Integrand size = 18, antiderivative size = 1137 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx =\text {Too large to display} \]
e^(1/3)*p^2*ln(d^(1/3)+e^(1/3)*x)^2/d^(1/3)+2*e^(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))/d^(1/3)-2 *(-1)^(1/3)*e^(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)/d^(1/3)-(-1)^(1/3)*e^(1/3)*p^2*ln (d^(1/3)-(-1)^(1/3)*e^(1/3)*x)^2/d^(1/3)+2*(-1)^(2/3)*e^(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)/d^(1/3)+2*(-1)^(2/3)*e^(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)/d^( 1/3)+(-1)^(2/3)*e^(1/3)*p^2*ln(d^(1/3)+(-1)^(2/3)*e^(1/3)*x)^2/d^(1/3)-2*( -1)^(2/3)*e^(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)) /d^(1/3)+2*e^(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))/d^(1/3)-2*(-1)^(2/3)*e^(1/3)*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)/(1-(-1)^(2/3))/d^(1/3))/d^(1/3)+2*(-1)^(1/3)*e^(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))/d^(1/3) -2*(-1)^(1/3)*e^(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))/d^(1/3)-2*e^(1/3)*p *ln(d^(1/3)+e^(1/3)*x)*ln(c*(e*x^3+d)^p)/d^(1/3)+2*(-1)^(1/3)*e^(1/3)*p...
Time = 0.35 (sec) , antiderivative size = 972, normalized size of antiderivative = 0.85 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}+6 e p \left (\frac {p \log ^2\left (-\sqrt [3]{d}-\sqrt [3]{e} x\right )}{6 \sqrt [3]{d} e^{2/3}}+\frac {p \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 )}{3 \sqrt [3]{d} e^{2/3}}+\frac {p \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 )}{3 \sqrt [3]{d} e^{2/3}}-\frac {\log \left (-\sqrt [3]{d}-\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac {(-1)^{2/3} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} p \left (\frac {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 )}{e^{2/3}}+\frac {\log ^2\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {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 )}{e^{2/3}}+\frac {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 )}{e^{2/3}}+\frac {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 )}{e^{2/3}}\right )}{6 \sqrt [3]{d}}+\frac {(-1)^{2/3} p \left (\frac {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 )}{e^{2/3}}+\frac {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 )}{e^{2/3}}+\frac {\log ^2\left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {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 )}{e^{2/3}}+\frac {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 )}{e^{2/3}}\right )}{6 \sqrt [3]{d}}\right ) \]
-(Log[c*(d + e*x^3)^p]^2/x) + 6*e*p*((p*Log[-d^(1/3) - e^(1/3)*x]^2)/(6*d^ (1/3)*e^(2/3)) + (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^(1/3)*e^(2/3)) + (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^(1/3)*e^(2/3)) - (Log[-d^(1/3) - e^(1/3)*x]*Lo g[c*(d + e*x^3)^p])/(3*d^(1/3)*e^(2/3)) + ((-1)^(1/3)*Log[-d^(1/3) + (-1)^ (1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*d^(1/3)*e^(2/3)) - ((-1)^(2/3)*L og[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*d^(1/3)*e^(2/ 3)) + (p*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(3* d^(1/3)*e^(2/3)) + (p*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d ^(1/3))])/(3*d^(1/3)*e^(2/3)) - ((-1)^(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^(2/3) + Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2/e^(2/3) + (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^(2/3) + (2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (2*PolyLog[2 , (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3)))/ (6*d^(1/3)) + ((-1)^(2/3)*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^(2/3) + (2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3)...
Time = 1.54 (sec) , antiderivative size = 1153, 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, 2926, 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.
