Integrand size = 24, antiderivative size = 93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e}{d \sqrt [3]{x}}\right ) \]
-3*(a+b*ln(c*(d+e/x^(1/3))^n))^2*ln(-e/d/x^(1/3))-6*b*n*(a+b*ln(c*(d+e/x^( 1/3))^n))*polylog(2,1+e/d/x^(1/3))+6*b^2*n^2*polylog(3,1+e/d/x^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(93)=186\).
Time = 0.15 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.18 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (\left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (1+\frac {e}{d \sqrt [3]{x}}\right )\right ) \log (x)+3 \operatorname {PolyLog}\left (2,-\frac {e}{d \sqrt [3]{x}}\right )\right )+3 b^2 n^2 \left (2 \log \left (\frac {e}{d}+\sqrt [3]{x}\right ) \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )-2 \left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (\frac {e}{d}+\sqrt [3]{x}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d \sqrt [3]{x}}{e}\right )+\frac {1}{81} \left (81 \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )+27 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \log (x)-27 \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right ) \log (x)-54 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log (x)+54 \log \left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log (x)+9 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log ^2(x)-9 \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log ^2(x)+\log ^3(x)-162 \operatorname {PolyLog}\left (3,1+\frac {d \sqrt [3]{x}}{e}\right )-162 \operatorname {PolyLog}\left (3,-\frac {d \sqrt [3]{x}}{e}\right )\right )\right ) \]
(a - b*n*Log[d + e/x^(1/3)] + b*Log[c*(d + e/x^(1/3))^n])^2*Log[x] + 2*b*n *(a - b*n*Log[d + e/x^(1/3)] + b*Log[c*(d + e/x^(1/3))^n])*((Log[d + e/x^( 1/3)] - Log[1 + e/(d*x^(1/3))])*Log[x] + 3*PolyLog[2, -(e/(d*x^(1/3)))]) + 3*b^2*n^2*(2*Log[e/d + x^(1/3)]*PolyLog[2, 1 + (d*x^(1/3))/e] - 2*(Log[d + e/x^(1/3)] - Log[e/d + x^(1/3)])*PolyLog[2, -((d*x^(1/3))/e)] + (81*Log[ e/d + x^(1/3)]^2*Log[-((d*x^(1/3))/e)] + 27*Log[d + e/x^(1/3)]^2*Log[x] - 27*Log[e/d + x^(1/3)]^2*Log[x] - 54*Log[d + e/x^(1/3)]*Log[1 + (d*x^(1/3)) /e]*Log[x] + 54*Log[e/d + x^(1/3)]*Log[1 + (d*x^(1/3))/e]*Log[x] + 9*Log[d + e/x^(1/3)]*Log[x]^2 - 9*Log[1 + (d*x^(1/3))/e]*Log[x]^2 + Log[x]^3 - 16 2*PolyLog[3, 1 + (d*x^(1/3))/e] - 162*PolyLog[3, -((d*x^(1/3))/e)])/81)
Time = 0.43 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2904, 2843, 2881, 2821, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-2 b e n \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}\right )\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-2 b n \int \sqrt [3]{x} \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-2 b n \left (b n \int \sqrt [3]{x} \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -3 \left (\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-2 b n \left (b n \operatorname {PolyLog}\left (3,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )-\operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )\right )\right )\) |
-3*((a + b*Log[c*(d + e/x^(1/3))^n])^2*Log[-(e/(d*x^(1/3)))] - 2*b*n*(-((a + b*Log[c/x^(n/3)])*PolyLog[2, (d + e/x^(1/3))/d]) + b*n*PolyLog[3, (d + e/x^(1/3))/d]))
3.5.100.3.1 Defintions of rubi rules used
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}}{x}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
integral((b^2*log(c*((d*x + e*x^(2/3))/x)^n)^2 + 2*a*b*log(c*((d*x + e*x^( 2/3))/x)^n) + a^2)/x, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{2}}{x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
b^2*log((d*x^(1/3) + e)^n)^2*log(x) - integrate(-1/3*(3*(b^2*d*x + b^2*e*x ^(2/3))*log(x^(1/3*n))^2 + 3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x - 2*(b^2*d*n*x*log(x) - 3*(b^2*d*log(c) + a*b*d)*x + 3*(b^2*d*x + b^2*e*x^( 2/3))*log(x^(1/3*n)) - 3*(b^2*e*log(c) + a*b*e)*x^(2/3))*log((d*x^(1/3) + e)^n) - 6*((b^2*d*log(c) + a*b*d)*x + (b^2*e*log(c) + a*b*e)*x^(2/3))*log( x^(1/3*n)) + 3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(2/3))/(d*x^2 + e*x^(5/3)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2}{x} \,d x \]