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

Optimal. Leaf size=597 \[ \frac{24 i a b x^{7/3} \text{PolyLog}\left (2,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac{84 a b x^2 \text{PolyLog}\left (3,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}-\frac{252 i a b x^{5/3} \text{PolyLog}\left (4,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac{630 a b x^{4/3} \text{PolyLog}\left (5,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac{1890 a b x^{2/3} \text{PolyLog}\left (7,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac{1260 i a b x \text{PolyLog}\left (6,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}-\frac{1890 i a b \sqrt [3]{x} \text{PolyLog}\left (8,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac{945 a b \text{PolyLog}\left (9,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}-\frac{84 i b^2 x^2 \text{PolyLog}\left (2,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac{252 b^2 x^{5/3} \text{PolyLog}\left (3,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac{630 i b^2 x^{4/3} \text{PolyLog}\left (4,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac{1890 i b^2 x^{2/3} \text{PolyLog}\left (6,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac{1260 b^2 x \text{PolyLog}\left (5,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac{1890 b^2 \sqrt [3]{x} \text{PolyLog}\left (7,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac{945 i b^2 \text{PolyLog}\left (8,-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac{a^2 x^3}{3}-\frac{6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac{2}{3} i a b x^3+\frac{24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac{3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac{3 i b^2 x^{8/3}}{d}-\frac{b^2 x^3}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.833384, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3747, 3722, 3719, 2190, 2531, 6609, 2282, 6589, 3720, 30} \[ \frac{a^2 x^3}{3}+\frac{24 i a b x^{7/3} \text{Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac{84 a b x^2 \text{Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}-\frac{252 i a b x^{5/3} \text{Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac{630 a b x^{4/3} \text{Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac{1890 a b x^{2/3} \text{Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac{1260 i a b x \text{Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}-\frac{1890 i a b \sqrt [3]{x} \text{Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac{945 a b \text{Li}_9\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}-\frac{6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac{2}{3} i a b x^3-\frac{84 i b^2 x^2 \text{Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac{252 b^2 x^{5/3} \text{Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac{630 i b^2 x^{4/3} \text{Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac{1890 i b^2 x^{2/3} \text{Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac{1260 b^2 x \text{Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac{1890 b^2 \sqrt [3]{x} \text{Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac{945 i b^2 \text{Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac{24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac{3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac{3 i b^2 x^{8/3}}{d}-\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3722

