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

Optimal. Leaf size=40 \[ \frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]

[Out]

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

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Rubi [A]  time = 0.0482115, antiderivative size = 40, normalized size of antiderivative = 1., 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 = {2374, 6589} \[ \frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*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 6589

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 = 0.0090543, size = 52, normalized size = 1.3 \[ -\frac{a \text{PolyLog}\left (2,-d f x^m\right )}{m}-\frac{b \log \left (c x^n\right ) \text{PolyLog}\left (2,-d f x^m\right )}{m}+\frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2} \]

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

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Maple [C]  time = 0.043, size = 308, normalized size = 7.7 \begin{align*} -{\frac{b\ln \left ( d \left ({d}^{-1}+f{x}^{m} \right ) \right ) n \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+b\ln \left ( x \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{m} \right ) \right ) \ln \left ({x}^{n} \right ) +{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}\ln \left ( df{x}^{m}+1 \right ) }{2}}-{\frac{bn\ln \left ( x \right ){\it polylog} \left ( 2,-df{x}^{m} \right ) }{m}}+{\frac{bn{\it polylog} \left ( 3,-df{x}^{m} \right ) }{{m}^{2}}}+{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) n\ln \left ( x \right ) }{m}}-{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) \ln \left ({x}^{n} \right ) }{m}}-b\ln \left ( x \right ) \ln \left ({x}^{n} \right ) \ln \left ( df{x}^{m}+1 \right ) +{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) }{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{m}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}}{m}}-{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) \ln \left ( c \right ) }{m}}-{\frac{{\it dilog} \left ( df{x}^{m}+1 \right ) a}{m}} \end{align*}

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)

[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(x^n)*ln(d*f*x^m+1)+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)-1/m*dilog(d*f*x^m+1)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left (d f x^{m} + 1\right ) - \int \frac{2 \, b d f m x^{m} \log \left (x\right ) \log \left (x^{n}\right ) -{\left (b d f m n \log \left (x\right )^{2} - 2 \,{\left (b d f m \log \left (c\right ) + a d f m\right )} \log \left (x\right )\right )} x^{m}}{2 \,{\left (d f x x^{m} + x\right )}}\,{d x} \end{align*}

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)

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Fricas [C]  time = 1.34605, size = 113, normalized size = 2.82 \begin{align*} \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{align*}

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

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

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + \frac{1}{d}\right )} d\right )}{x}\,{d x} \end{align*}

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)