*c^6 - (52920*a*b^2 - 52920*I*b^3)*c^5)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*dilog((I*a + b)*e^(2*I*d*x^(1
/3) + 2*I*c)/(-I*a + b)) - ((1260*a*b^2 - 1260*I*b^3)*c^7*cos(2*d*x^(1/3) + 2*c) + 1260*(I*a*b^2 + b^3)*c^7*si
n(2*d*x^(1/3) + 2*c) + (1260*a*b^2 + 1260*I*b^3)*c^7)*log((a^2 + b^2)*cos(2*d*x^(1/3) + 2*c)^2 + 4*a*b*sin(2*d
*x^(1/3) + 2*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^(1/3) + 2*c)) + (
5040*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^8 + (11520*a*b^2 + 11520*I*b^3 - 23040*(a^2*b + I*a*b^2)*c)*(d*x^(1/3)
+ c)^7 + (47040*(a^2*b + I*a*b^2)*c^2 - (47040*a*b^2 + 47040*I*b^3)*c)*(d*x^(1/3) + c)^6 - (56448*(a^2*b + I*a
*b^2)*c^3 - (84672*a*b^2 + 84672*I*b^3)*c^2)*(d*x^(1/3) + c)^5 + (44100*(a^2*b + I*a*b^2)*c^4 - (88200*a*b^2 +
88200*I*b^3)*c^3)*(d*x^(1/3) + c)^4 - (23520*(a^2*b + I*a*b^2)*c^5 - (58800*a*b^2 + 58800*I*b^3)*c^4)*(d*x^(1
/3) + c)^3 + (8820*(a^2*b + I*a*b^2)*c^6 - (26460*a*b^2 + 26460*I*b^3)*c^5)*(d*x^(1/3) + c)^2 - (2520*(a^2*b +
I*a*b^2)*c^7 - (8820*a*b^2 + 8820*I*b^3)*c^6)*(d*x^(1/3) + c) + (5040*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^8 + (
11520*a*b^2 - 11520*I*b^3 - 23040*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^7 + (47040*(a^2*b - I*a*b^2)*c^2 - (470
40*a*b^2 - 47040*I*b^3)*c)*(d*x^(1/3) + c)^6 - (56448*(a^2*b - I*a*b^2)*c^3 - (84672*a*b^2 - 84672*I*b^3)*c^2)
*(d*x^(1/3) + c)^5 + (44100*(a^2*b - I*a*b^2)*c^4 - (88200*a*b^2 - 88200*I*b^3)*c^3)*(d*x^(1/3) + c)^4 - (2352
0*(a^2*b - I*a*b^2)*c^5 - (58800*a*b^2 - 58800*I*b^3)*c^4)*(d*x^(1/3) + c)^3 + (8820*(a^2*b - I*a*b^2)*c^6 - (
26460*a*b^2 - 26460*I*b^3)*c^5)*(d*x^(1/3) + c)^2 - (2520*(a^2*b - I*a*b^2)*c^7 - (8820*a*b^2 - 8820*I*b^3)*c^
6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + ((5040*I*a^2*b + 5040*a*b^2)*(d*x^(1/3) + c)^8 + (11520*I*a*b^2 +
11520*b^3 + (-23040*I*a^2*b - 23040*a*b^2)*c)*(d*x^(1/3) + c)^7 + ((47040*I*a^2*b + 47040*a*b^2)*c^2 - 47040*
(I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^6 + ((-56448*I*a^2*b - 56448*a*b^2)*c^3 - 84672*(-I*a*b^2 - b^3)*c^2)*(d*x^
(1/3) + c)^5 + ((44100*I*a^2*b + 44100*a*b^2)*c^4 - 88200*(I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^4 + ((-23520*I*
a^2*b - 23520*a*b^2)*c^5 - 58800*(-I*a*b^2 - b^3)*c^4)*(d*x^(1/3) + c)^3 + ((8820*I*a^2*b + 8820*a*b^2)*c^6 -
26460*(I*a*b^2 + b^3)*c^5)*(d*x^(1/3) + c)^2 + ((-2520*I*a^2*b - 2520*a*b^2)*c^7 - 8820*(-I*a*b^2 - b^3)*c^6)*
(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*log(((a^2 + b^2)*cos(2*d*x^(1/3) + 2*c)^2 + 4*a*b*sin(2*d*x^(1/3) + 2
*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^(1/3) + 2*c))/(a^2 + b^2)) -
(1587600*a^2*b + 1587600*I*a*b^2 + 1587600*(a^2*b - I*a*b^2)*cos(2*d*x^(1/3) + 2*c) - (-1587600*I*a^2*b - 1587
600*a*b^2)*sin(2*d*x^(1/3) + 2*c))*polylog(9, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + (907200*I*a*b^
2 - 907200*b^3 + (3175200*I*a^2*b - 3175200*a*b^2)*(d*x^(1/3) + c) + (-1814400*I*a^2*b + 1814400*a*b^2)*c + (9
