3.1.67 \(\int \frac {(a+b \log (c x^n)) \log (d (\frac {1}{d}+f x^m))}{x} \, dx\) [67]

Optimal. Leaf size=40 \[ -\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2} \]

[Out]

-(a+b*ln(c*x^n))*polylog(2,-d*f*x^m)/m+b*n*polylog(3,-d*f*x^m)/m^2

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2421, 6724} \begin {gather*} \frac {b n \text {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {\text {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

-(((a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^m)])/m) + (b*n*PolyLog[3, -(d*f*x^m)])/m^2

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 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 {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {(b n) \int \frac {\text {Li}_2\left (-d f x^m\right )}{x} \, dx}{m}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 52, normalized size = 1.30 \begin {gather*} -\frac {a \text {Li}_2\left (-d f x^m\right )}{m}-\frac {b \log \left (c x^n\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^m)])/x,x]

[Out]

-((a*PolyLog[2, -(d*f*x^m)])/m) - (b*Log[c*x^n]*PolyLog[2, -(d*f*x^m)])/m + (b*n*PolyLog[3, -(d*f*x^m)])/m^2

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 308, normalized size = 7.70

method	result	size
risch	\(-\frac {b \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) n \ln \left (x \right )^{2}}{2}+b \ln \left (x \right ) \ln \left (d \left (\frac {1}{d}+f \,x^{m}\right )\right ) \ln \left (x^{n}\right )+\frac {b n \ln \left (x \right )^{2} \ln \left (d f \,x^{m}+1\right )}{2}-\frac {b n \ln \left (x \right ) \polylog \left (2, -d f \,x^{m}\right )}{m}+\frac {b n \polylog \left (3, -d f \,x^{m}\right )}{m^{2}}+\frac {b \dilog \left (d f \,x^{m}+1\right ) n \ln \left (x \right )}{m}-\frac {b \dilog \left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )}{m}-b \ln \left (x \right ) \ln \left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )+\frac {i \dilog \left (d f \,x^{m}+1\right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 m}-\frac {i \dilog \left (d f \,x^{m}+1\right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 m}-\frac {i \dilog \left (d f \,x^{m}+1\right ) b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 m}+\frac {i \dilog \left (d f \,x^{m}+1\right ) b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 m}-\frac {\dilog \left (d f \,x^{m}+1\right ) b \ln \left (c \right )}{m}-\frac {a \dilog \left (d f \,x^{m}+1\right )}{m}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^m))/x,x,method=_RETURNVERBOSE)

[Out]

-1/2*b*ln(d*(1/d+f*x^m))*n*ln(x)^2+b*ln(x)*ln(d*(1/d+f*x^m))*ln(x^n)+1/2*b*n*ln(x)^2*ln(d*f*x^m+1)-b*n/m*ln(x)
*polylog(2,-d*f*x^m)+b*n*polylog(3,-d*f*x^m)/m^2+b/m*dilog(d*f*x^m+1)*n*ln(x)-b/m*dilog(d*f*x^m+1)*ln(x^n)-b*l
n(x)*ln(d*f*x^m+1)*ln(x^n)+1/2*I/m*dilog(d*f*x^m+1)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I/m*dilog(d*f
*x^m+1)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I/m*dilog(d*f*x^m+1)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I/m*dilog
(d*f*x^m+1)*b*Pi*csgn(I*c*x^n)^3-1/m*dilog(d*f*x^m+1)*b*ln(c)-a/m*dilog(d*f*x^m+1)

________________________________________________________________________________________

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((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x, algorithm="maxima")

[Out]

-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*log(d*f*x^m + 1) - integrate(1/2*(2*b*d*f*
m*x^m*log(x)*log(x^n) - (b*d*f*m*n*log(x)^2 - 2*(b*d*f*m*log(c) + a*d*f*m)*log(x))*x^m)/(d*f*x*x^m + x), x)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 42, normalized size = 1.05 \begin {gather*} \frac {b n {\rm polylog}\left (3, -d f x^{m}\right ) - {\left (b m n \log \left (x\right ) + b m \log \left (c\right ) + a m\right )} {\rm Li}_2\left (-d f x^{m}\right )}{m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")

[Out]

(b*n*polylog(3, -d*f*x^m) - (b*m*n*log(x) + b*m*log(c) + a*m)*dilog(-d*f*x^m))/m^2

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(1/d+f*x**m))/x,x)

[Out]

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

________________________________________________________________________________________

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((a+b*log(c*x^n))*log(d*(1/d+f*x^m))/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*x^m + 1/d)*d)/x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n)))/x,x)

[Out]

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n)))/x, x)

________________________________________________________________________________________