\(\int \frac {\log ^2(c (d+e x^3)^p)}{x^3} \, dx\) [136]

Optimal result

Integrand size = 18, antiderivative size = 1170 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 766, normalized size of antiderivative = 0.65 \[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-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))
 

Rubi [A] (verified)

Time = 2.81 (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}\)

Input:

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

Output:

-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^(...
 

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 2921

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]))
 

Maple [C] (warning: unable to verify)

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

Time = 1.16 (sec) , antiderivative size = 1787, normalized size of antiderivative = 1.53

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

Input:

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

Output:

-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)*(9*_a 
lpha^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))+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,index=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*_a 
lpha+3*_alpha^2,index=1))+6*_alpha*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,inde 
x=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))+2*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,ind 
ex=2)*RootOf(_Z^2+3*_Z*_alpha+3*_alpha^2,index=1)*ln((RootOf(_Z^2+3*_Z*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^3} \, dx=\frac {-6 \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) e \,p^{2} x^{2}-12 e^{\frac {1}{3}} d^{\frac {5}{3}} \left (\int \frac {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}{e \,x^{6}+d \,x^{3}}d x \right ) p \,x^{2}-2 e^{\frac {1}{3}} d^{\frac {2}{3}} {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2}-6 e^{\frac {1}{3}} d^{\frac {2}{3}} \mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) p +9 \,\mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) e \,p^{2} x^{2}-3 \,\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) e p \,x^{2}}{4 e^{\frac {1}{3}} d^{\frac {2}{3}} x^{2}} \] Input:

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

Output:

( - 6*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*e*p**2*x* 
*2 - 12*e**(1/3)*d**(2/3)*int(log((d + e*x**3)**p*c)/(d*x**3 + e*x**6),x)* 
d*p*x**2 - 2*e**(1/3)*d**(2/3)*log((d + e*x**3)**p*c)**2 - 6*e**(1/3)*d**( 
2/3)*log((d + e*x**3)**p*c)*p + 9*log(d**(1/3) + e**(1/3)*x)*e*p**2*x**2 - 
 3*log((d + e*x**3)**p*c)*e*p*x**2)/(4*e**(1/3)*d**(2/3)*x**2)