07200*I*a*b^2 + 907200*b^3 + (3175200*I*a^2*b + 3175200*a*b^2)*(d*x^(1/3) + c) + (-1814400*I*a^2*b - 1814400*a
*b^2)*c)*cos(2*d*x^(1/3) + 2*c) - (907200*a*b^2 - 907200*I*b^3 + 3175200*(a^2*b - I*a*b^2)*(d*x^(1/3) + c) - 1
814400*(a^2*b - I*a*b^2)*c)*sin(2*d*x^(1/3) + 2*c))*polylog(8, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b))
+ (3175200*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^2 + 1058400*(a^2*b + I*a*b^2)*c^2 + (1814400*a*b^2 + 1814400*I*b
^3 - 3628800*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c) - (1058400*a*b^2 + 1058400*I*b^3)*c + (3175200*(a^2*b - I*a*
b^2)*(d*x^(1/3) + c)^2 + 1058400*(a^2*b - I*a*b^2)*c^2 + (1814400*a*b^2 - 1814400*I*b^3 - 3628800*(a^2*b - I*a
*b^2)*c)*(d*x^(1/3) + c) - (1058400*a*b^2 - 1058400*I*b^3)*c)*cos(2*d*x^(1/3) + 2*c) + ((3175200*I*a^2*b + 317
5200*a*b^2)*(d*x^(1/3) + c)^2 + (1058400*I*a^2*b + 1058400*a*b^2)*c^2 + (1814400*I*a*b^2 + 1814400*b^3 + (-362
8800*I*a^2*b - 3628800*a*b^2)*c)*(d*x^(1/3) + c) - 1058400*(I*a*b^2 + b^3)*c)*sin(2*d*x^(1/3) + 2*c))*polylog(
7, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + ((-2116800*I*a^2*b + 2116800*a*b^2)*(d*x^(1/3) + c)^3 + (
423360*I*a^2*b - 423360*a*b^2)*c^3 + (-1814400*I*a*b^2 + 1814400*b^3 + (3628800*I*a^2*b - 3628800*a*b^2)*c)*(d
*x^(1/3) + c)^2 - 635040*(I*a*b^2 - b^3)*c^2 + ((-2116800*I*a^2*b + 2116800*a*b^2)*c^2 - 2116800*(-I*a*b^2 + b
^3)*c)*(d*x^(1/3) + c) + ((-2116800*I*a^2*b - 2116800*a*b^2)*(d*x^(1/3) + c)^3 + (423360*I*a^2*b + 423360*a*b^
2)*c^3 + (-1814400*I*a*b^2 - 1814400*b^3 + (3628800*I*a^2*b + 3628800*a*b^2)*c)*(d*x^(1/3) + c)^2 - 635040*(I*
a*b^2 + b^3)*c^2 + ((-2116800*I*a^2*b - 2116800*a*b^2)*c^2 - 2116800*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c))*cos(
2*d*x^(1/3) + 2*c) + (2116800*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^3 - 423360*(a^2*b - I*a*b^2)*c^3 + (1814400*a*
b^2 - 1814400*I*b^3 - 3628800*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^2 + (635040*a*b^2 - 635040*I*b^3)*c^2 + (21
16800*(a^2*b - I*a*b^2)*c^2 - (2116800*a*b^2 - 2116800*I*b^3)*c)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*poly
log(6, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - (1058400*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^4 + 132300
*(a^2*b + I*a*b^2)*c^4 + (1209600*a*b^2 + 1209600*I*b^3 - 2419200*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c)^3 - (26
4600*a*b^2 + 264600*I*b^3)*c^3 + (2116800*(a^2*b + I*a*b^2)*c^2 - (2116800*a*b^2 + 2116800*I*b^3)*c)*(d*x^(1/3
) + c)^2 - (846720*(a^2*b + I*a*b^2)*c^3 - (1270080*a*b^2 + 1270080*I*b^3)*c^2)*(d*x^(1/3) + c) + (1058400*(a^
2*b - I*a*b^2)*(d*x^(1/3) + c)^4 + 132300*(a^2*b - I*a*b^2)*c^4 + (1209600*a*b^2 - 1209600*I*b^3 - 2419200*(a^
2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^3 - (264600*a*b^2 - 264600*I*b^3)*c^3 + (2116800*(a^2*b - I*a*b^2)*c^2 - (21
16800*a*b^2 - 2116800*I*b^3)*c)*(d*x^(1/3) + c)^2 - (846720*(a^2*b - I*a*b^2)*c^3 - (1270080*a*b^2 - 1270080*I
