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

Optimal result

Integrand size = 14, antiderivative size = 1304 \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx =\text {Too large to display} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.83 (sec) , antiderivative size = 1101, normalized size of antiderivative = 0.84 \[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(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...
 

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 1316, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2900, 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.

\(\displaystyle \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2900

\(\displaystyle x \log ^2\left (c \left (d+e x^3\right )^p\right )-6 e p \int \frac {x^3 \log \left (c \left (e x^3+d\right )^p\right )}{e x^3+d}dx\)

\(\Big \downarrow \) 2926

\(\displaystyle x \log ^2\left (c \left (d+e x^3\right )^p\right )-6 e p \int \left (\frac {\log \left (c \left (e x^3+d\right )^p\right )}{e}-\frac {d \log \left (c \left (e x^3+d\right )^p\right )}{e \left (e x^3+d\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x \log ^2\left (c \left (e x^3+d\right )^p\right )-6 e p \left (\frac {\sqrt [3]{d} p \log ^2\left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 e^{4/3}}+\frac {\sqrt [3]{d} 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 e^{4/3}}+\frac {\sqrt [3]{d} 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 e^{4/3}}-\frac {\sqrt [3]{d} \log \left (c \left (e x^3+d\right )^p\right ) \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )}{3 e^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} p \log ^2\left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 e^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} p \log ^2\left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )}{6 e^{4/3}}-\frac {3 p x}{e}-\frac {\sqrt {3} \sqrt [3]{d} p \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{e^{4/3}}+\frac {\sqrt [3]{d} p \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{e^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} 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 e^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} 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 e^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} 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 e^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} 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 e^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} 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 e^{4/3}}-\frac {\sqrt [3]{d} p \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right )}{2 e^{4/3}}+\frac {x \log \left (c \left (e x^3+d\right )^p\right )}{e}-\frac {(-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 )}{3 e^{4/3}}+\frac {\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 )}{3 e^{4/3}}+\frac {\sqrt [3]{d} p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{3 e^{4/3}}+\frac {\sqrt [3]{d} 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 e^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} 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 e^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} 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 e^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} 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 e^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} 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 e^{4/3}}\right )\)

Input:

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

Output:

x*Log[c*(d + e*x^3)^p]^2 - 6*e*p*((-3*p*x)/e - (Sqrt[3]*d^(1/3)*p*ArcTan[( 
d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/e^(4/3) + (d^(1/3)*p*Log[-d^(1/ 
3) - e^(1/3)*x]^2)/(6*e^(4/3)) + (d^(1/3)*p*Log[d^(1/3) + e^(1/3)*x])/e^(4 
/3) + (d^(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*e^(4/3)) + ((-1)^(2/3)*d^(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*e^(4/3)) + ((-1)^(2/3)*d^(1/3)*p*Log[-d^ 
(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/(6*e^(4/3)) - ((-1)^(1/3)*d^(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*e^(4/3)) - ((-1)^(1/3)*d^(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*e^(4/3)) - ((-1)^(1/3)*d^(1/3)*p*Log[- 
d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/(6*e^(4/3)) + ((-1)^(1/3)*d^(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*e^(4 
/3)) + (d^(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*e^(4/3)) + ((-1)^(2/3) 
*d^(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*e^(4/3)) - (d^( 
1/3)*p*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(2*e^(4/3)) + (x...
 

Defintions of rubi rules used

rule 2009

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

rule 2900

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo 
l] :> Simp[x*(a + b*Log[c*(d + e*x^n)^p])^q, x] - Simp[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 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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {6 d^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x}{d^{\frac {1}{3}} \sqrt {3}}\right ) p^{2}-9 d^{\frac {1}{3}} \mathrm {log}\left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right ) p^{2}+3 d^{\frac {1}{3}} \mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) p +6 e^{\frac {1}{3}} \left (\int \frac {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}{e \,x^{3}+d}d x \right ) d p +e^{\frac {1}{3}} {\mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right )}^{2} x -6 e^{\frac {1}{3}} \mathrm {log}\left (\left (e \,x^{3}+d \right )^{p} c \right ) p x +18 e^{\frac {1}{3}} p^{2} x}{e^{\frac {1}{3}}} \] Input:

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

Output:

(6*d**(1/3)*sqrt(3)*atan((d**(1/3) - 2*e**(1/3)*x)/(d**(1/3)*sqrt(3)))*p** 
2 - 9*d**(1/3)*log(d**(1/3) + e**(1/3)*x)*p**2 + 3*d**(1/3)*log((d + e*x** 
3)**p*c)*p + 6*e**(1/3)*int(log((d + e*x**3)**p*c)/(d + e*x**3),x)*d*p + e 
**(1/3)*log((d + e*x**3)**p*c)**2*x - 6*e**(1/3)*log((d + e*x**3)**p*c)*p* 
x + 18*e**(1/3)*p**2*x)/e**(1/3)