Integrand size = 18, antiderivative size = 1170 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx =\text {Too large to display} \]
-1/2*e^(2/3)*p^2*ln(-d^(1/3)-e^(1/3)*x)^2/d^(2/3)-e^(2/3)*p^2*ln(-d^(1/3)- e^(1/3)*x)*ln((-(-1)^(2/3)*d^(1/3)-e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/d^(2 /3)-(-1)^(2/3)*e^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)+e^(1/3)*x)/(1+(-1)^(1/3) )/d^(1/3))*ln(-d^(1/3)+(-1)^(1/3)*e^(1/3)*x)/d^(2/3)-1/2*(-1)^(2/3)*e^(2/3 )*p^2*ln(-d^(1/3)+(-1)^(1/3)*e^(1/3)*x)^2/d^(2/3)+(-1)^(1/3)*e^(2/3)*p^2*l n(-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))*ln(-d^(1/3)-(-1) ^(2/3)*e^(1/3)*x)/d^(2/3)+(-1)^(1/3)*e^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)-(- 1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))*ln(-d^(1/3)-(-1)^(2/3)*e^(1/3) *x)/d^(2/3)+1/2*(-1)^(1/3)*e^(2/3)*p^2*ln(-d^(1/3)-(-1)^(2/3)*e^(1/3)*x)^2 /d^(2/3)-(-1)^(1/3)*e^(2/3)*p^2*ln((-1)^(1/3)*(d^(1/3)-(-1)^(1/3)*e^(1/3)* x)/(1+(-1)^(1/3))/d^(1/3))*ln((d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1+(-1)^(1/3) )/d^(1/3))/d^(2/3)-e^(2/3)*p^2*ln(-d^(1/3)-e^(1/3)*x)*ln((-1)^(1/3)*(d^(1/ 3)+(-1)^(2/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/d^(2/3)-(-1)^(1/3)*e^(2/3 )*p^2*ln(-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))*ln((d^(1/ 3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/d^(2/3)+(-1)^(2/3)*e^(2/3 )*p^2*ln(-(-1)^(1/3)*((-1)^(2/3)*d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3) )*ln(-(-1)^(2/3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/d^ (2/3)-(-1)^(2/3)*e^(2/3)*p^2*ln(-d^(1/3)+(-1)^(1/3)*e^(1/3)*x)*ln(-(-1)^(2 /3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/d^(2/3)+e^(2/3) *p*ln(-d^(1/3)-e^(1/3)*x)*ln(c*(e*x^3+d)^p)/d^(2/3)+(-1)^(2/3)*e^(2/3)*...
Time = 0.64 (sec) , antiderivative size = 766, normalized size of antiderivative = 0.65 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}-\frac {e^{2/3} p \left (p \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{e} x}{\sqrt [3]{d}}}{3 i+\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 (-1)^{2/3} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+2 \sqrt [3]{-1} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+(-1)^{2/3} p \left (\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \left (2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )+2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \operatorname {PolyLog}\left (2,\frac {-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )-\sqrt [3]{-1} p \left (\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \left (2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 p \operatorname {PolyLog}\left (2,\frac {2 i \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 i+\sqrt {3}}\right )\right )}{2 d^{2/3}} \]
-1/2*Log[c*(d + e*x^3)^p]^2/x^2 - (e^(2/3)*p*(p*Log[-d^(1/3) - e^(1/3)*x]^ 2 + 2*p*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*d^(1/3) - e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*p*Log[-d^(1/3) - e^(1/3)*x]*Log[(I + Sqrt[3] - ((2*I)*e^(1/3)*x)/d^(1/3))/(3*I + Sqrt[3])] - 2*Log[-d^(1/3) - e^(1/3)*x ]*Log[c*(d + e*x^3)^p] - 2*(-1)^(2/3)*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x] *Log[c*(d + e*x^3)^p] + 2*(-1)^(1/3)*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]* Log[c*(d + e*x^3)^p] + 2*p*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/ 3))*d^(1/3))] + (-1)^(2/3)*p*(Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*(2*Log[ ((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))] + Log[-d^(1 /3) + (-1)^(1/3)*e^(1/3)*x] + 2*Log[((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1 /3)*x))/((-1 + (-1)^(2/3))*d^(1/3))]) + 2*PolyLog[2, (d^(1/3) - (-1)^(1/3) *e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*PolyLog[2, (-d^(1/3) + (-1)^(1 /3)*e^(1/3)*x)/((-1 + (-1)^(2/3))*d^(1/3))]) - (-1)^(1/3)*p*(Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*(2*Log[((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((-1 + ( -1)^(2/3))*d^(1/3))] + 2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x)) /((1 + (-1)^(1/3))*d^(1/3))] + Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]) + 2*P olyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2 *PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))]) + 2*p*PolyLog[2, ((2*I)*(1 + (e^(1/3)*x)/d^(1/3)))/(3*I + Sqrt[3])]))/(2*d ^(2/3))
Time = 1.61 (sec) , antiderivative size = 1183, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2907, 2921, 2009}
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.
