3.177 \(\int \frac{a+b \log (c x^n)}{x \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \log (\log (x)) \left (a+b \log \left (c x^n\right )\right )+b n \log (x)-b n \log (\log (x)) \log (x) \]

[Out]

b*n*Log[x] - b*n*Log[x]*Log[Log[x]] + (a + b*Log[c*x^n])*Log[Log[x]]

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Rubi [A]  time = 0.0531439, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2302, 29, 2366, 2521} \[ \log (\log (x)) \left (a+b \log \left (c x^n\right )\right )+b n \log (x)-b n \log (\log (x)) \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x*Log[x]),x]

[Out]

b*n*Log[x] - b*n*Log[x]*Log[Log[x]] + (a + b*Log[c*x^n])*Log[Log[x]]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2521

Int[((a_.) + Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)]*(b_.))/(x_), x_Symbol] :> Simp[(Log[d*x^n]*(a + b*Log[c*Lo
g[d*x^n]^p]))/n, x] - Simp[b*p*Log[x], x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \log (x)} \, dx &=\left (a+b \log \left (c x^n\right )\right ) \log (\log (x))-(b n) \int \frac{\log (\log (x))}{x} \, dx\\ &=b n \log (x)-b n \log (x) \log (\log (x))+\left (a+b \log \left (c x^n\right )\right ) \log (\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0172452, size = 28, normalized size = 0.97 \[ a \log (\log (x))+b \log (\log (x)) \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x*Log[x]),x]

[Out]

b*n*Log[x] + a*Log[Log[x]] + b*(-(n*Log[x]) + Log[c*x^n])*Log[Log[x]]

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Maple [C]  time = 0.04, size = 131, normalized size = 4.5 \begin{align*} -bn\ln \left ( x \right ) \ln \left ( \ln \left ( x \right ) \right ) +bn\ln \left ( x \right ) +\ln \left ( \ln \left ( x \right ) \right ) \ln \left ({x}^{n} \right ) b-{\frac{i}{2}}\ln \left ( \ln \left ( x \right ) \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) +{\frac{i}{2}}\ln \left ( \ln \left ( x \right ) \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{2}}\ln \left ( \ln \left ( x \right ) \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{2}}\ln \left ( \ln \left ( x \right ) \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+\ln \left ( \ln \left ( x \right ) \right ) b\ln \left ( c \right ) +\ln \left ( \ln \left ( x \right ) \right ) a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x/ln(x),x)

[Out]

-b*n*ln(x)*ln(ln(x))+b*n*ln(x)+ln(ln(x))*ln(x^n)*b-1/2*I*ln(ln(x))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/
2*I*ln(ln(x))*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*ln(ln(x))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*ln(ln(x))*
b*Pi*csgn(I*c*x^n)^3+ln(ln(x))*b*ln(c)+ln(ln(x))*a

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Maxima [A]  time = 1.14282, size = 43, normalized size = 1.48 \begin{align*} -{\left (\log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right )\right )} b n + b \log \left (c x^{n}\right ) \log \left (\log \left (x\right )\right ) + a \log \left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x/log(x),x, algorithm="maxima")

[Out]

-(log(x)*log(log(x)) - log(x))*b*n + b*log(c*x^n)*log(log(x)) + a*log(log(x))

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Fricas [A]  time = 0.817742, size = 55, normalized size = 1.9 \begin{align*} b n \log \left (x\right ) +{\left (b \log \left (c\right ) + a\right )} \log \left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*n*log(x) + (b*log(c) + a)*log(log(x))

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Sympy [A]  time = 9.6219, size = 32, normalized size = 1.1 \begin{align*} a \log{\left (\log{\left (x \right )} \right )} - b \left (n \left (\log{\left (x \right )} \log{\left (\log{\left (x \right )} \right )} - \log{\left (x \right )}\right ) - \log{\left (c x^{n} \right )} \log{\left (\log{\left (x \right )} \right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x/ln(x),x)

[Out]

a*log(log(x)) - b*(n*(log(x)*log(log(x)) - log(x)) - log(c*x**n)*log(log(x)))

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Giac [A]  time = 1.30365, size = 23, normalized size = 0.79 \begin{align*} b n \log \left (x\right ) +{\left (b \log \left (c\right ) + a\right )} \log \left ({\left | \log \left (x\right ) \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x/log(x),x, algorithm="giac")

[Out]

b*n*log(x) + (b*log(c) + a)*log(abs(log(x)))