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

Optimal. Leaf size=1155 \[ \text{result too large to display} \]

[Out]

((-6*I)*b^2*x^(5/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(5/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (
15*b^2*x^(4/3)*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^(5/3)*Lo
g[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^(5/3)*Log[1 +
((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((30*I)*b^2*x*PolyLog[2, -(((a - I*b)*E
^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (15*b*x^(4/3)*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) - (15*b^2*x^(4/3)*PolyLog[2, -(((a - I*b)*E^((2*I)*(c
+ d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (45*b^2*x^(2/3)*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(
1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (30*b*x*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) - ((30*I)*b^2*x*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))
])/((a^2 + b^2)^2*d^3) + ((45*I)*b^2*x^(1/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(
(a^2 + b^2)^2*d^5) - (45*b*x^(2/3)*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) + (45*b^2*x^(2/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^4) - (45*b^2*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6) - (45
*b*x^(1/3)*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) + ((45*
I)*b^2*x^(1/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) + (45*b*Pol
yLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(I*a - b)*(a - I*b)^2*d^6) - (45*b^2*PolyLog[6
, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6)

________________________________________________________________________________________

Rubi [A]  time = 2.0827, antiderivative size = 1155, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {3747, 3734, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ -\frac{2 x^2 b^2}{\left (a^2+b^2\right )^2}-\frac{6 i x^{5/3} b^2}{\left (a^2+b^2\right )^2 d}+\frac{6 x^{5/3} b^2}{(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{6 i x^{5/3} \log \left (\frac{e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) b^2}{\left (a^2+b^2\right )^2 d}+\frac{15 x^{4/3} \log \left (\frac{e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) b^2}{\left (a^2+b^2\right )^2 d^2}-\frac{15 x^{4/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^2}-\frac{30 i x \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}-\frac{30 i x \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}+\frac{45 x^{2/3} \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^4}+\frac{45 x^{2/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^4}+\frac{45 i \sqrt [3]{x} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^5}+\frac{45 i \sqrt [3]{x} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{\left (a^2+b^2\right )^2 d^5}-\frac{45 \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{2 \left (a^2+b^2\right )^2 d^6}-\frac{45 \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b^2}{2 \left (a^2+b^2\right )^2 d^6}+\frac{2 x^2 b}{(i a-b) (a-i b)^2}+\frac{6 x^{5/3} \log \left (\frac{e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) b}{(a-i b)^2 (a+i b) d}+\frac{15 x^{4/3} \text{Li}_2\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b}{(i a-b) (a-i b)^2 d^2}+\frac{30 x \text{Li}_3\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b}{(a-i b)^2 (a+i b) d^3}-\frac{45 x^{2/3} \text{Li}_4\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b}{(i a-b) (a-i b)^2 d^4}-\frac{45 \sqrt [3]{x} \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b}{(a-i b)^2 (a+i b) d^5}+\frac{45 \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) b}{2 (i a-b) (a-i b)^2 d^6}+\frac{x^2}{2 (a-i b)^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Tan[c + d*x^(1/3)])^2,x]

[Out]

((-6*I)*b^2*x^(5/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(5/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (
15*b^2*x^(4/3)*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^(5/3)*Lo
g[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^(5/3)*Log[1 +
((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((30*I)*b^2*x*PolyLog[2, -(((a - I*b)*E
^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (15*b*x^(4/3)*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) - (15*b^2*x^(4/3)*PolyLog[2, -(((a - I*b)*E^((2*I)*(c
+ d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (45*b^2*x^(2/3)*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(
1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (30*b*x*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) - ((30*I)*b^2*x*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))
])/((a^2 + b^2)^2*d^3) + ((45*I)*b^2*x^(1/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(
(a^2 + b^2)^2*d^5) - (45*b*x^(2/3)*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) + (45*b^2*x^(2/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^4) - (45*b^2*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6) - (45
*b*x^(1/3)*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) + ((45*
I)*b^2*x^(1/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) + (45*b*Pol
yLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(I*a - b)*(a - I*b)^2*d^6) - (45*b^2*PolyLog[6
, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6)