\(\displaystyle \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2907 |
\(\displaystyle 6 e p \int \frac {x \log \left (c \left (e x^3+d\right )^p\right )}{e x^3+d}dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle 6 e p \int \left (-\frac {\log \left (c \left (e x^3+d\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac {(-1)^{2/3} \log \left (c \left (e x^3+d\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (e x^3+d\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}\right )dx-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 e p \left (\frac {p \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 \sqrt [3]{d} e^{2/3}}+\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 \sqrt [3]{d} e^{2/3}}+\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 \sqrt [3]{d} e^{2/3}}-\frac {\log \left (c \left (e x^3+d\right )^p\right ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} p \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{6 \sqrt [3]{d} e^{2/3}}+\frac {(-1)^{2/3} p \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{6 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} 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]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac {(-1)^{2/3} 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 \sqrt [3]{d} e^{2/3}}+\frac {(-1)^{2/3} 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 \sqrt [3]{d} e^{2/3}}-\frac {(-1)^{2/3} 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 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\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 \sqrt [3]{d} e^{2/3}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (e x^3+d\right )^p\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac {(-1)^{2/3} \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 \sqrt [3]{d} e^{2/3}}+\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 \sqrt [3]{d} e^{2/3}}+\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 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} 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 \sqrt [3]{d} e^{2/3}}-\frac {\sqrt [3]{-1} 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 \sqrt [3]{d} e^{2/3}}-\frac {(-1)^{2/3} 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 \sqrt [3]{d} e^{2/3}}+\frac {(-1)^{2/3} 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 \sqrt [3]{d} e^{2/3}}\right )-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{x}\) |
-(Log[c*(d + e*x^3)^p]^2/x) + 6*e*p*((p*Log[d^(1/3) + e^(1/3)*x]^2)/(6*d^( 1/3)*e^(2/3)) + (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^(1/3)*e^(2/3)) - ((-1)^(1/3)*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^(1/3)*e^(2/3)) - ((-1)^(1/3)*p*Log[d^( 1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(6*d^(1/3)*e^(2/3)) + ((-1)^(2/3)*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^(1/3)*e^(2/3)) + ((-1)^(2/3)*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^(1/3)*e^(2/3)) + ((-1)^(2/3)*p*Log[d^(1 /3) + (-1)^(2/3)*e^(1/3)*x]^2)/(6*d^(1/3)*e^(2/3)) - ((-1)^(2/3)*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^(1/3)* e^(2/3)) + (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^(1/3)*e^(2/3)) - ((-1)^(1 /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^(1/3)*e^(2/3)) - (L og[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*d^(1/3)*e^(2/3)) + ((-1)^ (1/3)*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(3*d^(1/3) *e^(2/3)) - ((-1)^(2/3)*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + ...
3.2.35.3.1 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_.)*(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.64 (sec) , antiderivative size = 1787, normalized size of antiderivative = 1.57
-1/x*ln((e*x^3+d)^p)^2+2*p^2/(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln(e*x^3+d)-2*p /(d/e)^(1/3)*ln(x+(d/e)^(1/3))*ln((e*x^3+d)^p)-p^2/(d/e)^(1/3)*ln(x^2-(d/e )^(1/3)*x+(d/e)^(2/3))*ln(e*x^3+d)+p/(d/e)^(1/3)*ln(x^2-(d/e)^(1/3)*x+(d/e )^(2/3))*ln((e*x^3+d)^p)-2*p^2*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/( d/e)^(1/3)*x-1))*ln(e*x^3+d)+2*p*3^(1/2)/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2 /(d/e)^(1/3)*x-1))*ln((e*x^3+d)^p)+p^2*sum(1/_alpha*(2*ln(x-_alpha)*ln(e*x ^3+d)-e*(1/_alpha^2/e*ln(x-_alpha)^2+2*_alpha*ln(x-_alpha)*(2*RootOf(_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)/RootOf(_Z^2+3* _Z*_alpha+3*_alpha^2,index=1))+2*RootOf(_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=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*_alph a+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((RootOf(_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,index=2)*ln ((RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_...
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\int \frac {\log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^2} \,d x \]