\(\int \frac {x^{-1+m} \log ^2(f x^p)}{d+e x^m} \, dx\) [620]

Optimal result

Integrand size = 23, antiderivative size = 75 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{e m^2}-\frac {2 p^2 \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{e m^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(75)=150\).

Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.93 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {p^2 \log ^3(x)+3 p \log ^2(x) \left (-p \log (x)+\log \left (f x^p\right )\right )+3 \log (x) \left (-p \log (x)+\log \left (f x^p\right )\right )^2-\frac {3 \left (-p \log (x)+\log \left (f x^p\right )\right )^2 \left (\log \left (x^m\right )-\log \left (d m \left (d+e x^m\right )\right )\right )}{m}-\frac {6 p \left (-p \log (x)+\log \left (f x^p\right )\right ) \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{m^2}+\frac {3 p^2 \left (m^2 \log ^2(x) \log \left (1+\frac {d x^{-m}}{e}\right )-2 m \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )-2 \operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )\right )}{m^3}}{3 e} \] Input:

Integrate[(x^(-1 + m)*Log[f*x^p]^2)/(d + e*x^m),x]
 

Output:

(p^2*Log[x]^3 + 3*p*Log[x]^2*(-(p*Log[x]) + Log[f*x^p]) + 3*Log[x]*(-(p*Lo 
g[x]) + Log[f*x^p])^2 - (3*(-(p*Log[x]) + Log[f*x^p])^2*(Log[x^m] - Log[d* 
m*(d + e*x^m)]))/m - (6*p*(-(p*Log[x]) + Log[f*x^p])*((m^2*Log[x]^2)/2 + ( 
-(m*Log[x]) + Log[-((e*x^m)/d)])*Log[d + e*x^m] + PolyLog[2, 1 + (e*x^m)/d 
]))/m^2 + (3*p^2*(m^2*Log[x]^2*Log[1 + d/(e*x^m)] - 2*m*Log[x]*PolyLog[2, 
-(d/(e*x^m))] - 2*PolyLog[3, -(d/(e*x^m))]))/m^3)/(3*e)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2775, 2821, 7143}

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.

\(\displaystyle \int \frac {x^{m-1} \log ^2\left (f x^p\right )}{d+e x^m} \, dx\)

\(\Big \downarrow \) 2775

\(\displaystyle \frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 p \int \frac {\log \left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{x}dx}{e m}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 p \left (\frac {p \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{x}dx}{m}-\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 p \left (\frac {p \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\)

Input:

Int[(x^(-1 + m)*Log[f*x^p]^2)/(d + e*x^m),x]
 

Output:

(Log[f*x^p]^2*Log[1 + (e*x^m)/d])/(e*m) - (2*p*(-((Log[f*x^p]*PolyLog[2, - 
((e*x^m)/d)])/m) + (p*PolyLog[3, -((e*x^m)/d)])/m^2))/(e*m)
 

Defintions of rubi rules used

rule 2775

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) 
+ (e_.)*(x_)^(r_)), x_Symbol] :> Simp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c* 
x^n])^p/(e*r)), x] - Simp[b*f^m*n*(p/(e*r))   Int[Log[1 + e*(x^r/d)]*((a + 
b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] & 
& EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

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

Time = 2.61 (sec) , antiderivative size = 496, normalized size of antiderivative = 6.61

method	result	size
risch	\(\frac {\ln \left (d +e \,x^{m}\right ) \ln \left (x \right )^{2} p^{2}}{m e}-\frac {2 \ln \left (d +e \,x^{m}\right ) \ln \left (x \right ) \ln \left (x^{p}\right ) p}{m e}+\frac {\ln \left (d +e \,x^{m}\right ) \ln \left (x^{p}\right )^{2}}{m e}+\frac {p^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e \,x^{m}}{d}\right )}{m e}+\frac {2 p^{2} \ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e \,x^{m}}{d}\right )}{m^{2} e}-\frac {2 p^{2} \operatorname {polylog}\left (3, -\frac {e \,x^{m}}{d}\right )}{e \,m^{3}}-\frac {2 p^{2} \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x \right )}{m^{2} e}+\frac {2 p \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x^{p}\right )}{m^{2} e}-\frac {2 p^{2} \ln \left (x \right )^{2} \ln \left (\frac {d +e \,x^{m}}{d}\right )}{m e}+\frac {2 p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x^{p}\right )}{m e}+\left (i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right ) \operatorname {csgn}\left (i f \right )-i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{3}+i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{2} \operatorname {csgn}\left (i f \right )+2 \ln \left (f \right )\right ) \left (\frac {\left (\ln \left (x^{p}\right )-p \ln \left (x \right )\right ) \ln \left (d +e \,x^{m}\right )}{m e}+\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right )}{m^{2} e}+\frac {p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right )}{m e}\right )+\frac {{\left (i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right ) \operatorname {csgn}\left (i f \right )-i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{3}+i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{2} \operatorname {csgn}\left (i f \right )+2 \ln \left (f \right )\right )}^{2} \ln \left (d +e \,x^{m}\right )}{4 m e}\)	\(496\)

