\(\int \log ^2(c (d+e x^3)^p) \, dx\) [134]

Optimal result

Integrand size = 14, antiderivative size = 1304 \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=18 p^2 x+\frac {6 \sqrt {3} \sqrt [3]{d} p^2 \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {\sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} p^2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} p^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 )}{\sqrt [3]{e}}-\frac {(-1)^{2/3} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p^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 ) \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p^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 )}{\sqrt [3]{e}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} p^2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p^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 ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} p^2 \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \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 )}{\sqrt [3]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {2 \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} p^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 )}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p^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 )}{\sqrt [3]{e}} \]

[Out]

2*(-1)^(2/3)*d^(1/3)*p^2*polylog(2,-(-1)^(2/3)*(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/e^(1/3)-
2*(-1)^(1/3)*d^(1/3)*p^2*polylog(2,(-1)^(1/3)*(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(1/3)-2
*(-1)^(2/3)*d^(1/3)*p^2*polylog(2,(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(1/3)-2*d^(1/3)*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))/e^(1/3)+2*d^(1/3)
*p*ln(-d^(1/3)-e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(1/3)+6*d^(1/3)*p^2*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*3^(
1/2))*3^(1/2)/e^(1/3)-2*(-1)^(1/3)*d^(1/3)*p^2*polylog(2,-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3
))/e^(1/3)-2*d^(1/3)*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))/e^(
1/3)+18*p^2*x-6*p*x*ln(c*(e*x^3+d)^p)+3*d^(1/3)*p^2*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/e^(1/3)-2*d^(1/3
)*p^2*polylog(2,2*(d^(1/3)+e^(1/3)*x)/d^(1/3)/(3-I*3^(1/2)))/e^(1/3)-6*d^(1/3)*p^2*ln(d^(1/3)+e^(1/3)*x)/e^(1/
3)-2*d^(1/3)*p^2*polylog(2,(d^(1/3)+e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/e^(1/3)-d^(1/3)*p^2*ln(-d^(1/3)-e^(1/3)
*x)^2/e^(1/3)-2*(-1)^(2/3)*d^(1/3)*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)/e^(1/3)+2*(-1)^(1/3)*d^(1/3)*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)/e^(1/3)+2*(-1)^(1/3)*d^(1/3)*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)/e^(1/3)-2*(-1)^(1/3)*d^(1/3)*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
))/e^(1/3)-2*(-1)^(1/3)*d^(1/3)*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))/e^(1/3)+2*(-1)^(2/3)*d^(1/3)*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))/e^(1
/3)-2*(-1)^(2/3)*d^(1/3)*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))/e^(1/3)+2*(-1)^(2/3)*d^(1/3)*p*ln(-d^(1/3)+(-1)^(1/3)*e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(1
/3)-2*(-1)^(1/3)*d^(1/3)*p*ln(-d^(1/3)-(-1)^(2/3)*e^(1/3)*x)*ln(c*(e*x^3+d)^p)/e^(1/3)+(-1)^(1/3)*d^(1/3)*p^2*
ln(-d^(1/3)-(-1)^(2/3)*e^(1/3)*x)^2/e^(1/3)-(-1)^(2/3)*d^(1/3)*p^2*ln(-d^(1/3)+(-1)^(1/3)*e^(1/3)*x)^2/e^(1/3)
+x*ln(c*(e*x^3+d)^p)^2

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 1310, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {2500, 2526, 2498, 327, 206, 31, 648, 631, 210, 642, 2521, 2512, 266, 2463, 2437, 2338, 2441, 2440, 2438, 12} \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=-\frac {\sqrt [3]{d} \log ^2\left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log ^2\left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log ^2\left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}+18 x p^2+\frac {6 \sqrt {3} \sqrt [3]{d} \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {6 \sqrt [3]{d} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \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 ) p^2}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}+\frac {3 \sqrt [3]{d} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-\frac {2 (-1)^{2/3} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \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 ) p^2}{\sqrt [3]{e}}-6 x \log \left (c \left (e x^3+d\right )^p\right ) p+\frac {2 \sqrt [3]{d} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{\sqrt [3]{e}}+x \log ^2\left (c \left (e x^3+d\right )^p\right ) \]

[In]

