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

Optimal. Leaf size=511 \[ -\frac{12 i b x^{7/3} \text{PolyLog}\left (2,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac{42 b x^2 \text{PolyLog}\left (3,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^3 \left (a^2+b^2\right )}+\frac{126 i b x^{5/3} \text{PolyLog}\left (4,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^4 \left (a^2+b^2\right )}-\frac{315 b x^{4/3} \text{PolyLog}\left (5,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^5 \left (a^2+b^2\right )}+\frac{945 b x^{2/3} \text{PolyLog}\left (7,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^7 \left (a^2+b^2\right )}-\frac{630 i b x \text{PolyLog}\left (6,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^6 \left (a^2+b^2\right )}+\frac{945 i b \sqrt [3]{x} \text{PolyLog}\left (8,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^8 \left (a^2+b^2\right )}-\frac{945 b \text{PolyLog}\left (9,-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d^9 \left (a^2+b^2\right )}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac{x^3}{3 (a+i b)} \]

[Out]

x^3/(3*(a + I*b)) + (3*b*x^(8/3)*Log[1 + ((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d)
- ((12*I)*b*x^(7/3)*PolyLog[2, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (
42*b*x^2*PolyLog[3, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + ((126*I)*b*x^
(5/3)*PolyLog[4, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^4) - (315*b*x^(4/3)*P
olyLog[5, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^5) - ((630*I)*b*x*PolyLog[6,
-(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^6) + (945*b*x^(2/3)*PolyLog[7, -(((a^
2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^7) + ((945*I)*b*x^(1/3)*PolyLog[8, -(((a^2 +
b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^8) - (945*b*PolyLog[9, -(((a^2 + b^2)*E^((2*I)*
(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^9)

________________________________________________________________________________________

Rubi [A]  time = 0.593475, antiderivative size = 511, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3747, 3732, 2190, 2531, 6609, 2282, 6589} \[ -\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^3 \left (a^2+b^2\right )}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^4 \left (a^2+b^2\right )}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^5 \left (a^2+b^2\right )}+\frac{945 b x^{2/3} \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^7 \left (a^2+b^2\right )}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^6 \left (a^2+b^2\right )}+\frac{945 i b \sqrt [3]{x} \text{Li}_8\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^8 \left (a^2+b^2\right )}-\frac{945 b \text{Li}_9\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d^9 \left (a^2+b^2\right )}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac{x^3}{3 (a+i b)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Tan[c + d*x^(1/3)]),x]

[Out]

x^3/(3*(a + I*b)) + (3*b*x^(8/3)*Log[1 + ((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d)
- ((12*I)*b*x^(7/3)*PolyLog[2, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (
42*b*x^2*PolyLog[3, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + ((126*I)*b*x^
(5/3)*PolyLog[4, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^4) - (315*b*x^(4/3)*P
olyLog[5, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^5) - ((630*I)*b*x*PolyLog[6,
-(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^6) + (945*b*x^(2/3)*PolyLog[7, -(((a^
2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^7) + ((945*I)*b*x^(1/3)*PolyLog[8, -(((a^2 +
b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^8) - (945*b*PolyLog[9, -(((a^2 + b^2)*E^((2*I)*
(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^9)

Rule 3747

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

Rule 3732

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(d*
(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[((c + d*x)^m*E^Simp[2*I*(e + f*x), x])/((a + I*b)^2 + (a^2 + b^2)*E^S
imp[2*I*(e + f*x), x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 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 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]

