\(\int \frac {x^2}{a+b \tan (c+d \sqrt [3]{x})} \, dx\) [57]

Optimal result

Integrand size = 20, antiderivative size = 511 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {x^3}{3 (a+i b)}+\frac {3 b x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {12 i b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}+\frac {945 i b \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^8}-\frac {945 b \operatorname {PolyLog}\left (9,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^9} \]

[Out]

1/3*x^3/(a+I*b)+3*b*x^(8/3)*ln(1+(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d-12*I*b*x^(7/3)*polylo
g(2,-(a^2+b^2)*exp(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)*exp(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)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2
)/d^4-315*b*x^(4/3)*polylog(5,-(a^2+b^2)*exp(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)*exp(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)*exp(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)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2
)/d^8-945/2*b*polylog(9,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^9

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3832, 3813, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=-\frac {945 b \operatorname {PolyLog}\left (9,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d^9 \left (a^2+b^2\right )}+\frac {945 i b \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^8 \left (a^2+b^2\right )}+\frac {945 b x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^7 \left (a^2+b^2\right )}-\frac {630 i b x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^6 \left (a^2+b^2\right )}-\frac {315 b x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^5 \left (a^2+b^2\right )}+\frac {126 i b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^4 \left (a^2+b^2\right )}+\frac {42 b x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^3 \left (a^2+b^2\right )}-\frac {12 i b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac {3 b x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac {x^3}{3 (a+i b)} \]

[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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3813

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^Si
mp[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 3832

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 6724

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 6744

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*} \text {integral}& = 3 \text {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) \text {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) \text {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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {(84 i b) \text {Subst}\left (\int x^6 \operatorname {PolyLog}\left (2,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}-\frac {(252 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (3,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {(630 i b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (4,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}+\frac {(1260 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (5,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {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 )}{\left (a^2+b^2\right ) d^6}+\frac {(1890 i b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (6,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {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 )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^7}-\frac {(1890 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (7,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {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 )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^7}+\frac {945 i b \sqrt [3]{x} \operatorname {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 )}{\left (a^2+b^2\right ) d^8}-\frac {(945 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (8,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {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 )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^7}+\frac {945 i b \sqrt [3]{x} \operatorname {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 )}{\left (a^2+b^2\right ) d^8}-\frac {(945 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (8,-\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} \operatorname {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 )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {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 )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {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 )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {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 )}{\left (a^2+b^2\right ) d^7}+\frac {945 i b \sqrt [3]{x} \operatorname {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 )}{\left (a^2+b^2\right ) d^8}-\frac {945 b \operatorname {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 \left (a^2+b^2\right ) d^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {2 a d^9 x^3+2 i b d^9 x^3+18 b d^8 x^{8/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+72 i b d^7 x^{7/3} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+252 b d^6 x^2 \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-756 i b d^5 x^{5/3} \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-1890 b d^4 x^{4/3} \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3780 i b d^3 x \operatorname {PolyLog}\left (6,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+5670 b d^2 x^{2/3} \operatorname {PolyLog}\left (7,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-5670 i b d \sqrt [3]{x} \operatorname {PolyLog}\left (8,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-2835 b \operatorname {PolyLog}\left (9,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{6 \left (a^2+b^2\right ) d^9} \]

[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
)

Maple [F]

\[\int \frac {x^{2}}{a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )}d x\]

[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)

Fricas [F]

\[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int { \frac {x^{2}}{b \tan \left (d x^{\frac {1}{3}} + c\right ) + a} \,d x } \]

[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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1315 vs. \(2 (430) = 860\).

Time = 0.68 (sec) , antiderivative size = 1315, normalized size of antiderivative = 2.57 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\text {Too large to display} \]

[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 - 12*(420*I*(d*x^(1/3) + c)^8*b - 1920*I*(d*x^(1/3) + c)^7*b*c + 3920*I*(d*x^(1/3) + c)^6*b*c^2 - 470
4*I*(d*x^(1/3) + c)^5*b*c^3 + 3675*I*(d*x^(1/3) + c)^4*b*c^4 - 1960*I*(d*x^(1/3) + c)^3*b*c^5 + 735*I*(d*x^(1/
3) + c)^2*b*c^6 - 210*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)) - 1260*(16*I*(d*x^(1/3) + c)^7*b - 64*I*(d*x^(1/3) + c)^6*b*c + 112*I*(d*x^(1/3) + c)^5*b*c^2 - 112*I
*(d*x^(1/3) + c)^4*b*c^3 + 70*I*(d*x^(1/3) + c)^3*b*c^4 - 28*I*(d*x^(1/3) + c)^2*b*c^5 + 7*I*(d*x^(1/3) + c)*b
*c^6 - 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)*co
s(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)) - 226800*(-7*I*(d*x^(1/3) + c)*b + 4*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)) - 30240*(35*I*(d*x^(1/3) + c)^3*b - 60*I*(d*x^(1/3) + c)^2*b*c + 35*I*(d*x^(1/3) + c)*b*c^2
- 7*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)) - 1260*(-168*I*(d*x^(1/3) + c)^5*b + 480*I*(d*x^(1/3) + c)^4*b*c - 56
0*I*(d*x^(1/3) + c)^3*b*c^2 + 336*I*(d*x^(1/3) + c)^2*b*c^3 - 105*I*(d*x^(1/3) + c)*b*c^4 + 14*I*b*c^5)*polylo
g(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

Giac [F]

\[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int { \frac {x^{2}}{b \tan \left (d x^{\frac {1}{3}} + c\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\int \frac {x^2}{a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )} \,d x \]

[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)