Int[Log[c*(d + e*x^3)^p]^2,x]

[Out]

18*p^2*x + (6*Sqrt[3]*d^(1/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/e^(1/3) - (d^(1/3)*p^2*Lo
g[-d^(1/3) - e^(1/3)*x]^2)/e^(1/3) - (6*d^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/e^(1/3) - (2*d^(1/3)*p^2*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)))])/e^(1/3) - (2*(-1)^(2/3
)*d^(1/3)*p^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])/e^(1/3) - ((-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/e^(1/3) + (2*(-1)^(1/3)*d^(
1/3)*p^2*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])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^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])/e^(1/3) + ((-1)^(1/3)*d^(1/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e
^(1/3)*x]^2)/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^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)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*d^(1/3)*p^2*Lo
g[-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))])/e^(1/3)
- (2*(-1)^(2/3)*d^(1/3)*p^2*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)))])/e^(1/3) + (3*d^(1/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])
/e^(1/3) - 6*p*x*Log[c*(d + e*x^3)^p] + (2*d^(1/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) +
(2*(-1)^(2/3)*d^(1/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) - (2*(-1)^(1/3)*d^
(1/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + x*Log[c*(d + e*x^3)^p]^2 - (2*d^(
1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*d^(1/3)*p^2*PolyLog[2, (2*
(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(1/3
)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[
2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2
, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p
^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(1/3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2500

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[x^n*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 2521

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]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 e p) \int \frac {x^3 \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx \\ & = x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 e p) \int \left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^3\right )^p\right )}{e \left (d+e x^3\right )}\right ) \, dx \\ & = x \log ^2\left (c \left (d+e x^3\right )^p\right )-(6 p) \int \log \left (c \left (d+e x^3\right )^p\right ) \, dx+(6 d p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx \\ & = -6 p x \log \left (c \left (d+e x^3\right )^p\right )+x \log ^2\left (c \left (d+e x^3\right )^p\right )+(6 d p) \int \left (-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx+\left (18 e p^2\right ) \int \frac {x^3}{d+e x^3} \, dx \\ & = 18 p^2 x-6 p x \log \left (c \left (d+e x^3\right )^p\right )+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (2 \sqrt [3]{d} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}-\sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x} \, dx-\left (18 d p^2\right ) \int \frac {1}{d+e x^3} \, dx \\ & = 18 p^2 x-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (6 \sqrt [3]{d} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (6 \sqrt [3]{d} p^2\right ) \int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx-\left (6 \sqrt [3]{d} e^{2/3} p^2\right ) \int \frac {x^2 \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{d+e x^3} \, dx+\left (6 \sqrt [3]{-1} \sqrt [3]{d} e^{2/3} p^2\right ) \int \frac {x^2 \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx-\left (6 (-1)^{2/3} \sqrt [3]{d} e^{2/3} p^2\right ) \int \frac {x^2 \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx \\ & = 18 p^2 x-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (9 d^{2/3} p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx+\frac {\left (3 \sqrt [3]{d} p^2\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{\sqrt [3]{e}}-\left (6 \sqrt [3]{d} e^{2/3} p^2\right ) \int \left (\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx+\left (6 \sqrt [3]{-1} \sqrt [3]{d} e^{2/3} p^2\right ) \int \left (\frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx-\left (6 (-1)^{2/3} \sqrt [3]{d} e^{2/3} p^2\right ) \int \left (\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx \\ & = 18 p^2 x-\frac {6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}+\frac {3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}-6 p x \log \left (c \left (d+e x^3\right )^p\right )+\frac {2 \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+\frac {2 (-1)^{2/3} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{e}}+x \log ^2\left (c \left (d+e x^3\right )^p\right )-\left (2 \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx+\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx+\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx+\left (2 \sqrt [3]{-1} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\left (2 (-1)^{2/3} \sqrt [3]{d} p^2\right ) \int \frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx-\frac {\left (18 \sqrt [3]{d} p^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 1101, normalized size of antiderivative = 0.84 \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {18 \sqrt [3]{e} p^2 x+6 \sqrt {3} \sqrt [3]{d} p^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )-\sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )-2 \sqrt [3]{d} p^2 \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 )-6 \sqrt [3]{d} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-2 (-1)^{2/3} \sqrt [3]{d} p^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 )-(-1)^{2/3} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )+2 \sqrt [3]{-1} \sqrt [3]{d} p^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 ) \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )+2 \sqrt [3]{-1} \sqrt [3]{d} p^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 )+\sqrt [3]{-1} \sqrt [3]{d} p^2 \log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )-2 (-1)^{2/3} \sqrt [3]{d} p^2 \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \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 )-2 \sqrt [3]{d} p^2 \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 )+3 \sqrt [3]{d} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-6 \sqrt [3]{e} p x \log \left (c \left (d+e x^3\right )^p\right )+2 \sqrt [3]{d} p \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} \sqrt [3]{d} p \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} \sqrt [3]{d} p \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )+\sqrt [3]{e} x \log ^2\left (c \left (d+e x^3\right )^p\right )-2 \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )-2 (-1)^{2/3} \sqrt [3]{d} p^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 (-1)^{2/3} \sqrt [3]{d} p^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 )+2 \sqrt [3]{-1} \sqrt [3]{d} p^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 \sqrt [3]{-1} \sqrt [3]{d} p^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 )-2 \sqrt [3]{d} p^2 \operatorname {PolyLog}\left (2,\frac {2 i \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 i+\sqrt {3}}\right )}{\sqrt [3]{e}} \]

