\(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2}{x^2} \, dx\) [523]

Optimal result

Integrand size = 24, antiderivative size = 361 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {8 b^2 n^2}{9 x}+\frac {32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac {32 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}+\frac {4 i b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac {4 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac {4 b d^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac {4 i b^2 d^{3/2} n^2 \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}} \] Output:

-8/9*b^2*n^2/x+32/3*b^2*d*n^2/e/x^(1/3)+32/3*b^2*d^(3/2)*n^2*arctan(d^(1/2 
)*x^(1/3)/e^(1/2))/e^(3/2)+4*I*b^2*d^(3/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1 
/2))^2/e^(3/2)-8*b^2*d^(3/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1/2))*ln(2-2*e^ 
(1/2)/(e^(1/2)-I*d^(1/2)*x^(1/3)))/e^(3/2)+4/3*b*n*(a+b*ln(c*(d+e/x^(2/3)) 
^n))/x-4*b*d*n*(a+b*ln(c*(d+e/x^(2/3))^n))/e/x^(1/3)-4*b*d^(3/2)*n*arctan( 
d^(1/2)*x^(1/3)/e^(1/2))*(a+b*ln(c*(d+e/x^(2/3))^n))/e^(3/2)-(a+b*ln(c*(d+ 
e/x^(2/3))^n))^2/x+4*I*b^2*d^(3/2)*n^2*polylog(2,-1+2*e^(1/2)/(e^(1/2)-I*d 
^(1/2)*x^(1/3)))/e^(3/2)
 

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx=-\frac {9 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {b n \left (36 a d \sqrt {e} x^{2/3}-72 b d \sqrt {e} n x^{2/3}+72 b d^{3/2} n x \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )+8 b n \left (\sqrt {e} \left (e-3 d x^{2/3}\right )+3 d^{3/2} x \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )\right )+36 b d \sqrt {e} x^{2/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-12 e^{3/2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+18 \sqrt {-d} d x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+18 (-d)^{3/2} x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+9 b (-d)^{3/2} n x \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )+9 b \sqrt {-d} d n x \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{e^{3/2}}}{9 x} \] Input:

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

Output:

-1/9*(9*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + (b*n*(36*a*d*Sqrt[e]*x^(2/3) 
- 72*b*d*Sqrt[e]*n*x^(2/3) + 72*b*d^(3/2)*n*x*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1 
/3))] + 8*b*n*(Sqrt[e]*(e - 3*d*x^(2/3)) + 3*d^(3/2)*x*ArcTan[Sqrt[e]/(Sqr 
t[d]*x^(1/3))]) + 36*b*d*Sqrt[e]*x^(2/3)*Log[c*(d + e/x^(2/3))^n] - 12*e^( 
3/2)*(a + b*Log[c*(d + e/x^(2/3))^n]) + 18*Sqrt[-d]*d*x*(a + b*Log[c*(d + 
e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 18*(-d)^(3/2)*x*(a + b*Lo 
g[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 9*b*(-d)^(3/2)*n 
*x*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 2*L 
og[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]] 
) - 4*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 2*PolyLog[2, 1/2 - (Sqr 
t[-d]*x^(1/3))/(2*Sqrt[e])]) + 9*b*Sqrt[-d]*d*n*x*(Log[Sqrt[e] + Sqrt[-d]* 
x^(1/3)]*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 2*Log[1/2 - (Sqrt[-d]*x^(1/3)) 
/(2*Sqrt[e])] - 4*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])]) + 2*PolyLog[2, (1 + 
(Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[ 
e]])))/e^(3/2))/x
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2908, 2907, 2005, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx\)

\(\Big \downarrow \) 2908

\(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{4/3}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 2907

\(\displaystyle 3 \left (-\frac {4}{3} b e n \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{\left (d+\frac {e}{x^{2/3}}\right ) x^2}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 x}\right )\)

\(\Big \downarrow \) 2005

\(\displaystyle 3 \left (-\frac {4}{3} b e n \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{\left (x^{2/3} d+e\right ) x^{4/3}}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 x}\right )\)

\(\Big \downarrow \) 2926

\(\displaystyle 3 \left (-\frac {4}{3} b e n \int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) d^2}{e^2 \left (x^{2/3} d+e\right )}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) d}{e^2 x^{2/3}}+\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{e x^{4/3}}\right )d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 x}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 x}-\frac {4}{3} b e n \left (\frac {d^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{5/2}}+\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^2 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 e x}-\frac {i b d^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{5/2}}-\frac {8 b d^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{5/2}}+\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{5/2}}-\frac {i b d^{3/2} n \operatorname {PolyLog}\left (2,\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{e^{5/2}}-\frac {8 b d n}{3 e^2 \sqrt [3]{x}}+\frac {2 b n}{9 e x}\right )\right )\)

Input:

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

Output:

3*(-1/3*(a + b*Log[c*(d + e/x^(2/3))^n])^2/x - (4*b*e*n*((2*b*n)/(9*e*x) - 
 (8*b*d*n)/(3*e^2*x^(1/3)) - (8*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[ 
e]])/(3*e^(5/2)) - (I*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^( 
5/2) + (2*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e] 
)/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(5/2) - (a + b*Log[c*(d + e/x^(2/3))^n 
])/(3*e*x) + (d*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e^2*x^(1/3)) + (d^(3/2) 
*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(5/ 
2) - (I*b*d^(3/2)*n*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/ 
3))])/e^(5/2)))/3)
 

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

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 2908

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*(x_)^(m_ 
.), x_Symbol] :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, 
 b, c, d, e, m, p, q}, x] && FractionQ[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]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{2}}{x^{2}}d x\]

Input:

int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^2,x)
 

Output:

int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^2,x)
 

Fricas [F]

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

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

Output:

integral((b^2*log(c*((d*x + e*x^(1/3))/x)^n)^2 + 2*a*b*log(c*((d*x + e*x^( 
1/3))/x)^n) + a^2)/x^2, x)
 

Sympy [F(-1)]

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

integrate((a+b*ln(c*(d+e/x**(2/3))**n))**2/x**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((a+b*log(c*(d+e/x^(2/3))^n))^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]

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

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

Output:

integrate((b*log(c*(d + e/x^(2/3))^n) + a)^2/x^2, x)
 

Mupad [F(-1)]

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

int((a + b*log(c*(d + e/x^(2/3))^n))^2/x^2,x)
 

Output:

int((a + b*log(c*(d + e/x^(2/3))^n))^2/x^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx=\frac {-12 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) a b d n x -12 x^{\frac {2}{3}} a b d e n -4 \left (\int \frac {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {8}{3}} d +e \,x^{2}}d x \right ) b^{2} e^{3} n x -3 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} e^{2}-6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,e^{2}-3 a^{2} e^{2}+4 a b \,e^{2} n}{3 e^{2} x} \] Input:

int((a+b*log(c*(d+e/x^(2/3))^n))^2/x^2,x)
 

Output:

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