Rubi steps

\begin{align*} \int \frac{x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^8}{a+b \tan (c+d x)} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{x^3}{3 (a+i b)}+(6 i b) \operatorname{Subst}\left (\int \frac{e^{2 i (c+d x)} x^8}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{(24 b) \operatorname{Subst}\left (\int x^7 \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{(84 i b) \operatorname{Subst}\left (\int x^6 \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^2}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}-\frac{(252 b) \operatorname{Subst}\left (\int x^5 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^3}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{(630 i b) \operatorname{Subst}\left (\int x^4 \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^4}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}+\frac{(1260 b) \operatorname{Subst}\left (\int x^3 \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^5}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac{(1890 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^6}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac{945 b x^{2/3} \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}-\frac{(1890 b) \operatorname{Subst}\left (\int x \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^7}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac{945 b x^{2/3} \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}+\frac{945 i b \sqrt [3]{x} \text{Li}_8\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^8}-\frac{(945 i b) \operatorname{Subst}\left (\int \text{Li}_8\left (-\frac{\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right ) d^8}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac{945 b x^{2/3} \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}+\frac{945 i b \sqrt [3]{x} \text{Li}_8\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^8}-\frac{(945 b) \operatorname{Subst}\left (\int \frac{\text{Li}_8\left (-\frac{\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{2 \left (a^2+b^2\right ) d^9}\\ &=\frac{x^3}{3 (a+i b)}+\frac{3 b x^{8/3} \log \left (1+\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac{12 i b x^{7/3} \text{Li}_2\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac{42 b x^2 \text{Li}_3\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac{126 i b x^{5/3} \text{Li}_4\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac{315 b x^{4/3} \text{Li}_5\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac{630 i b x \text{Li}_6\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac{945 b x^{2/3} \text{Li}_7\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}+\frac{945 i b \sqrt [3]{x} \text{Li}_8\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^8}-\frac{945 b \text{Li}_9\left (-\frac{\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^9}\\ \end{align*}

Mathematica [A]  time = 1.80589, size = 451, normalized size = 0.88 \[ \frac{72 i b d^7 x^{7/3} \text{PolyLog}\left (2,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+252 b d^6 x^2 \text{PolyLog}\left (3,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-756 i b d^5 x^{5/3} \text{PolyLog}\left (4,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-1890 b d^4 x^{4/3} \text{PolyLog}\left (5,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+5670 b d^2 x^{2/3} \text{PolyLog}\left (7,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3780 i b d^3 x \text{PolyLog}\left (6,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-5670 i b d \sqrt [3]{x} \text{PolyLog}\left (8,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-2835 b \text{PolyLog}\left (9,\frac{(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+18 b d^8 x^{8/3} \log \left (1+\frac{(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+2 a d^9 x^3+2 i b d^9 x^3}{6 d^9 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*d^9*x^3 + (2*I)*b*d^9*x^3 + 18*b*d^8*x^(8/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (
72*I)*b*d^7*x^(7/3)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 252*b*d^6*x^2*PolyLog[3, (-
a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (756*I)*b*d^5*x^(5/3)*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2
*I)*(c + d*x^(1/3))))] - 1890*b*d^4*x^(4/3)*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (37
80*I)*b*d^3*x*PolyLog[6, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 5670*b*d^2*x^(2/3)*PolyLog[7, (-a
- I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (5670*I)*b*d*x^(1/3)*PolyLog[8, (-a - I*b)/((a - I*b)*E^((2*I
)*(c + d*x^(1/3))))] - 2835*b*PolyLog[9, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))])/(6*(a^2 + b^2)*d^9
)

