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

Optimal. Leaf size=1691 \[ \frac {4 b x^3}{3 (i a-b) (a-i b)^2}+\frac {x^3}{3 (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {6 b \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{8/3}}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{8/3}}{\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) (i a+b)^2 d \left (i a+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}-b\right )}+\frac {24 b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{7/3}}{\left (a^2+b^2\right )^2 d^2}+\frac {24 b \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{7/3}}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{7/3}}{\left (a^2+b^2\right )^2 d^2}-\frac {84 i b^2 \text {Li}_2\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{\left (a^2+b^2\right )^2 d^3}+\frac {84 b \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{(a-i b)^2 (a+i b) d^3}-\frac {84 i b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{\left (a^2+b^2\right )^2 d^3}+\frac {252 b^2 \text {Li}_3\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{\left (a^2+b^2\right )^2 d^4}-\frac {252 b \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{(i a-b) (a-i b)^2 d^4}+\frac {252 b^2 \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{\left (a^2+b^2\right )^2 d^4}+\frac {630 i b^2 \text {Li}_4\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{\left (a^2+b^2\right )^2 d^5}-\frac {630 b \text {Li}_5\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{(a-i b)^2 (a+i b) d^5}+\frac {630 i b^2 \text {Li}_5\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{\left (a^2+b^2\right )^2 d^5}-\frac {1260 b^2 \text {Li}_5\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^6}+\frac {1260 b \text {Li}_6\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{(i a-b) (a-i b)^2 d^6}-\frac {1260 b^2 \text {Li}_6\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^6}-\frac {1890 i b^2 \text {Li}_6\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{\left (a^2+b^2\right )^2 d^7}+\frac {1890 b \text {Li}_7\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{(a-i b)^2 (a+i b) d^7}-\frac {1890 i b^2 \text {Li}_7\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{\left (a^2+b^2\right )^2 d^7}+\frac {1890 b^2 \text {Li}_7\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{\left (a^2+b^2\right )^2 d^8}-\frac {1890 b \text {Li}_8\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{(i a-b) (a-i b)^2 d^8}+\frac {1890 b^2 \text {Li}_8\left (-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{\left (a^2+b^2\right )^2 d^8}+\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 {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} \]

[Out]

6*b^2*x^(8/3)/(a+I*b)/(I*a+b)^2/d/(I*a-b+(I*a+b)*exp(2*I*(c+d*x^(1/3))))+24*b*x^(7/3)*polylog(2,-(a-I*b)*exp(2
*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^2+84*b*x^2*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(
a-I*b)^2/(a+I*b)/d^3-252*b*x^(5/3)*polylog(4,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^4-24
*b^2*x^(7/3)*polylog(2,-(a-I*b)*exp(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)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^4+252*b^2*x^(5/3)*polylog(4,-(a-I*b)*exp(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)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^6-1260*b
^2*x*polylog(6,-(a-I*b)*exp(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)*ex
p(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^8+1890*b^2*x^(1/3)*polylog(8,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*
b))/(a^2+b^2)^2/d^8-945*b*polylog(9,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^9-6*I*b^2*x^(
8/3)/(a^2+b^2)^2/d-4/3*b^2*x^3/(a^2+b^2)^2+630*I*b^2*x^(4/3)*polylog(4,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)
)/(a^2+b^2)^2/d^5+630*I*b^2*x^(4/3)*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^5+24*b^2*
x^(7/3)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^2+1/3*x^3/(a-I*b)^2-630*b*x^(4/3)*polylog(5
,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^5+1260*b*x*polylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/
3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^6+1890*b*x^(2/3)*polylog(7,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^
2/(a+I*b)/d^7+945*I*b^2*polylog(8,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^9-1890*b*x^(1/3)*poly
log(8,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^8+945*I*b^2*polylog(9,-(a-I*b)*exp(2*I*(c+d
*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^9-84*I*b^2*x^2*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^
2/d^3-84*I*b^2*x^2*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^3-1890*I*b^2*x^(2/3)*polyl
og(6,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^7-1890*I*b^2*x^(2/3)*polylog(7,-(a-I*b)*exp(2*I*(c
+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^7+6*b*x^(8/3)*ln(1+(a-I*b)*exp(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)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d+4/3*b*x^3/(I*a-b)/(a-I*b)^2

________________________________________________________________________________________

Rubi [A]  time = 2.89, antiderivative size = 1691, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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 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]

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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 5.74, 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 {Li}_2\left (\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 {Li}_3\left (\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 {Li}_4\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^4-4 i x \text {Li}_5\left (\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 {Li}_6\left (\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 {Li}_7\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text {Li}_8\left (\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 {Li}_2\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^7+28 x^2 \text {Li}_3\left (\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 {Li}_4\left (\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 {Li}_5\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d^4-4 i x \text {Li}_6\left (\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 {Li}_7\left (\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 {Li}_8\left (\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text {Li}_9\left (\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)

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{b^{2} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 2 \, a b \tan \left (d x^{\frac {1}{3}} + c\right ) + a^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [F]  time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )^{2}}\, dx \]

