Integrand size = 14, antiderivative size = 69 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )-\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {\operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2} \] Output:
1/2*ln(x)^2*ln(d+e*x^m)-1/2*ln(x)^2*ln(1+e*x^m/d)-ln(x)*polylog(2,-e*x^m/d )/m+polylog(3,-e*x^m/d)/m^2
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=-\frac {1}{6} \log ^2(x) \left (m \log (x)+3 \log \left (1+\frac {d x^{-m}}{e}\right )-3 \log \left (d+e x^m\right )\right )+\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{m}+\frac {\operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )}{m^2} \] Input:
Integrate[(Log[x]*Log[d + e*x^m])/x,x]
Output:
-1/6*(Log[x]^2*(m*Log[x] + 3*Log[1 + d/(e*x^m)] - 3*Log[d + e*x^m])) + (Lo g[x]*PolyLog[2, -(d/(e*x^m))])/m + PolyLog[3, -(d/(e*x^m))]/m^2
Time = 0.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2822, 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.
\(\displaystyle \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx\) |
\(\Big \downarrow \) 2822 |
\(\displaystyle \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} e m \int \frac {x^{m-1} \log ^2(x)}{e x^m+d}dx\) |
\(\Big \downarrow \) 2775 |
\(\displaystyle \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} e m \left (\frac {\log ^2(x) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 \int \frac {\log (x) \log \left (\frac {e x^m}{d}+1\right )}{x}dx}{e m}\right )\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} e m \left (\frac {\log ^2(x) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 \left (\frac {\int \frac {\operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{x}dx}{m}-\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} e m \left (\frac {\log ^2(x) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}\right )}{e m}\right )\) |
Input:
Int[(Log[x]*Log[d + e*x^m])/x,x]
Output:
(Log[x]^2*Log[d + e*x^m])/2 - (e*m*((Log[x]^2*Log[1 + (e*x^m)/d])/(e*m) - (2*(-((Log[x]*PolyLog[2, -((e*x^m)/d)])/m) + PolyLog[3, -((e*x^m)/d)]/m^2) )/(e*m)))/2
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]
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]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_ .)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[ c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[f*m*(r/(b*n*(p + 1))) Int[x^(m - 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
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]
Time = 2.61 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2} \ln \left (d +e \,x^{m}\right )}{2}-\frac {\ln \left (x \right )^{2} \ln \left (1+\frac {e \,x^{m}}{d}\right )}{2}-\frac {\ln \left (x \right ) \operatorname {polylog}\left (2, -\frac {e \,x^{m}}{d}\right )}{m}+\frac {\operatorname {polylog}\left (3, -\frac {e \,x^{m}}{d}\right )}{m^{2}}\) | \(66\) |
Input:
int(ln(x)*ln(d+e*x^m)/x,x,method=_RETURNVERBOSE)
Output:
1/2*ln(x)^2*ln(d+e*x^m)-1/2*ln(x)^2*ln(1+e*x^m/d)-ln(x)*polylog(2,-e*x^m/d )/m+polylog(3,-e*x^m/d)/m^2
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\frac {m^{2} \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} - m^{2} \log \left (x\right )^{2} \log \left (\frac {e x^{m} + d}{d}\right ) - 2 \, m {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) \log \left (x\right ) + 2 \, {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{2 \, m^{2}} \] Input:
integrate(log(x)*log(d+e*x^m)/x,x, algorithm="fricas")
Output:
1/2*(m^2*log(e*x^m + d)*log(x)^2 - m^2*log(x)^2*log((e*x^m + d)/d) - 2*m*d ilog(-(e*x^m + d)/d + 1)*log(x) + 2*polylog(3, -e*x^m/d))/m^2
Exception generated. \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(ln(x)*ln(d+e*x**m)/x,x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
\[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int { \frac {\log \left (e x^{m} + d\right ) \log \left (x\right )}{x} \,d x } \] Input:
integrate(log(x)*log(d+e*x^m)/x,x, algorithm="maxima")
Output:
-1/6*m*log(x)^3 + d*m*integrate(1/2*log(x)^2/(e*x*x^m + d*x), x) + 1/2*log (e*x^m + d)*log(x)^2
\[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int { \frac {\log \left (e x^{m} + d\right ) \log \left (x\right )}{x} \,d x } \] Input:
integrate(log(x)*log(d+e*x^m)/x,x, algorithm="giac")
Output:
integrate(log(e*x^m + d)*log(x)/x, x)
Timed out. \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int \frac {\ln \left (d+e\,x^m\right )\,\ln \left (x\right )}{x} \,d x \] Input:
int((log(d + e*x^m)*log(x))/x,x)
Output:
int((log(d + e*x^m)*log(x))/x, x)
\[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int \frac {\mathrm {log}\left (x^{m} e +d \right ) \mathrm {log}\left (x \right )}{x}d x \] Input:
int(log(x)*log(d+e*x^m)/x,x)
Output:
int((log(x**m*e + d)*log(x))/x,x)