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Maple [F]  time = 0.173, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\tan \left ( c+d\sqrt [3]{x} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 5.84093, size = 1769, normalized size = 3.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(315*(2*(d*x^(1/3) + c)*a/(a^2 + b^2) + 2*b*log(b*tan(d*x^(1/3) + c) + a)/(a^2 + b^2) - b*log(tan(d*x^(1
/3) + c)^2 + 1)/(a^2 + b^2))*c^8 + 2*(35*(d*x^(1/3) + c)^9*(a - I*b) - 315*(d*x^(1/3) + c)^8*(a - I*b)*c + 126
0*(d*x^(1/3) + c)^7*(a - I*b)*c^2 - 2940*(d*x^(1/3) + c)^6*(a - I*b)*c^3 + 4410*(d*x^(1/3) + c)^5*(a - I*b)*c^
4 - 4410*(d*x^(1/3) + c)^4*(a - I*b)*c^5 + 2940*(d*x^(1/3) + c)^3*(a - I*b)*c^6 - 1260*(d*x^(1/3) + c)^2*(a -
I*b)*c^7 + (-5040*I*(d*x^(1/3) + c)^8*b + 23040*I*(d*x^(1/3) + c)^7*b*c - 47040*I*(d*x^(1/3) + c)^6*b*c^2 + 56
448*I*(d*x^(1/3) + c)^5*b*c^3 - 44100*I*(d*x^(1/3) + c)^4*b*c^4 + 23520*I*(d*x^(1/3) + c)^3*b*c^5 - 8820*I*(d*
x^(1/3) + c)^2*b*c^6 + 2520*I*(d*x^(1/3) + c)*b*c^7)*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)) + (-20160*I*(d*x^(1/3) + c)^7*b + 80640*I*(d*x^(1/3) + c)^6*b*c - 141120*I*(d*x^(1/3) + c)^5*b*
c^2 + 141120*I*(d*x^(1/3) + c)^4*b*c^3 - 88200*I*(d*x^(1/3) + c)^3*b*c^4 + 35280*I*(d*x^(1/3) + c)^2*b*c^5 - 8
820*I*(d*x^(1/3) + c)*b*c^6 + 1260*I*b*c^7)*dilog((I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + 6*(420*(d*
x^(1/3) + c)^8*b - 1920*(d*x^(1/3) + c)^7*b*c + 3920*(d*x^(1/3) + c)^6*b*c^2 - 4704*(d*x^(1/3) + c)^5*b*c^3 +
3675*(d*x^(1/3) + c)^4*b*c^4 - 1960*(d*x^(1/3) + c)^3*b*c^5 + 735*(d*x^(1/3) + c)^2*b*c^6 - 210*(d*x^(1/3) + c
)*b*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))/(a^2 + b^2)) - 793800*b*polylog(9, (I*a + b)*e^
(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + (1587600*I*(d*x^(1/3) + c)*b - 907200*I*b*c)*polylog(8, (I*a + b)*e^(2*I
*d*x^(1/3) + 2*I*c)/(-I*a + b)) + 75600*(21*(d*x^(1/3) + c)^2*b - 24*(d*x^(1/3) + c)*b*c + 7*b*c^2)*polylog(7,
(I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + (-1058400*I*(d*x^(1/3) + c)^3*b + 1814400*I*(d*x^(1/3) + c)
^2*b*c - 1058400*I*(d*x^(1/3) + c)*b*c^2 + 211680*I*b*c^3)*polylog(6, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*
a + b)) - 1890*(280*(d*x^(1/3) + c)^4*b - 640*(d*x^(1/3) + c)^3*b*c + 560*(d*x^(1/3) + c)^2*b*c^2 - 224*(d*x^(
1/3) + c)*b*c^3 + 35*b*c^4)*polylog(5, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + (211680*I*(d*x^(1/3)
+ c)^5*b - 604800*I*(d*x^(1/3) + c)^4*b*c + 705600*I*(d*x^(1/3) + c)^3*b*c^2 - 423360*I*(d*x^(1/3) + c)^2*b*c^
3 + 132300*I*(d*x^(1/3) + c)*b*c^4 - 17640*I*b*c^5)*polylog(4, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b))
+ 630*(112*(d*x^(1/3) + c)^6*b - 384*(d*x^(1/3) + c)^5*b*c + 560*(d*x^(1/3) + c)^4*b*c^2 - 448*(d*x^(1/3) + c
)^3*b*c^3 + 210*(d*x^(1/3) + c)^2*b*c^4 - 56*(d*x^(1/3) + c)*b*c^5 + 7*b*c^6)*polylog(3, (I*a + b)*e^(2*I*d*x^
(1/3) + 2*I*c)/(-I*a + b)))/(a^2 + b^2))/d^9

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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