[In]

Integrate[Log[c*(d + e*x^3)^p]^2,x]

[Out]

(18*e^(1/3)*p^2*x + 6*Sqrt[3]*d^(1/3)*p^2*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - d^(1/3)*p^2*Log[-d^(1/
3) - e^(1/3)*x]^2 - 2*d^(1/3)*p^2*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))] - 6*d^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x] - 2*(-1)^(2/3)*d^(1/3)*p^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] - (-1)^(2/3)*d^(1/3)*p^2*Log[-d^(
1/3) + (-1)^(1/3)*e^(1/3)*x]^2 + 2*(-1)^(1/3)*d^(1/3)*p^2*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] + 2*(-1)^(1/3)*d^(1/3)*p^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] + (-1)^(1/3)*d^(1/3)*p^2*
Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2 - 2*(-1)^(2/3)*d^(1/3)*p^2*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))] - 2*d^(1/3)*p^2*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])] + 3*d^(1/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/
3)*x + e^(2/3)*x^2] - 6*e^(1/3)*p*x*Log[c*(d + e*x^3)^p] + 2*d^(1/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*
x^3)^p] + 2*(-1)^(2/3)*d^(1/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p] - 2*(-1)^(1/3)*d^(1
/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p] + e^(1/3)*x*Log[c*(d + e*x^3)^p]^2 - 2*d^(1/3)
*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] - 2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/
3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] - 2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, (-d^(1/3) + (-1)^
(1/3)*e^(1/3)*x)/((-1 + (-1)^(2/3))*d^(1/3))] + 2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1
/3)*x)/((1 + (-1)^(1/3))*d^(1/3))] + 2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1
- (-1)^(2/3))*d^(1/3))] - 2*d^(1/3)*p^2*PolyLog[2, ((2*I)*(1 + (e^(1/3)*x)/d^(1/3)))/(3*I + Sqrt[3])])/e^(1/3)

Maple [F]

\[\int {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2}d x\]

[In]

int(ln(c*(e*x^3+d)^p)^2,x)

[Out]

int(ln(c*(e*x^3+d)^p)^2,x)

Fricas [F]

\[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int { \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

integrate(log(c*(e*x^3+d)^p)^2,x, algorithm="fricas")

[Out]

integral(log((e*x^3 + d)^p*c)^2, x)

Sympy [F]

\[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}\, dx \]

[In]

integrate(ln(c*(e*x**3+d)**p)**2,x)

[Out]

Integral(log(c*(d + e*x**3)**p)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(log(c*(e*x^3+d)^p)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int { \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

integrate(log(c*(e*x^3+d)^p)^2,x, algorithm="giac")

[Out]

integrate(log((e*x^3 + d)^p*c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2 \,d x \]

[In]

int(log(c*(d + e*x^3)^p)^2,x)

[Out]

int(log(c*(d + e*x^3)^p)^2, x)