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

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, 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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 4.65186, size = 599, normalized size = 1. \[ \frac{1}{3} \left (b \left (\frac{9 a \left (-8 i d^7 x^{7/3} \text{PolyLog}\left (2,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-28 d^6 x^2 \text{PolyLog}\left (3,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+84 i d^5 x^{5/3} \text{PolyLog}\left (4,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+105 \left (2 d^4 x^{4/3} \text{PolyLog}\left (5,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-6 d^2 x^{2/3} \text{PolyLog}\left (7,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-4 i d^3 x \text{PolyLog}\left (6,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+6 i d \sqrt [3]{x} \text{PolyLog}\left (8,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+3 \text{PolyLog}\left (9,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )\right )\right )}{d^9}+\frac{63 b \left (4 i d^6 x^2 \text{PolyLog}\left (2,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+12 d^5 x^{5/3} \text{PolyLog}\left (3,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-15 i \left (2 d^4 x^{4/3} \text{PolyLog}\left (4,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-6 d^2 x^{2/3} \text{PolyLog}\left (6,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-4 i d^3 x \text{PolyLog}\left (5,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+6 i d \sqrt [3]{x} \text{PolyLog}\left (7,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+3 \text{PolyLog}\left (8,-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )\right )\right )}{d^9}-\frac{18 a x^{8/3} \log \left (1+e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac{4 i a x^3}{1+e^{2 i c}}+\frac{72 b x^{7/3} \log \left (1+e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac{18 i b x^{8/3}}{d+e^{2 i c} d}\right )+x^3 \left (a^2+2 a b \tan (c)-b^2\right )+\frac{9 b^2 x^{8/3} \sec (c) \sin \left (d \sqrt [3]{x}\right ) \sec \left (c+d \sqrt [3]{x}\right )}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\tan \left ( c+d\sqrt [3]{x} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 5.4707, size = 6309, normalized size = 10.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((d*x^(1/3) + c)^9*a^2 - 9*(d*x^(1/3) + c)^8*a^2*c + 36*(d*x^(1/3) + c)^7*a^2*c^2 - 84*(d*x^(1/3) + c)^6*a
^2*c^3 + 126*(d*x^(1/3) + c)^5*a^2*c^4 - 126*(d*x^(1/3) + c)^4*a^2*c^5 + 84*(d*x^(1/3) + c)^3*a^2*c^6 - 36*(d*
x^(1/3) + c)^2*a^2*c^7 + 9*(d*x^(1/3) + c)*a^2*c^8 + 18*a*b*c^8*log(sec(d*x^(1/3) + c)) - 9*(-315*I*(d*x^(1/3)
+ c)*b^2*c^8 - (70*a*b + 35*I*b^2)*(d*x^(1/3) + c)^9 + (630*a*b + 315*I*b^2)*(d*x^(1/3) + c)^8*c - (2520*a*b
+ 1260*I*b^2)*(d*x^(1/3) + c)^7*c^2 + (5880*a*b + 2940*I*b^2)*(d*x^(1/3) + c)^6*c^3 - (8820*a*b + 4410*I*b^2)*
(d*x^(1/3) + c)^5*c^4 + (8820*a*b + 4410*I*b^2)*(d*x^(1/3) + c)^4*c^5 - (5880*a*b + 2940*I*b^2)*(d*x^(1/3) + c