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}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^5}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{x^5}{(a-i b)^2}-\frac{4 b^2 x^5}{(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^5}{(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^2}{2 (a-i b)^2}+\frac{(12 b) \operatorname{Subst}\left (\int \frac{x^5}{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^5}{\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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}+\frac{\left (12 b^2\right ) \operatorname{Subst}\left (\int \frac{x^5}{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^5}{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^5}{\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^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b x^{5/3} \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^5}{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{(30 b) \operatorname{Subst}\left (\int x^4 \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 (30 b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{6 b x^{5/3} \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^{5/3} \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{15 b x^{4/3} \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{(60 b) \operatorname{Subst}\left (\int x^3 \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 (30 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i c+2 i d x} x^4}{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 (30 i b^2\right ) \operatorname{Subst}\left (\int x^4 \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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{30 b x \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{(90 b) \operatorname{Subst}\left (\int x^2 \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 (60 b^2\right ) \operatorname{Subst}\left (\int x^3 \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 (60 b^2\right ) \operatorname{Subst}\left (\int x^3 \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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{30 i b^2 x \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{30 b x \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{30 i b^2 x \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{45 b x^{2/3} \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{(90 b) \operatorname{Subst}\left (\int x \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 (90 i b^2\right ) \operatorname{Subst}\left (\int x^2 \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 (90 i b^2\right ) \operatorname{Subst}\left (\int x^2 \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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{30 i b^2 x \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{45 b^2 x^{2/3} \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{30 b x \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{30 i b^2 x \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{45 b x^{2/3} \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{45 b^2 x^{2/3} \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{45 b \sqrt [3]{x} \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{(45 b) \operatorname{Subst}\left (\int \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 (90 b^2\right ) \operatorname{Subst}\left (\int x \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 (90 b^2\right ) \operatorname{Subst}\left (\int x \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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{30 i b^2 x \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{45 b^2 x^{2/3} \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{30 b x \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{30 i b^2 x \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{45 i b^2 \sqrt [3]{x} \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{45 b x^{2/3} \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{45 b^2 x^{2/3} \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{45 b \sqrt [3]{x} \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{45 i b^2 \sqrt [3]{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^5}+\frac{(45 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{2 (i a-b) (a-i b)^2 d^6}-\frac{\left (45 i b^2\right ) \operatorname{Subst}\left (\int \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 (45 i b^2\right ) \operatorname{Subst}\left (\int \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^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{30 i b^2 x \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{45 b^2 x^{2/3} \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{30 b x \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{30 i b^2 x \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{45 i b^2 \sqrt [3]{x} \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{45 b x^{2/3} \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{45 b^2 x^{2/3} \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{45 b \sqrt [3]{x} \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{45 i b^2 \sqrt [3]{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^5}+\frac{45 b \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 d^6}-\frac{\left (45 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{2 \left (a^2+b^2\right )^2 d^6}-\frac{\left (45 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (-\frac{(a-i b) x}{a+i b}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{2 \left (a^2+b^2\right )^2 d^6}\\ &=-\frac{6 i b^2 x^{5/3}}{\left (a^2+b^2\right )^2 d}-\frac{6 b^2 x^{5/3}}{(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^2}{2 (a-i b)^2}+\frac{2 b x^2}{(i a-b) (a-i b)^2}-\frac{2 b^2 x^2}{\left (a^2+b^2\right )^2}+\frac{15 b^2 x^{4/3} \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^{5/3} \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^{5/3} \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{30 i b^2 x \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{15 b x^{4/3} \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{15 b^2 x^{4/3} \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{45 b^2 x^{2/3} \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{30 b x \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{30 i b^2 x \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{45 i b^2 \sqrt [3]{x} \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{45 b x^{2/3} \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{45 b^2 x^{2/3} \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{45 b^2 \text{Li}_5\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 d^6}-\frac{45 b \sqrt [3]{x} \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{45 i b^2 \sqrt [3]{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^5}+\frac{45 b \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 d^6}-\frac{45 b^2 \text{Li}_6\left (-\frac{(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 d^6}\\ \end{align*}