*b^3)*c^2)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - ((-1058400*I*a^2*b - 1058400*a*b^2)*(d*x^(1/3) + c)^4 + (
-132300*I*a^2*b - 132300*a*b^2)*c^4 + (-1209600*I*a*b^2 - 1209600*b^3 + (2419200*I*a^2*b + 2419200*a*b^2)*c)*(
d*x^(1/3) + c)^3 - 264600*(-I*a*b^2 - b^3)*c^3 + ((-2116800*I*a^2*b - 2116800*a*b^2)*c^2 - 2116800*(-I*a*b^2 -
b^3)*c)*(d*x^(1/3) + c)^2 + ((846720*I*a^2*b + 846720*a*b^2)*c^3 - 1270080*(I*a*b^2 + b^3)*c^2)*(d*x^(1/3) +
c))*sin(2*d*x^(1/3) + 2*c))*polylog(5, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + ((423360*I*a^2*b - 42
3360*a*b^2)*(d*x^(1/3) + c)^5 + (-35280*I*a^2*b + 35280*a*b^2)*c^5 + (604800*I*a*b^2 - 604800*b^3 + (-1209600*
I*a^2*b + 1209600*a*b^2)*c)*(d*x^(1/3) + c)^4 - 88200*(-I*a*b^2 + b^3)*c^4 + ((1411200*I*a^2*b - 1411200*a*b^2
)*c^2 - 1411200*(I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^3 + ((-846720*I*a^2*b + 846720*a*b^2)*c^3 - 1270080*(-I*a*b
^2 + b^3)*c^2)*(d*x^(1/3) + c)^2 + ((264600*I*a^2*b - 264600*a*b^2)*c^4 - 529200*(I*a*b^2 - b^3)*c^3)*(d*x^(1/
3) + c) + ((423360*I*a^2*b + 423360*a*b^2)*(d*x^(1/3) + c)^5 + (-35280*I*a^2*b - 35280*a*b^2)*c^5 + (604800*I*
a*b^2 + 604800*b^3 + (-1209600*I*a^2*b - 1209600*a*b^2)*c)*(d*x^(1/3) + c)^4 - 88200*(-I*a*b^2 - b^3)*c^4 + ((
1411200*I*a^2*b + 1411200*a*b^2)*c^2 - 1411200*(I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^3 + ((-846720*I*a^2*b - 8467
20*a*b^2)*c^3 - 1270080*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^2 + ((264600*I*a^2*b + 264600*a*b^2)*c^4 - 52920
0*(I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (423360*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^5 -
35280*(a^2*b - I*a*b^2)*c^5 + (604800*a*b^2 - 604800*I*b^3 - 1209600*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^4 +
(88200*a*b^2 - 88200*I*b^3)*c^4 + (1411200*(a^2*b - I*a*b^2)*c^2 - (1411200*a*b^2 - 1411200*I*b^3)*c)*(d*x^(1
/3) + c)^3 - (846720*(a^2*b - I*a*b^2)*c^3 - (1270080*a*b^2 - 1270080*I*b^3)*c^2)*(d*x^(1/3) + c)^2 + (264600*
(a^2*b - I*a*b^2)*c^4 - (529200*a*b^2 - 529200*I*b^3)*c^3)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(4,
(I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + (141120*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^6 + 8820*(a^2*b +
I*a*b^2)*c^6 + (241920*a*b^2 + 241920*I*b^3 - 483840*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c)^5 - (26460*a*b^2 +
26460*I*b^3)*c^5 + (705600*(a^2*b + I*a*b^2)*c^2 - (705600*a*b^2 + 705600*I*b^3)*c)*(d*x^(1/3) + c)^4 - (56448
0*(a^2*b + I*a*b^2)*c^3 - (846720*a*b^2 + 846720*I*b^3)*c^2)*(d*x^(1/3) + c)^3 + (264600*(a^2*b + I*a*b^2)*c^4
- (529200*a*b^2 + 529200*I*b^3)*c^3)*(d*x^(1/3) + c)^2 - (70560*(a^2*b + I*a*b^2)*c^5 - (176400*a*b^2 + 17640
0*I*b^3)*c^4)*(d*x^(1/3) + c) + (141120*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^6 + 8820*(a^2*b - I*a*b^2)*c^6 + (24
1920*a*b^2 - 241920*I*b^3 - 483840*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^5 - (26460*a*b^2 - 26460*I*b^3)*c^5 +