)^3*c^6 + (2520*a*b + 1260*I*b^2)*(d*x^(1/3) + c)^2*c^7 - 630*b^2*c^8 + (10080*(d*x^(1/3) + c)^8*a*b + 2520*b^
2*c^7 - 23040*(2*a*b*c + b^2)*(d*x^(1/3) + c)^7 + 94080*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^6 - 56448*(2*a*b*c^3
+ 3*b^2*c^2)*(d*x^(1/3) + c)^5 + 88200*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c)^4 - 23520*(2*a*b*c^5 + 5*b^2*c^4
)*(d*x^(1/3) + c)^3 + 17640*(a*b*c^6 + 3*b^2*c^5)*(d*x^(1/3) + c)^2 - 2520*(2*a*b*c^7 + 7*b^2*c^6)*(d*x^(1/3)
+ c) + 24*(420*(d*x^(1/3) + c)^8*a*b + 105*b^2*c^7 - 960*(2*a*b*c + b^2)*(d*x^(1/3) + c)^7 + 3920*(a*b*c^2 + b
^2*c)*(d*x^(1/3) + c)^6 - 2352*(2*a*b*c^3 + 3*b^2*c^2)*(d*x^(1/3) + c)^5 + 3675*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/
3) + c)^4 - 980*(2*a*b*c^5 + 5*b^2*c^4)*(d*x^(1/3) + c)^3 + 735*(a*b*c^6 + 3*b^2*c^5)*(d*x^(1/3) + c)^2 - 105*
(2*a*b*c^7 + 7*b^2*c^6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (10080*I*(d*x^(1/3) + c)^8*a*b + 2520*I*b^2*
c^7 + (-46080*I*a*b*c - 23040*I*b^2)*(d*x^(1/3) + c)^7 + (94080*I*a*b*c^2 + 94080*I*b^2*c)*(d*x^(1/3) + c)^6 +
(-112896*I*a*b*c^3 - 169344*I*b^2*c^2)*(d*x^(1/3) + c)^5 + (88200*I*a*b*c^4 + 176400*I*b^2*c^3)*(d*x^(1/3) +
c)^4 + (-47040*I*a*b*c^5 - 117600*I*b^2*c^4)*(d*x^(1/3) + c)^3 + (17640*I*a*b*c^6 + 52920*I*b^2*c^5)*(d*x^(1/3
) + c)^2 + (-5040*I*a*b*c^7 - 17640*I*b^2*c^6)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*arctan2(sin(2*d*x^(1/3
) + 2*c), cos(2*d*x^(1/3) + 2*c) + 1) - ((70*a*b + 35*I*b^2)*(d*x^(1/3) + c)^9 - (630*b^2 + (630*a*b + 315*I*b
^2)*c)*(d*x^(1/3) + c)^8 + (5040*b^2*c + (2520*a*b + 1260*I*b^2)*c^2)*(d*x^(1/3) + c)^7 - (17640*b^2*c^2 + (58
80*a*b + 2940*I*b^2)*c^3)*(d*x^(1/3) + c)^6 + (35280*b^2*c^3 + (8820*a*b + 4410*I*b^2)*c^4)*(d*x^(1/3) + c)^5
- (44100*b^2*c^4 + (8820*a*b + 4410*I*b^2)*c^5)*(d*x^(1/3) + c)^4 + (35280*b^2*c^5 + (5880*a*b + 2940*I*b^2)*c
^6)*(d*x^(1/3) + c)^3 - (17640*b^2*c^6 + (2520*a*b + 1260*I*b^2)*c^7)*(d*x^(1/3) + c)^2 - (-315*I*b^2*c^8 - 50
40*b^2*c^7)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (40320*(d*x^(1/3) + c)^7*a*b - 2520*a*b*c^7 - 8820*b^2*c
^6 - 80640*(2*a*b*c + b^2)*(d*x^(1/3) + c)^6 + 282240*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^5 - 141120*(2*a*b*c^3
+ 3*b^2*c^2)*(d*x^(1/3) + c)^4 + 176400*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c)^3 - 35280*(2*a*b*c^5 + 5*b^2*c^4
)*(d*x^(1/3) + c)^2 + 17640*(a*b*c^6 + 3*b^2*c^5)*(d*x^(1/3) + c) + 1260*(32*(d*x^(1/3) + c)^7*a*b - 2*a*b*c^7
- 7*b^2*c^6 - 64*(2*a*b*c + b^2)*(d*x^(1/3) + c)^6 + 224*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^5 - 112*(2*a*b*c^3
+ 3*b^2*c^2)*(d*x^(1/3) + c)^4 + 140*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c)^3 - 28*(2*a*b*c^5 + 5*b^2*c^4)*(d*
x^(1/3) + c)^2 + 14*(a*b*c^6 + 3*b^2*c^5)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (-40320*I*(d*x^(1/3) + c)^
7*a*b + 2520*I*a*b*c^7 + 8820*I*b^2*c^6 + (161280*I*a*b*c + 80640*I*b^2)*(d*x^(1/3) + c)^6 + (-282240*I*a*b*c^
2 - 282240*I*b^2*c)*(d*x^(1/3) + c)^5 + (282240*I*a*b*c^3 + 423360*I*b^2*c^2)*(d*x^(1/3) + c)^4 + (-176400*I*a
*b*c^4 - 352800*I*b^2*c^3)*(d*x^(1/3) + c)^3 + (70560*I*a*b*c^5 + 176400*I*b^2*c^4)*(d*x^(1/3) + c)^2 + (-1764
0*I*a*b*c^6 - 52920*I*b^2*c^5)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*dilog(-e^(2*I*d*x^(1/3) + 2*I*c)) + (-