Mathematica [A]  time = 5.19955, size = 820, normalized size = 0.71 \[ \frac{\frac{6 x^{5/3} \sin \left (d \sqrt [3]{x}\right ) b^2}{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{\left (\frac{4 a d x^2}{a-i b}+\frac{12 b x^{5/3}}{a-i b}+\frac{12 a \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (\frac{e^{-2 i \left (c+d \sqrt [3]{x}\right )} (a+i b)}{a-i b}+1\right ) x^{5/3}}{(a+i b) (i a+b)}+\frac{30 b \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \log \left (\frac{e^{-2 i \left (c+d \sqrt [3]{x}\right )} (a+i b)}{a-i b}+1\right ) x^{4/3}}{(a+i b) (i a+b) d}+\frac{15 b \left (b \left (-1+e^{2 i c}\right )+i a \left (1+e^{2 i c}\right )\right ) \left (-4 i x \text{PolyLog}\left (2,\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 (3,\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 (4,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text{PolyLog}\left (5,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )}{\left (a^2+b^2\right ) d^5}+\frac{15 a \left (a \left (1+e^{2 i c}\right )-i b \left (-1+e^{2 i c}\right )\right ) \left (2 x^{4/3} \text{PolyLog}\left (2,\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 (3,\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 (4,\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 (5,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right ) d+3 \text{PolyLog}\left (6,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )}{\left (a^2+b^2\right ) d^5}\right ) b}{d \left (-e^{2 i c} b+b-i a \left (1+e^{2 i c}\right )\right )}+\frac{x^2 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}}{2 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*((12*b*x^(5/3))/(a - I*b) + (4*a*d*x^2)/(a - I*b) + (30*b*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c))
)*x^(4/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))])/((a + I*b)*(I*a + b)*d) + (12*a*((-I)*b*(-
1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(5/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))])/((a
+ I*b)*(I*a + b)) + (15*b*(b*(-1 + E^((2*I)*c)) + I*a*(1 + E^((2*I)*c)))*((-4*I)*d^3*x*PolyLog[2, (-a - I*b)/(
(a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3)
)))] + (6*I)*d*x^(1/3)*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 3*PolyLog[5, (-a - I*b)/
((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]))/((a^2 + b^2)*d^5) + (15*a*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I
)*c)))*(2*d^4*x^(4/3)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (4*I)*d^3*x*PolyLog[3, (-
a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c +
d*x^(1/3))))] + (6*I)*d*x^(1/3)*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 3*PolyLog[6, (
-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]))/((a^2 + b^2)*d^5)))/(d*(b - b*E^((2*I)*c) - I*a*(1 + E^((2*
I)*c)))) + (x^2*(a*Cos[c] - b*Sin[c]))/(a*Cos[c] + b*Sin[c]) + (6*b^2*x^(5/3)*Sin[d*x^(1/3)])/(d*(a*Cos[c] + b
*Sin[c])*(a*Cos[c + d*x^(1/3)] + b*Sin[c + d*x^(1/3)])))/(2*(a^2 + b^2))

