3.26 \(\int \frac{(a+b \log (c x^n)) \log (d (\frac{1}{d}+f x^2))}{x} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-((a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^2)])/2 + (b*n*PolyLog[3, -(d*f*x^2)])/4

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Rubi [A]  time = 0.032042, antiderivative size = 39, 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} \[ \frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right )-\frac{1}{2} \text{PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x,x]

[Out]

-((a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^2)])/2 + (b*n*PolyLog[3, -(d*f*x^2)])/4

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^2\right )\right )}{x} \, dx &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )+\frac{1}{2} (b n) \int \frac{\text{Li}_2\left (-d f x^2\right )}{x} \, dx\\ &=-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^2\right )+\frac{1}{4} b n \text{Li}_3\left (-d f x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0103969, size = 50, normalized size = 1.28 \[ -\frac{1}{2} a \text{PolyLog}\left (2,-d f x^2\right )-\frac{1}{2} b \log \left (c x^n\right ) \text{PolyLog}\left (2,-d f x^2\right )+\frac{1}{4} b n \text{PolyLog}\left (3,-d f x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)])/x,x]

[Out]

-(a*PolyLog[2, -(d*f*x^2)])/2 - (b*Log[c*x^n]*PolyLog[2, -(d*f*x^2)])/2 + (b*n*PolyLog[3, -(d*f*x^2)])/4

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) }{x}}\, dx \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^2))/x,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2))/x,x)

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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^{2} + 1\right ) - \int -\frac{b d f n x \log \left (x\right )^{2} - 2 \, b d f x \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b d f \log \left (c\right ) + a d f\right )} x \log \left (x\right )}{d f x^{2} + 1}\,{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^2))/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^2 + 1) - integrate(-(b*d*f*n*x*l
og(x)^2 - 2*b*d*f*x*log(x)*log(x^n) - 2*(b*d*f*log(c) + a*d*f)*x*log(x))/(d*f*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a \log \left (d f x^{2} + 1\right )}{x}, x\right ) \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^2))/x,x, algorithm="fricas")

[Out]

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

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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**2))/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^{2} + \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^2))/x,x, algorithm="giac")

[Out]

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