(705600*(a^2*b - I*a*b^2)*c^2 - (705600*a*b^2 - 705600*I*b^3)*c)*(d*x^(1/3) + c)^4 - (564480*(a^2*b - I*a*b^2)
*c^3 - (846720*a*b^2 - 846720*I*b^3)*c^2)*(d*x^(1/3) + c)^3 + (264600*(a^2*b - I*a*b^2)*c^4 - (529200*a*b^2 -
529200*I*b^3)*c^3)*(d*x^(1/3) + c)^2 - (70560*(a^2*b - I*a*b^2)*c^5 - (176400*a*b^2 - 176400*I*b^3)*c^4)*(d*x^
(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + ((141120*I*a^2*b + 141120*a*b^2)*(d*x^(1/3) + c)^6 + (8820*I*a^2*b + 8820
*a*b^2)*c^6 + (241920*I*a*b^2 + 241920*b^3 + (-483840*I*a^2*b - 483840*a*b^2)*c)*(d*x^(1/3) + c)^5 - 26460*(I*
a*b^2 + b^3)*c^5 + ((705600*I*a^2*b + 705600*a*b^2)*c^2 - 705600*(I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^4 + ((-564
480*I*a^2*b - 564480*a*b^2)*c^3 - 846720*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^3 + ((264600*I*a^2*b + 264600*a
*b^2)*c^4 - 529200*(I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^2 + ((-70560*I*a^2*b - 70560*a*b^2)*c^5 - 176400*(-I*a
*b^2 - b^3)*c^4)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(3, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a
+ b)) + ((35*I*a^3 + 105*a^2*b - 105*I*a*b^2 - 35*b^3)*(d*x^(1/3) + c)^9 + (630*a*b^2 - 630*I*b^3 + (-315*I*a
^3 - 945*a^2*b + 945*I*a*b^2 + 315*b^3)*c)*(d*x^(1/3) + c)^8 - (5040*a*b^2 - 5040*I*b^3)*(d*x^(1/3) + c)*c^7 +
((1260*I*a^3 + 3780*a^2*b - 3780*I*a*b^2 - 1260*b^3)*c^2 - (5040*a*b^2 - 5040*I*b^3)*c)*(d*x^(1/3) + c)^7 + (
(-2940*I*a^3 - 8820*a^2*b + 8820*I*a*b^2 + 2940*b^3)*c^3 + (17640*a*b^2 - 17640*I*b^3)*c^2)*(d*x^(1/3) + c)^6
+ ((4410*I*a^3 + 13230*a^2*b - 13230*I*a*b^2 - 4410*b^3)*c^4 - (35280*a*b^2 - 35280*I*b^3)*c^3)*(d*x^(1/3) + c
)^5 + ((-4410*I*a^3 - 13230*a^2*b + 13230*I*a*b^2 + 4410*b^3)*c^5 + (44100*a*b^2 - 44100*I*b^3)*c^4)*(d*x^(1/3
) + c)^4 + ((2940*I*a^3 + 8820*a^2*b - 8820*I*a*b^2 - 2940*b^3)*c^6 - (35280*a*b^2 - 35280*I*b^3)*c^5)*(d*x^(1
/3) + c)^3 + ((-1260*I*a^3 - 3780*a^2*b + 3780*I*a*b^2 + 1260*b^3)*c^7 + (17640*a*b^2 - 17640*I*b^3)*c^6)*(d*x
^(1/3) + c)^2)*sin(2*d*x^(1/3) + 2*c))/(315*a^5 + 315*I*a^4*b + 630*a^3*b^2 + 630*I*a^2*b^3 + 315*a*b^4 + 315*
I*b^5 + (315*a^5 - 315*I*a^4*b + 630*a^3*b^2 - 630*I*a^2*b^3 + 315*a*b^4 - 315*I*b^5)*cos(2*d*x^(1/3) + 2*c) +
(315*I*a^5 + 315*a^4*b + 630*I*a^3*b^2 + 630*a^2*b^3 + 315*I*a*b^4 + 315*b^5)*sin(2*d*x^(1/3) + 2*c)))/d^9

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{b^{2} \tan \left (d x^{\frac{1}{3}} + c\right )^{2} + 2 \, a b \tan \left (d x^{\frac{1}{3}} + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/(b^2*tan(d*x^(1/3) + c)^2 + 2*a*b*tan(d*x^(1/3) + c) + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b \tan{\left (c + d \sqrt [3]{x} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(a + b*tan(c + d*x**(1/3)))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b \tan \left (d x^{\frac{1}{3}} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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