________________________________________________________________________________________

Maple [F]  time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{x \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/(a+b*tan(c+d*x^(1/3)))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 8.61927, size = 5889, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(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^5 - ((5*a^3 - 5*I*a^2*b + 5*a*b^2 - 5*I*b^3)*(d*x^(1/3) + c)^6 - (30*a^3 - 30*I*a^2*b + 30*a*b^
2 - 30*I*b^3)*(d*x^(1/3) + c)^5*c + (75*a^3 - 75*I*a^2*b + 75*a*b^2 - 75*I*b^3)*(d*x^(1/3) + c)^4*c^2 - (100*a
^3 - 100*I*a^2*b + 100*a*b^2 - 100*I*b^3)*(d*x^(1/3) + c)^3*c^3 + (75*a^3 - 75*I*a^2*b + 75*a*b^2 - 75*I*b^3)*
(d*x^(1/3) + c)^2*c^4 - (150*(-I*a*b^2 - b^3)*c^4*cos(2*d*x^(1/3) + 2*c) + (150*a*b^2 - 150*I*b^3)*c^4*sin(2*d
*x^(1/3) + 2*c) + 150*(-I*a*b^2 + b^3)*c^4)*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) + ((-192*I*a^2*b + 192*a*b^2)*(d*x^(1/3) + c)^5 + (-3
00*I*a*b^2 + 300*b^3 + (600*I*a^2*b - 600*a*b^2)*c)*(d*x^(1/3) + c)^4 + ((-800*I*a^2*b + 800*a*b^2)*c^2 - 800*
(-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^3 + ((600*I*a^2*b - 600*a*b^2)*c^3 - 900*(I*a*b^2 - b^3)*c^2)*(d*x^(1/3) +
c)^2 + ((-300*I*a^2*b + 300*a*b^2)*c^4 - 600*(-I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c) + ((-192*I*a^2*b - 192*a*b
^2)*(d*x^(1/3) + c)^5 + (-300*I*a*b^2 - 300*b^3 + (600*I*a^2*b + 600*a*b^2)*c)*(d*x^(1/3) + c)^4 + ((-800*I*a^
2*b - 800*a*b^2)*c^2 - 800*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^3 + ((600*I*a^2*b + 600*a*b^2)*c^3 - 900*(I*a*b
^2 + b^3)*c^2)*(d*x^(1/3) + c)^2 + ((-300*I*a^2*b - 300*a*b^2)*c^4 - 600*(-I*a*b^2 - b^3)*c^3)*(d*x^(1/3) + c)
)*cos(2*d*x^(1/3) + 2*c) + (192*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^5 + (300*a*b^2 - 300*I*b^3 - 600*(a^2*b - I*
a*b^2)*c)*(d*x^(1/3) + c)^4 + (800*(a^2*b - I*a*b^2)*c^2 - (800*a*b^2 - 800*I*b^3)*c)*(d*x^(1/3) + c)^3 - (600
*(a^2*b - I*a*b^2)*c^3 - (900*a*b^2 - 900*I*b^3)*c^2)*(d*x^(1/3) + c)^2 + (300*(a^2*b - I*a*b^2)*c^4 - (600*a*
b^2 - 600*I*b^3)*c^3)*(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)) + ((5*a^3 - 15*I*a^2*b - 15*a*b^2 + 5*I*b^3)*(d*x^(1/3) + c)^6 + (-60*I*a*b^2 - 60*b^
3 - (30*a^3 - 90*I*a^2*b - 90*a*b^2 + 30*I*b^3)*c)*(d*x^(1/3) + c)^5 - 300*(I*a*b^2 + b^3)*(d*x^(1/3) + c)*c^4
+ ((75*a^3 - 225*I*a^2*b - 225*a*b^2 + 75*I*b^3)*c^2 - 300*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^4 - ((100*a^3
- 300*I*a^2*b - 300*a*b^2 + 100*I*b^3)*c^3 + 600*(I*a*b^2 + b^3)*c^2)*(d*x^(1/3) + c)^3 + ((75*a^3 - 225*I*a^2
*b - 225*a*b^2 + 75*I*b^3)*c^4 - 600*(-I*a*b^2 - b^3)*c^3)*(d*x^(1/3) + c)^2)*cos(2*d*x^(1/3) + 2*c) + ((-480*
I*a^2*b + 480*a*b^2)*(d*x^(1/3) + c)^4 + (-150*I*a^2*b + 150*a*b^2)*c^4 + (-600*I*a*b^2 + 600*b^3 + (1200*I*a^
