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

Optimal. Leaf size=69 \[ \frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )-\frac {\log (x) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2} \]

[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

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Rubi [A]
time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2422, 2375, 2421, 6724} \begin {gather*} \frac {\text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {\log (x) \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[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*} \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} (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*}

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Mathematica [A]
time = 0.04, size = 75, normalized size = 1.09 \begin {gather*} -\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) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}+\frac {\text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2} \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 0.69, size = 66, normalized size = 0.96

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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/(d*x + x*e^(m*log(x) + 1)), x) + 1/2*log(d + e^(m*log(x) + 1))*lo
g(x)^2

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Fricas [A]
time = 0.36, size = 80, normalized size = 1.16 \begin {gather*} \frac {m^{2} \log \left (x^{m} e + d\right ) \log \left (x\right )^{2} - m^{2} \log \left (x\right )^{2} \log \left (\frac {x^{m} e + d}{d}\right ) - 2 \, m {\rm Li}_2\left (-\frac {x^{m} e + d}{d} + 1\right ) \log \left (x\right ) + 2 \, {\rm polylog}\left (3, -\frac {x^{m} e}{d}\right )}{2 \, m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(x^m*e + d)*log(x)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d+e\,x^m\right )\,\ln \left (x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d + e*x^m)*log(x))/x,x)

[Out]

int((log(d + e*x^m)*log(x))/x, x)

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