3.42.48 \(\int \frac {5 \log (5)+(5+x) \log ^2(x)}{x \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ x-5 \left (-\frac {11}{5}-x+\log \left (\frac {e^x}{x}\right )+\frac {\log (5)}{\log (x)}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 14, normalized size of antiderivative = 0.54, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6742, 43, 2302, 30} \begin {gather*} x+5 \log (x)-\frac {5 \log (5)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*Log[5] + (5 + x)*Log[x]^2)/(x*Log[x]^2),x]

[Out]

x - (5*Log[5])/Log[x] + 5*Log[x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5+x}{x}+\frac {5 \log (5)}{x \log ^2(x)}\right ) \, dx\\ &=(5 \log (5)) \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {5+x}{x} \, dx\\ &=(5 \log (5)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \left (1+\frac {5}{x}\right ) \, dx\\ &=x-\frac {5 \log (5)}{\log (x)}+5 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.54 \begin {gather*} x-\frac {5 \log (5)}{\log (x)}+5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*Log[5] + (5 + x)*Log[x]^2)/(x*Log[x]^2),x]

[Out]

x - (5*Log[5])/Log[x] + 5*Log[x]

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fricas [A]  time = 0.57, size = 20, normalized size = 0.77 \begin {gather*} \frac {x \log \relax (x) + 5 \, \log \relax (x)^{2} - 5 \, \log \relax (5)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5+x)*log(x)^2+5*log(5))/x/log(x)^2,x, algorithm="fricas")

[Out]

(x*log(x) + 5*log(x)^2 - 5*log(5))/log(x)

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giac [A]  time = 0.23, size = 14, normalized size = 0.54 \begin {gather*} x - \frac {5 \, \log \relax (5)}{\log \relax (x)} + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5+x)*log(x)^2+5*log(5))/x/log(x)^2,x, algorithm="giac")

[Out]

x - 5*log(5)/log(x) + 5*log(x)

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maple [A]  time = 0.02, size = 15, normalized size = 0.58

method	result	size
default	\(x +5 \ln \relax (x )-\frac {5 \ln \relax (5)}{\ln \relax (x )}\)	\(15\)
risch	\(x +5 \ln \relax (x )-\frac {5 \ln \relax (5)}{\ln \relax (x )}\)	\(15\)
norman	\(\frac {x \ln \relax (x )+5 \ln \relax (x )^{2}-5 \ln \relax (5)}{\ln \relax (x )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5+x)*ln(x)^2+5*ln(5))/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

x+5*ln(x)-5*ln(5)/ln(x)

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maxima [A]  time = 0.37, size = 14, normalized size = 0.54 \begin {gather*} x - \frac {5 \, \log \relax (5)}{\log \relax (x)} + 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5+x)*log(x)^2+5*log(5))/x/log(x)^2,x, algorithm="maxima")

[Out]

x - 5*log(5)/log(x) + 5*log(x)

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mupad [B]  time = 2.99, size = 14, normalized size = 0.54 \begin {gather*} x+5\,\ln \relax (x)-\frac {5\,\ln \relax (5)}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*log(5) + log(x)^2*(x + 5))/(x*log(x)^2),x)

[Out]

x + 5*log(x) - (5*log(5))/log(x)

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sympy [A]  time = 0.09, size = 14, normalized size = 0.54 \begin {gather*} x + 5 \log {\relax (x )} - \frac {5 \log {\relax (5 )}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5+x)*ln(x)**2+5*ln(5))/x/ln(x)**2,x)

[Out]

x + 5*log(x) - 5*log(5)/log(x)

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