\(\int \frac {\log (x) \log (d+e x^m)}{x} \, dx\) [389]

Optimal result

Integrand size = 14, antiderivative size = 69 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )-\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {\operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2422, 2375, 2421, 6724} \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (\frac {e x^m}{d}+1\right ) \]

[In]

Int[(Log[x]*Log[d + e*x^m])/x,x]

[Out]

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

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[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 2421

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] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[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 2422

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] - Dist[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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} (e m) \int \frac {x^{-1+m} \log ^2(x)}{d+e x^m} \, 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 )+\int \frac {\log (x) \log \left (1+\frac {e x^m}{d}\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) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {\int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m} \\ & = \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) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=-\frac {1}{6} \log ^2(x) \left (m \log (x)+3 \log \left (1+\frac {d x^{-m}}{e}\right )-3 \log \left (d+e x^m\right )\right )+\frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{m}+\frac {\operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )}{m^2} \]

[In]

Integrate[(Log[x]*Log[d + e*x^m])/x,x]

[Out]

-1/6*(Log[x]^2*(m*Log[x] + 3*Log[1 + d/(e*x^m)] - 3*Log[d + e*x^m])) + (Log[x]*PolyLog[2, -(d/(e*x^m))])/m + P
olyLog[3, -(d/(e*x^m))]/m^2

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {\ln \left (x \right )^{2} \ln \left (d +e \,x^{m}\right )}{2}-\frac {\ln \left (x \right )^{2} \ln \left (1+\frac {e \,x^{m}}{d}\right )}{2}-\frac {\ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e \,x^{m}}{d}\right )}{m}+\frac {\operatorname {Li}_{3}\left (-\frac {e \,x^{m}}{d}\right )}{m^{2}}\)	\(66\)

[In]

int(ln(x)*ln(d+e*x^m)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\frac {m^{2} \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} - m^{2} \log \left (x\right )^{2} \log \left (\frac {e x^{m} + d}{d}\right ) - 2 \, m {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) \log \left (x\right ) + 2 \, {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{2 \, m^{2}} \]

[In]

integrate(log(x)*log(d+e*x^m)/x,x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(ln(x)*ln(d+e*x**m)/x,x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

\[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int { \frac {\log \left (e x^{m} + d\right ) \log \left (x\right )}{x} \,d x } \]

[In]

integrate(log(x)*log(d+e*x^m)/x,x, algorithm="maxima")

[Out]

-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

Giac [F]

\[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int { \frac {\log \left (e x^{m} + d\right ) \log \left (x\right )}{x} \,d x } \]

[In]

integrate(log(x)*log(d+e*x^m)/x,x, algorithm="giac")

[Out]

integrate(log(e*x^m + d)*log(x)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx=\int \frac {\ln \left (d+e\,x^m\right )\,\ln \left (x\right )}{x} \,d x \]

[In]

int((log(d + e*x^m)*log(x))/x,x)

[Out]

int((log(d + e*x^m)*log(x))/x, x)