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)

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maxima [B]  time = 5.43, size = 8147, normalized size = 4.82 \[ \text {result too large to display} \]

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 - 2520*b^3)*c^7*cos(2*d*x^(1/3) + 2
*c) + 2520*(a*b^2 - I*b^3)*c^7*sin(2*d*x^(1/3) + 2*c) + (-2520*I*a*b^2 + 2520*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) + ((-1008
0*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 - 94080*b^3)*c)*(d*x^(1/3) + c)^6 + ((1
12896*I*a^2*b - 112896*a*b^2)*c^3 + (-169344*I*a*b^2 + 169344*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 - 176400*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 + 117600*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
- 52920*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 + 17640*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 + ((-94080*I*a^2*b - 94080*a*b^2)*c^2 + (94080*I*a*b^2 + 94080*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 - 169344*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 + 176400*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 - 117600*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 + 52920*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 - 17640*b^3)*c^6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + 24*(420*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^8 + 9
60*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^7 + 3920*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c
)*(d*x^(1/3) + c)^6 - 2352*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((a^2*b
- I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^4 - 980*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c
^4)*(d*x^(1/3) + c)^3 + 735*((a^2*b - I*a*b^2)*c^6 - 3*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 105*(2*(a^2*b
- I*a*b^2)*c^7 - 7*(a*b^2 - 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 + 5040*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
+ 5040*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
- 17640*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 + 35280*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 - 44100*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 + (352
80*I*a*b^2 + 35280*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 - 17640*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 + 8820*b^3)*c^6 + ((-282240*I*a^2*b + 282240*a*b^2)*c^2 + (282240*I*
a*b^2 - 282240*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 + 423360*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 - 352800*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 + 176400*b^3)*c^4)*(d*x^(1/3) + c)^2 + ((-1
7640*I*a^2*b + 17640*a*b^2)*c^6 + (52920*I*a*b^2 - 52920*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 - 80640*b^3 + (161280*I*a^2*b + 1
61280*a*b^2)*c)*(d*x^(1/3) + c)^6 + (-8820*I*a*b^2 - 8820*b^3)*c^6 + ((-282240*I*a^2*b - 282240*a*b^2)*c^2 + (
282240*I*a*b^2 + 282240*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 -
423360*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 + 352800*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 - 176400*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 + 52920*b^3)*c^5)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) +
2*c) + 1260*(32*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^7 - 2*(a^2*b - I*a*b^2)*c^7 + 64*(a*b^2 - I*b^3 - 2*(a^2*b
- I*a*b^2)*c)*(d*x^(1/3) + c)^6 + 7*(a*b^2 - I*b^3)*c^6 + 224*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x
^(1/3) + c)^5 - 112*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^4 + 140*((a^2*b - I*a*b^
2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^3 - 28*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c^4)*(d*x^
(1/3) + c)^2 + 14*((a^2*b - I*a*b^2)*c^6 - 3*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*dil
og((I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - (1260*(a*b^2 - I*b^3)*c^7*cos(2*d*x^(1/3) + 2*c) - (-1260
*I*a*b^2 - 1260*b^3)*c^7*sin(2*d*x^(1/3) + 2*c) + 1260*(a*b^2 + 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 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)
*(d*x^(1/3) + c)^7 + 47040*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*x^(1/3) + c)^6 - 28224*(2*(a^2*b + I
*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 44100*((a^2*b + I*a*b^2)*c^4 - 2*(a*b^2 + I*b^3)*c^3)
*(d*x^(1/3) + c)^4 - 11760*(2*(a^2*b + I*a*b^2)*c^5 - 5*(a*b^2 + I*b^3)*c^4)*(d*x^(1/3) + c)^3 + 8820*((a^2*b
+ I*a*b^2)*c^6 - 3*(a*b^2 + I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 1260*(2*(a^2*b + I*a*b^2)*c^7 - 7*(a*b^2 + I*b^3)*
c^6)*(d*x^(1/3) + c) + 12*(420*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^8 + 960*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*
c)*(d*x^(1/3) + c)^7 + 3920*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^6 - 2352*(2*(a^2*b - I
*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*
(d*x^(1/3) + c)^4 - 980*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((a^2*b - I*
a*b^2)*c^6 - 3*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 105*(2*(a^2*b - I*a*b^2)*c^7 - 7*(a*b^2 - 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 + 11
520*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 - 47040*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 + 84672*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 - 88200*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 + 58800*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 - 26460*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 + 8820*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 - 1587600*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 + (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)*cos(2*d*x^(1/3) + 2*c) - 453600*(2*a*b^2 - 2*I*b^3 + 7*(a^2*b - I*a*b