2*b - 1200*a*b^2)*c)*(d*x^(1/3) + c)^3 - 300*(-I*a*b^2 + b^3)*c^3 + ((-1200*I*a^2*b + 1200*a*b^2)*c^2 - 1200*(
-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^2 + ((600*I*a^2*b - 600*a*b^2)*c^3 - 900*(I*a*b^2 - b^3)*c^2)*(d*x^(1/3) +
c) + ((-480*I*a^2*b - 480*a*b^2)*(d*x^(1/3) + c)^4 + (-150*I*a^2*b - 150*a*b^2)*c^4 + (-600*I*a*b^2 - 600*b^3
+ (1200*I*a^2*b + 1200*a*b^2)*c)*(d*x^(1/3) + c)^3 - 300*(-I*a*b^2 - b^3)*c^3 + ((-1200*I*a^2*b - 1200*a*b^2)*
c^2 - 1200*(-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^2 + ((600*I*a^2*b + 600*a*b^2)*c^3 - 900*(I*a*b^2 + b^3)*c^2)*(
d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (480*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^4 + 150*(a^2*b - I*a*b^2)*c^4
+ (600*a*b^2 - 600*I*b^3 - 1200*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^3 - (300*a*b^2 - 300*I*b^3)*c^3 + (1200*(
a^2*b - I*a*b^2)*c^2 - (1200*a*b^2 - 1200*I*b^3)*c)*(d*x^(1/3) + c)^2 - (600*(a^2*b - I*a*b^2)*c^3 - (900*a*b^
2 - 900*I*b^3)*c^2)*(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)) + ((75*a*b^2 - 75*I*b^3)*c^4*cos(2*d*x^(1/3) + 2*c) - 75*(-I*a*b^2 - b^3)*c^4*sin(2*d*x^(1/3) + 2*c) + (7
5*a*b^2 + 75*I*b^3)*c^4)*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)) + (96*(a^2*b + I*a*b^2)*(d*x^(1/
3) + c)^5 + (150*a*b^2 + 150*I*b^3 - 300*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c)^4 + (400*(a^2*b + I*a*b^2)*c^2 -
(400*a*b^2 + 400*I*b^3)*c)*(d*x^(1/3) + c)^3 - (300*(a^2*b + I*a*b^2)*c^3 - (450*a*b^2 + 450*I*b^3)*c^2)*(d*x
^(1/3) + c)^2 + (150*(a^2*b + I*a*b^2)*c^4 - (300*a*b^2 + 300*I*b^3)*c^3)*(d*x^(1/3) + c) + (96*(a^2*b - I*a*b
^2)*(d*x^(1/3) + c)^5 + (150*a*b^2 - 150*I*b^3 - 300*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^4 + (400*(a^2*b - I*
a*b^2)*c^2 - (400*a*b^2 - 400*I*b^3)*c)*(d*x^(1/3) + c)^3 - (300*(a^2*b - I*a*b^2)*c^3 - (450*a*b^2 - 450*I*b^
3)*c^2)*(d*x^(1/3) + c)^2 + (150*(a^2*b - I*a*b^2)*c^4 - (300*a*b^2 - 300*I*b^3)*c^3)*(d*x^(1/3) + c))*cos(2*d
*x^(1/3) + 2*c) + ((96*I*a^2*b + 96*a*b^2)*(d*x^(1/3) + c)^5 + (150*I*a*b^2 + 150*b^3 + (-300*I*a^2*b - 300*a*
b^2)*c)*(d*x^(1/3) + c)^4 + ((400*I*a^2*b + 400*a*b^2)*c^2 - 400*(I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^3 + ((-300
*I*a^2*b - 300*a*b^2)*c^3 - 450*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^2 + ((150*I*a^2*b + 150*a*b^2)*c^4 - 300
*(I*a*b^2 + b^3)*c^3)*(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)) + (-720*I*a^2*b + 720*a*b^2 + (-720*I*a^2*b - 720*a*b^2)*cos(2*d*x^(1/3) + 2*c) + 720*(a^2
*b - I*a*b^2)*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)) - (450*a*b^2
+ 450*I*b^3 + 1440*(a^2*b + I*a*b^2)*(d*x^(1/3) + c) - 900*(a^2*b + I*a*b^2)*c + (450*a*b^2 - 450*I*b^3 + 1440