5040*I*(d*x^(1/3) + c)^8*a*b - 1260*I*b^2*c^7 + (23040*I*a*b*c + 11520*I*b^2)*(d*x^(1/3) + c)^7 + (-47040*I*a*
b*c^2 - 47040*I*b^2*c)*(d*x^(1/3) + c)^6 + (56448*I*a*b*c^3 + 84672*I*b^2*c^2)*(d*x^(1/3) + c)^5 + (-44100*I*a
*b*c^4 - 88200*I*b^2*c^3)*(d*x^(1/3) + c)^4 + (23520*I*a*b*c^5 + 58800*I*b^2*c^4)*(d*x^(1/3) + c)^3 + (-8820*I
*a*b*c^6 - 26460*I*b^2*c^5)*(d*x^(1/3) + c)^2 + (2520*I*a*b*c^7 + 8820*I*b^2*c^6)*(d*x^(1/3) + c) + (-5040*I*(
d*x^(1/3) + c)^8*a*b - 1260*I*b^2*c^7 + (23040*I*a*b*c + 11520*I*b^2)*(d*x^(1/3) + c)^7 + (-47040*I*a*b*c^2 -
47040*I*b^2*c)*(d*x^(1/3) + c)^6 + (56448*I*a*b*c^3 + 84672*I*b^2*c^2)*(d*x^(1/3) + c)^5 + (-44100*I*a*b*c^4 -
88200*I*b^2*c^3)*(d*x^(1/3) + c)^4 + (23520*I*a*b*c^5 + 58800*I*b^2*c^4)*(d*x^(1/3) + c)^3 + (-8820*I*a*b*c^6
- 26460*I*b^2*c^5)*(d*x^(1/3) + c)^2 + (2520*I*a*b*c^7 + 8820*I*b^2*c^6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2
*c) + 12*(420*(d*x^(1/3) + c)^8*a*b + 105*b^2*c^7 - 960*(2*a*b*c + b^2)*(d*x^(1/3) + c)^7 + 3920*(a*b*c^2 + b^
2*c)*(d*x^(1/3) + c)^6 - 2352*(2*a*b*c^3 + 3*b^2*c^2)*(d*x^(1/3) + c)^5 + 3675*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3
) + c)^4 - 980*(2*a*b*c^5 + 5*b^2*c^4)*(d*x^(1/3) + c)^3 + 735*(a*b*c^6 + 3*b^2*c^5)*(d*x^(1/3) + c)^2 - 105*(
2*a*b*c^7 + 7*b^2*c^6)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*log(cos(2*d*x^(1/3) + 2*c)^2 + sin(2*d*x^(1/3)
+ 2*c)^2 + 2*cos(2*d*x^(1/3) + 2*c) + 1) + (1587600*I*a*b*cos(2*d*x^(1/3) + 2*c) - 1587600*a*b*sin(2*d*x^(1/3
) + 2*c) + 1587600*I*a*b)*polylog(9, -e^(2*I*d*x^(1/3) + 2*I*c)) + (3175200*(d*x^(1/3) + c)*a*b - 1814400*a*b*
c - 907200*b^2 + 453600*(7*(d*x^(1/3) + c)*a*b - 4*a*b*c - 2*b^2)*cos(2*d*x^(1/3) + 2*c) + (3175200*I*(d*x^(1/
3) + c)*a*b - 1814400*I*a*b*c - 907200*I*b^2)*sin(2*d*x^(1/3) + 2*c))*polylog(8, -e^(2*I*d*x^(1/3) + 2*I*c)) +
(-3175200*I*(d*x^(1/3) + c)^2*a*b - 1058400*I*a*b*c^2 - 1058400*I*b^2*c + (3628800*I*a*b*c + 1814400*I*b^2)*(
d*x^(1/3) + c) + (-3175200*I*(d*x^(1/3) + c)^2*a*b - 1058400*I*a*b*c^2 - 1058400*I*b^2*c + (3628800*I*a*b*c +
1814400*I*b^2)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + 151200*(21*(d*x^(1/3) + c)^2*a*b + 7*a*b*c^2 + 7*b^2*
c - 12*(2*a*b*c + b^2)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(7, -e^(2*I*d*x^(1/3) + 2*I*c)) - (2116
800*(d*x^(1/3) + c)^3*a*b - 423360*a*b*c^3 - 635040*b^2*c^2 - 1814400*(2*a*b*c + b^2)*(d*x^(1/3) + c)^2 + 2116
800*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c) + 30240*(70*(d*x^(1/3) + c)^3*a*b - 14*a*b*c^3 - 21*b^2*c^2 - 60*(2*a*b*
c + b^2)*(d*x^(1/3) + c)^2 + 70*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (-2116800*I*(d*x^(
1/3) + c)^3*a*b + 423360*I*a*b*c^3 + 635040*I*b^2*c^2 + (3628800*I*a*b*c + 1814400*I*b^2)*(d*x^(1/3) + c)^2 +
(-2116800*I*a*b*c^2 - 2116800*I*b^2*c)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(6, -e^(2*I*d*x^(1/3) +
2*I*c)) + (1058400*I*(d*x^(1/3) + c)^4*a*b + 132300*I*a*b*c^4 + 264600*I*b^2*c^3 + (-2419200*I*a*b*c - 120960
0*I*b^2)*(d*x^(1/3) + c)^3 + (2116800*I*a*b*c^2 + 2116800*I*b^2*c)*(d*x^(1/3) + c)^2 + (-846720*I*a*b*c^3 - 12
70080*I*b^2*c^2)*(d*x^(1/3) + c) + (1058400*I*(d*x^(1/3) + c)^4*a*b + 132300*I*a*b*c^4 + 264600*I*b^2*c^3 + (-
