\(\int \frac {\log ^2(c (d+e x^3)^p)}{x^2} \, dx\) [135]

Optimal result

Integrand size = 18, antiderivative size = 1137 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx =\text {Too large to display} \] Output:

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*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*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^(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*ln(d^(1/3)-(-1)^(1/3)*e^(1/3)*x)*ln(c*( 
e*x^3+d)^p)/d^(1/3)-2*(-1)^(2/3)*e^(1/3)*p*ln(d^(1/3)+(-1)^(2/3)*e^(1/3)*x 
)*ln(c*(e*x^3+d)^p)/d^(1/3)-ln(c*(e*x^3+d)^p)^2/x+2*e^(1/3)*p^2*polylog(2, 
(d^(1/3)+e^(1/3)*x)/(1+(-1)^(1/3))/d^(1/3))/d^(1/3)-2*(-1)^(2/3)*e^(1/3)*p 
^2*polylog(2,-(-1)^(2/3)*(d^(1/3)+e^(1/3)*x)/(1-(-1)^(2/3))/d^(1/3))/d^...
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 972, normalized size of antiderivative = 0.85 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx =\text {Too large to display} \] Input:

Integrate[Log[c*(d + e*x^3)^p]^2/x^2,x]
 

Output:

-(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)...
 

Rubi [A] (verified)

Time = 2.84 (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}\)

Input:

Int[Log[c*(d + e*x^3)^p]^2/x^2,x]
 

Output:

-(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 + ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2907

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]
 

rule 2926

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.10 (sec) , antiderivative size = 1787, normalized size of antiderivative = 1.57

method	result	size
risch	\(\text {Expression too large to display}\)	\(1787\)

Input:

int(ln(c*(e*x^3+d)^p)^2/x^2,x,method=_RETURNVERBOSE)
 

Output:

-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)*(9*_alpha^2*ln( 
(RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)-x+_alpha)/RootOf(_Z^2+3*_Z*_a 
lpha+3*_alpha^2,index=1))+9*_alpha^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))+6*_alpha 
*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))+ 
3*_alpha*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,in 
dex=2))+3*_alpha*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*_al 
pha^2,index=1))+6*_alpha*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((R 
ootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)-x+_alpha)/RootOf(_Z^2+3*_Z*_alp 
ha+3*_alpha^2,index=2))+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=2)*Root 
Of(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((RootOf(_Z^2+3*_Z*_alpha+3*_...
 

Fricas [F]

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

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

Output:

integral(log((e*x^3 + d)^p*c)^2/x^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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
 

Giac [F]

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

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

Output:

integrate(log((e*x^3 + d)^p*c)^2/x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx=\frac {-6 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) e \,p^{2} x -6 e^{\frac {2}{3}} d^{\frac {4}{3}} \left (\int \frac {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}{e \,x^{5}+d \,x^{2}}d x \right ) p x -e^{\frac {2}{3}} d^{\frac {1}{3}} {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2}-6 e^{\frac {2}{3}} d^{\frac {1}{3}} \mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) p -9 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) e \,p^{2} x +3 \,\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) e p x}{e^{\frac {2}{3}} d^{\frac {1}{3}} x} \] Input:

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

Output:

( - 6*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*e*p**2*x 
- 6*e**(2/3)*d**(1/3)*int(log((d + e*x**3)**p*c)/(d*x**2 + e*x**5),x)*d*p* 
x - e**(2/3)*d**(1/3)*log((d + e*x**3)**p*c)**2 - 6*e**(2/3)*d**(1/3)*log( 
(d + e*x**3)**p*c)*p - 9*log(d**(1/3) + e**(1/3)*x)*e*p**2*x + 3*log((d + 
e*x**3)**p*c)*e*p*x)/(e**(2/3)*d**(1/3)*x)