*(a^2*b - I*a*b^2)*(d*x^(1/3) + c) - 900*(a^2*b - I*a*b^2)*c)*cos(2*d*x^(1/3) + 2*c) - (-450*I*a*b^2 - 450*b^3
+ (-1440*I*a^2*b - 1440*a*b^2)*(d*x^(1/3) + c) + (900*I*a^2*b + 900*a*b^2)*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)) + ((1440*I*a^2*b - 1440*a*b^2)*(d*x^(1/3) + c)^2 + (600*I*
a^2*b - 600*a*b^2)*c^2 + (900*I*a*b^2 - 900*b^3 + (-1800*I*a^2*b + 1800*a*b^2)*c)*(d*x^(1/3) + c) - 600*(I*a*b
^2 - b^3)*c + ((1440*I*a^2*b + 1440*a*b^2)*(d*x^(1/3) + c)^2 + (600*I*a^2*b + 600*a*b^2)*c^2 + (900*I*a*b^2 +
900*b^3 + (-1800*I*a^2*b - 1800*a*b^2)*c)*(d*x^(1/3) + c) - 600*(I*a*b^2 + b^3)*c)*cos(2*d*x^(1/3) + 2*c) - (1
440*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^2 + 600*(a^2*b - I*a*b^2)*c^2 + (900*a*b^2 - 900*I*b^3 - 1800*(a^2*b - I
*a*b^2)*c)*(d*x^(1/3) + c) - (600*a*b^2 - 600*I*b^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)) + (960*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^3 - 300*(a^2*b + I*a*b^2)*c^3 + (900*a*b
^2 + 900*I*b^3 - 1800*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c)^2 + (450*a*b^2 + 450*I*b^3)*c^2 + (1200*(a^2*b + I*
a*b^2)*c^2 - (1200*a*b^2 + 1200*I*b^3)*c)*(d*x^(1/3) + c) + (960*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^3 - 300*(a^
2*b - I*a*b^2)*c^3 + (900*a*b^2 - 900*I*b^3 - 1800*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^2 + (450*a*b^2 - 450*I
*b^3)*c^2 + (1200*(a^2*b - I*a*b^2)*c^2 - (1200*a*b^2 - 1200*I*b^3)*c)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c)
+ ((960*I*a^2*b + 960*a*b^2)*(d*x^(1/3) + c)^3 + (-300*I*a^2*b - 300*a*b^2)*c^3 + (900*I*a*b^2 + 900*b^3 + (-
1800*I*a^2*b - 1800*a*b^2)*c)*(d*x^(1/3) + c)^2 - 450*(-I*a*b^2 - b^3)*c^2 + ((1200*I*a^2*b + 1200*a*b^2)*c^2
- 1200*(I*a*b^2 + b^3)*c)*(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)) + ((5*I*a^3 + 15*a^2*b - 15*I*a*b^2 - 5*b^3)*(d*x^(1/3) + c)^6 + (60*a*b^2 - 60*I*b^3 + (-30*I
*a^3 - 90*a^2*b + 90*I*a*b^2 + 30*b^3)*c)*(d*x^(1/3) + c)^5 + (300*a*b^2 - 300*I*b^3)*(d*x^(1/3) + c)*c^4 + ((
75*I*a^3 + 225*a^2*b - 225*I*a*b^2 - 75*b^3)*c^2 - (300*a*b^2 - 300*I*b^3)*c)*(d*x^(1/3) + c)^4 + ((-100*I*a^3
- 300*a^2*b + 300*I*a*b^2 + 100*b^3)*c^3 + (600*a*b^2 - 600*I*b^3)*c^2)*(d*x^(1/3) + c)^3 + ((75*I*a^3 + 225*
a^2*b - 225*I*a*b^2 - 75*b^3)*c^4 - (600*a*b^2 - 600*I*b^3)*c^3)*(d*x^(1/3) + c)^2)*sin(2*d*x^(1/3) + 2*c))/(3
0*a^5 + 30*I*a^4*b + 60*a^3*b^2 + 60*I*a^2*b^3 + 30*a*b^4 + 30*I*b^5 + (30*a^5 - 30*I*a^4*b + 60*a^3*b^2 - 60*
I*a^2*b^3 + 30*a*b^4 - 30*I*b^5)*cos(2*d*x^(1/3) + 2*c) + (30*I*a^5 + 30*a^4*b + 60*I*a^3*b^2 + 60*a^2*b^3 + 3
0*I*a*b^4 + 30*b^5)*sin(2*d*x^(1/3) + 2*c)))/d^6

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

[Out]

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

[Out]

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

[Out]

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