2419200*I*a*b*c - 1209600*I*b^2)*(d*x^(1/3) + c)^3 + (2116800*I*a*b*c^2 + 2116800*I*b^2*c)*(d*x^(1/3) + c)^2 +
(-846720*I*a*b*c^3 - 1270080*I*b^2*c^2)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - 3780*(280*(d*x^(1/3) + c)^4
*a*b + 35*a*b*c^4 + 70*b^2*c^3 - 320*(2*a*b*c + b^2)*(d*x^(1/3) + c)^3 + 560*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)
^2 - 112*(2*a*b*c^3 + 3*b^2*c^2)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(5, -e^(2*I*d*x^(1/3) + 2*I*c
)) + (423360*(d*x^(1/3) + c)^5*a*b - 35280*a*b*c^5 - 88200*b^2*c^4 - 604800*(2*a*b*c + b^2)*(d*x^(1/3) + c)^4
+ 1411200*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^3 - 423360*(2*a*b*c^3 + 3*b^2*c^2)*(d*x^(1/3) + c)^2 + 264600*(a*b
*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c) + 2520*(168*(d*x^(1/3) + c)^5*a*b - 14*a*b*c^5 - 35*b^2*c^4 - 240*(2*a*b*c +
b^2)*(d*x^(1/3) + c)^4 + 560*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^3 - 168*(2*a*b*c^3 + 3*b^2*c^2)*(d*x^(1/3) + c
)^2 + 105*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (423360*I*(d*x^(1/3) + c)^5*a*b - 35
280*I*a*b*c^5 - 88200*I*b^2*c^4 + (-1209600*I*a*b*c - 604800*I*b^2)*(d*x^(1/3) + c)^4 + (1411200*I*a*b*c^2 + 1
411200*I*b^2*c)*(d*x^(1/3) + c)^3 + (-846720*I*a*b*c^3 - 1270080*I*b^2*c^2)*(d*x^(1/3) + c)^2 + (264600*I*a*b*
c^4 + 529200*I*b^2*c^3)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(4, -e^(2*I*d*x^(1/3) + 2*I*c)) + (-14
1120*I*(d*x^(1/3) + c)^6*a*b - 8820*I*a*b*c^6 - 26460*I*b^2*c^5 + (483840*I*a*b*c + 241920*I*b^2)*(d*x^(1/3) +
c)^5 + (-705600*I*a*b*c^2 - 705600*I*b^2*c)*(d*x^(1/3) + c)^4 + (564480*I*a*b*c^3 + 846720*I*b^2*c^2)*(d*x^(1
/3) + c)^3 + (-264600*I*a*b*c^4 - 529200*I*b^2*c^3)*(d*x^(1/3) + c)^2 + (70560*I*a*b*c^5 + 176400*I*b^2*c^4)*(
d*x^(1/3) + c) + (-141120*I*(d*x^(1/3) + c)^6*a*b - 8820*I*a*b*c^6 - 26460*I*b^2*c^5 + (483840*I*a*b*c + 24192
0*I*b^2)*(d*x^(1/3) + c)^5 + (-705600*I*a*b*c^2 - 705600*I*b^2*c)*(d*x^(1/3) + c)^4 + (564480*I*a*b*c^3 + 8467
20*I*b^2*c^2)*(d*x^(1/3) + c)^3 + (-264600*I*a*b*c^4 - 529200*I*b^2*c^3)*(d*x^(1/3) + c)^2 + (70560*I*a*b*c^5
+ 176400*I*b^2*c^4)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + 1260*(112*(d*x^(1/3) + c)^6*a*b + 7*a*b*c^6 + 21
*b^2*c^5 - 192*(2*a*b*c + b^2)*(d*x^(1/3) + c)^5 + 560*(a*b*c^2 + b^2*c)*(d*x^(1/3) + c)^4 - 224*(2*a*b*c^3 +
3*b^2*c^2)*(d*x^(1/3) + c)^3 + 210*(a*b*c^4 + 2*b^2*c^3)*(d*x^(1/3) + c)^2 - 28*(2*a*b*c^5 + 5*b^2*c^4)*(d*x^(
1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(3, -e^(2*I*d*x^(1/3) + 2*I*c)) - (35*(2*I*a*b - b^2)*(d*x^(1/3) + c
)^9 - (630*I*b^2 - 315*(-2*I*a*b + b^2)*c)*(d*x^(1/3) + c)^8 - (-5040*I*b^2*c - 1260*(2*I*a*b - b^2)*c^2)*(d*x
^(1/3) + c)^7 - (17640*I*b^2*c^2 - 2940*(-2*I*a*b + b^2)*c^3)*(d*x^(1/3) + c)^6 - (-35280*I*b^2*c^3 - 4410*(2*
I*a*b - b^2)*c^4)*(d*x^(1/3) + c)^5 - (44100*I*b^2*c^4 - 4410*(-2*I*a*b + b^2)*c^5)*(d*x^(1/3) + c)^4 - (-3528
0*I*b^2*c^5 - 2940*(2*I*a*b - b^2)*c^6)*(d*x^(1/3) + c)^3 - (17640*I*b^2*c^6 - 1260*(-2*I*a*b + b^2)*c^7)*(d*x
^(1/3) + c)^2 - 315*(b^2*c^8 - 16*I*b^2*c^7)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))/(-315*I*cos(2*d*x^(1/3)
+ 2*c) + 315*sin(2*d*x^(1/3) + 2*c) - 315*I))/d^9

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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