Input:

int(x^(m-1)*ln(f*x^p)^2/(d+e*x^m),x,method=_RETURNVERBOSE)
 

Output:

1/m*ln(d+e*x^m)/e*ln(x)^2*p^2-2/m*ln(d+e*x^m)/e*ln(x)*ln(x^p)*p+1/m*ln(d+e 
*x^m)/e*ln(x^p)^2+1/m*p^2/e*ln(x)^2*ln(1+e*x^m/d)+2/m^2*p^2/e*ln(x)*polylo 
g(2,-e*x^m/d)-2*p^2*polylog(3,-e*x^m/d)/e/m^3-2/m^2*p^2*dilog((d+e*x^m)/d) 
/e*ln(x)+2/m^2*p*dilog((d+e*x^m)/d)/e*ln(x^p)-2/m*p^2*ln(x)^2*ln((d+e*x^m) 
/d)/e+2/m*p*ln(x)*ln((d+e*x^m)/d)/e*ln(x^p)+(I*Pi*csgn(I*x^p)*csgn(I*f*x^p 
)^2-I*Pi*csgn(I*x^p)*csgn(I*f*x^p)*csgn(I*f)-I*Pi*csgn(I*f*x^p)^3+I*Pi*csg 
n(I*f*x^p)^2*csgn(I*f)+2*ln(f))*(1/m*(ln(x^p)-p*ln(x))*ln(d+e*x^m)/e+1/m^2 
*p*dilog((d+e*x^m)/d)/e+1/m*p*ln(x)*ln((d+e*x^m)/d)/e)+1/4*(I*Pi*csgn(I*x^ 
p)*csgn(I*f*x^p)^2-I*Pi*csgn(I*x^p)*csgn(I*f*x^p)*csgn(I*f)-I*Pi*csgn(I*f* 
x^p)^3+I*Pi*csgn(I*f*x^p)^2*csgn(I*f)+2*ln(f))^2/m*ln(d+e*x^m)/e
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {m^{2} \log \left (e x^{m} + d\right ) \log \left (f\right )^{2} - 2 \, p^{2} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right ) + 2 \, {\left (m p^{2} \log \left (x\right ) + m p \log \left (f\right )\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (m^{2} p^{2} \log \left (x\right )^{2} + 2 \, m^{2} p \log \left (f\right ) \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{e m^{3}} \] Input:

integrate(x^(-1+m)*log(f*x^p)^2/(d+e*x^m),x, algorithm="fricas")
 

Output:

(m^2*log(e*x^m + d)*log(f)^2 - 2*p^2*polylog(3, -e*x^m/d) + 2*(m*p^2*log(x 
) + m*p*log(f))*dilog(-(e*x^m + d)/d + 1) + (m^2*p^2*log(x)^2 + 2*m^2*p*lo 
g(f)*log(x))*log((e*x^m + d)/d))/(e*m^3)
 

Sympy [F]

\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int \frac {x^{m - 1} \log {\left (f x^{p} \right )}^{2}}{d + e x^{m}}\, dx \] Input:

integrate(x**(-1+m)*ln(f*x**p)**2/(d+e*x**m),x)
 

Output:

Integral(x**(m - 1)*log(f*x**p)**2/(d + e*x**m), x)
 

Maxima [F]

\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int { \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d} \,d x } \] Input:

integrate(x^(-1+m)*log(f*x^p)^2/(d+e*x^m),x, algorithm="maxima")
 

Output:

integrate(x^(m - 1)*log(f*x^p)^2/(e*x^m + d), x)
 

Giac [F]

\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int { \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d} \,d x } \] Input:

integrate(x^(-1+m)*log(f*x^p)^2/(d+e*x^m),x, algorithm="giac")
 

Output:

integrate(x^(m - 1)*log(f*x^p)^2/(e*x^m + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int \frac {x^{m-1}\,{\ln \left (f\,x^p\right )}^2}{d+e\,x^m} \,d x \] Input:

int((x^(m - 1)*log(f*x^p)^2)/(d + e*x^m),x)
 

Output:

int((x^(m - 1)*log(f*x^p)^2)/(d + e*x^m), x)
 

Reduce [F]

\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int \frac {x^{m} \mathrm {log}\left (x^{p} f \right )^{2}}{x^{m} e x +d x}d x \] Input:

int(x^(-1+m)*log(f*x^p)^2/(d+e*x^m),x)
 

Output:

int((x**m*log(x**p*f)**2)/(x**m*e*x + d*x),x)