^2)*(d*x^(1/3) + c) - 4*(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 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c) - 1058400*(a*b^2 + I*b^3)*c + 151200*(21*(a^2*b - I*a*b^2
)*(d*x^(1/3) + c)^2 + 7*(a^2*b - I*a*b^2)*c^2 + 12*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c) - 7
*(a*b^2 - I*b^3)*c)*cos(2*d*x^(1/3) + 2*c) + ((3175200*I*a^2*b + 3175200*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 + (-3628800*I*a^2*b - 3628800*a*b^2)*c)*(d*x^(1/3
) + c) + (-1058400*I*a*b^2 - 1058400*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 + 635040*b^3)*c^2 + ((-2116800*I*a^2*b + 2116800*a*b^2)*c^2 + (2116800*I*a*b^2 - 2116800*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 - 635040*b^
3)*c^2 + ((-2116800*I*a^2*b - 2116800*a*b^2)*c^2 + (2116800*I*a*b^2 + 2116800*b^3)*c)*(d*x^(1/3) + c))*cos(2*d
*x^(1/3) + 2*c) + 30240*(70*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^3 - 14*(a^2*b - I*a*b^2)*c^3 + 60*(a*b^2 - I*b^3
- 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^2 + 21*(a*b^2 - I*b^3)*c^2 + 70*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*
b^3)*c)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(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 + I*b^3 - 2*(a^2*
b + I*a*b^2)*c)*(d*x^(1/3) + c)^3 - 264600*(a*b^2 + I*b^3)*c^3 + 2116800*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b
^3)*c)*(d*x^(1/3) + c)^2 - 423360*(2*(a^2*b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*x^(1/3) + c) + 3780*(28
0*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^4 + 35*(a^2*b - I*a*b^2)*c^4 + 320*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)
*(d*x^(1/3) + c)^3 - 70*(a*b^2 - I*b^3)*c^3 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^
2 - 112*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - 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 - 12096
00*b^3 + (2419200*I*a^2*b + 2419200*a*b^2)*c)*(d*x^(1/3) + c)^3 + (264600*I*a*b^2 + 264600*b^3)*c^3 + ((-21168
00*I*a^2*b - 2116800*a*b^2)*c^2 + (2116800*I*a*b^2 + 2116800*b^3)*c)*(d*x^(1/3) + c)^2 + ((846720*I*a^2*b + 84
6720*a*b^2)*c^3 + (-1270080*I*a*b^2 - 1270080*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 - 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 - 88200*b^3)*c^4 + ((1411200*I*a^2*b - 1411200*a*b^2)*c^2 + (-1411200*I*a*b^2 + 1411200*
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 - 1270080*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 + 529200*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 + 6048
00*b^3 + (-1209600*I*a^2*b - 1209600*a*b^2)*c)*(d*x^(1/3) + c)^4 + (88200*I*a*b^2 + 88200*b^3)*c^4 + ((1411200
*I*a^2*b + 1411200*a*b^2)*c^2 + (-1411200*I*a*b^2 - 1411200*b^3)*c)*(d*x^(1/3) + c)^3 + ((-846720*I*a^2*b - 84
6720*a*b^2)*c^3 + (1270080*I*a*b^2 + 1270080*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 - 529200*b^3)*c^3)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - 2520*(168*(a^2*b - I*a*b^2)*
(d*x^(1/3) + c)^5 - 14*(a^2*b - I*a*b^2)*c^5 + 240*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^4 +
35*(a*b^2 - I*b^3)*c^4 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^3 - 168*(2*(a^2*b -
I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^2 + 105*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - 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 + I*b^3 - 2*(a^2*b + I*a*b^2)
*c)*(d*x^(1/3) + c)^5 - 26460*(a*b^2 + I*b^3)*c^5 + 705600*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*x^(1
/3) + c)^4 - 282240*(2*(a^2*b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*x^(1/3) + c)^3 + 264600*((a^2*b + I*a
*b^2)*c^4 - 2*(a*b^2 + I*b^3)*c^3)*(d*x^(1/3) + c)^2 - 35280*(2*(a^2*b + I*a*b^2)*c^5 - 5*(a*b^2 + I*b^3)*c^4)
*(d*x^(1/3) + c) + 1260*(112*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^6 + 7*(a^2*b - I*a*b^2)*c^6 + 192*(a*b^2 - I*b^
3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^5 - 21*(a*b^2 - I*b^3)*c^5 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 -
I*b^3)*c)*(d*x^(1/3) + c)^4 - 224*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^3 + 210*((
a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^2 - 28*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - 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 - 26460*b^3)*c^5 + ((705600*I*a^2*b + 705600*a*b^2)*c^2 + (-705600*I*a*b^2 - 705600*b^3)*
c)*(d*x^(1/3) + c)^4 + ((-564480*I*a^2*b - 564480*a*b^2)*c^3 + (846720*I*a*b^2 + 846720*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 - 529200*b^3)*c^3)*(d*x^(1/3) + c)^2 + ((-7056
0*I*a^2*b - 70560*a*b^2)*c^5 + (176400*I*a*b^2 + 176400*b^3)*c^4)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*pol
ylog(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 - 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 - 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
- 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 -
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 -
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 - 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 - 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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2} \,d x \]

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)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

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]

Exception raised: HeuristicGCDFailed

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