| \(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2907 |
| \(\displaystyle 3 e p \int \frac {\log \left (c \left (e x^3+d\right )^p\right )}{e x^3+d}dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle 3 e p \int \left (-\frac {\log \left (c \left (e x^3+d\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}-\frac {\log \left (c \left (e x^3+d\right )^p\right )}{3 d^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}-\frac {\log \left (c \left (e x^3+d\right )^p\right )}{3 d^{2/3} \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )}\right )dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 e p \left (-\frac {p \log ^2\left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {p \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {p \log \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {\log \left (c \left (e x^3+d\right )^p\right ) \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} p \log ^2\left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 d^{2/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} p \log ^2\left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} p \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {\sqrt [3]{-1} p \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} p \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {\sqrt [3]{-1} \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {p \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} p \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac {\sqrt [3]{-1} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} p \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 d^{2/3} \sqrt [3]{e}}\right )-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{2 x^2}\) |
-1/2*Log[c*(d + e*x^3)^p]^2/x^2 + 3*e*p*(-1/6*(p*Log[-d^(1/3) - e^(1/3)*x] ^2)/(d^(2/3)*e^(1/3)) - (p*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^( 1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/(3*d^(2/3)*e^(1/3)) - ((-1 )^(2/3)*p*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3) )]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/(3*d^(2/3)*e^(1/3)) - ((-1)^(2/3) *p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/(6*d^(2/3)*e^(1/3)) + ((-1)^(1/ 3)*p*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))] *Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/(3*d^(2/3)*e^(1/3)) + ((-1)^(1/3)*p *Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/ 3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/(3*d^(2/3)*e^(1/3)) + ((-1)^(1/ 3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/(6*d^(2/3)*e^(1/3)) - ((-1)^( 1/3)*p*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3)) *d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3)) ])/(3*d^(2/3)*e^(1/3)) - (p*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^( 1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(3*d^(2/3)*e^(1 /3)) - ((-1)^(2/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3 )*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(3*d^(2/ 3)*e^(1/3)) + (Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*d^(2/3)* e^(1/3)) + ((-1)^(2/3)*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x ^3)^p])/(3*d^(2/3)*e^(1/3)) - ((-1)^(1/3)*Log[-d^(1/3) - (-1)^(2/3)*e^(...
3.2.36.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*( x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q /(f*(m + 1))), x] - Simp[b*e*n*p*(q/(f^n*(m + 1))) Int[(f*x)^(m + n)*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d , e, f, m, p}, x] && IGtQ[q, 1] && IntegerQ[n] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.77 (sec) , antiderivative size = 1787, normalized size of antiderivative = 1.53
-1/2/x^2*ln((e*x^3+d)^p)^2-p^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*ln(e*x^3+d)+p /(d/e)^(2/3)*ln(x+(d/e)^(1/3))*ln((e*x^3+d)^p)+1/2*p^2/(d/e)^(2/3)*ln(x^2- (d/e)^(1/3)*x+(d/e)^(2/3))*ln(e*x^3+d)-1/2*p/(d/e)^(2/3)*ln(x^2-(d/e)^(1/3 )*x+(d/e)^(2/3))*ln((e*x^3+d)^p)-p^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2 )*(2/(d/e)^(1/3)*x-1))*ln(e*x^3+d)+p/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2 )*(2/(d/e)^(1/3)*x-1))*ln((e*x^3+d)^p)+1/2*p^2*sum(1/_alpha^2*(2*ln(x-_alp ha)*ln(e*x^3+d)-e*(1/_alpha^2/e*ln(x-_alpha)^2+2*_alpha*ln(x-_alpha)*(2*Ro otOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha ^2,index=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/Root Of(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1))+2*RootOf(_Z^2+3*_Z*_alpha+3*_alph a^2,index=1)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*ln((RootOf(_Z^2+3 *_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha^ 2,index=2))+3*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((RootOf(_Z^2+ 3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+3*_alpha ^2,index=1))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((Root Of(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alpha+ 3*_alpha^2,index=2))*_alpha+6*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)* ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z *_alpha+3*_alpha^2,index=1))*_alpha+3*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,i ndex=2)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootO...
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^3} \,d x \]