3.21.85 \(\int \frac {15-21 x+21 x \log (x)}{x \log ^2(x)} \, dx\)

Optimal. Leaf size=14 \[ \frac {26 x-5 (3+x)}{\log (x)} \]

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Rubi [A]  time = 0.22, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6741, 12, 6742, 2353, 2297, 2298, 2302, 30} \begin {gather*} \frac {21 x}{\log (x)}-\frac {15}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 - 21*x + 21*x*Log[x])/(x*Log[x]^2),x]

[Out]

-15/Log[x] + (21*x)/Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 6741

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

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 \frac {3 (5-7 x+7 x \log (x))}{x \log ^2(x)} \, dx\\ &=3 \int \frac {5-7 x+7 x \log (x)}{x \log ^2(x)} \, dx\\ &=3 \int \left (\frac {5-7 x}{x \log ^2(x)}+\frac {7}{\log (x)}\right ) \, dx\\ &=3 \int \frac {5-7 x}{x \log ^2(x)} \, dx+21 \int \frac {1}{\log (x)} \, dx\\ &=21 \text {li}(x)+3 \int \left (-\frac {7}{\log ^2(x)}+\frac {5}{x \log ^2(x)}\right ) \, dx\\ &=21 \text {li}(x)+15 \int \frac {1}{x \log ^2(x)} \, dx-21 \int \frac {1}{\log ^2(x)} \, dx\\ &=\frac {21 x}{\log (x)}+21 \text {li}(x)+15 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )-21 \int \frac {1}{\log (x)} \, dx\\ &=-\frac {15}{\log (x)}+\frac {21 x}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {15}{\log (x)}+\frac {21 x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 - 21*x + 21*x*Log[x])/(x*Log[x]^2),x]

[Out]

-15/Log[x] + (21*x)/Log[x]

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fricas [A]  time = 0.69, size = 11, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (7 \, x - 5\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((21*x*log(x)-21*x+15)/x/log(x)^2,x, algorithm="fricas")

[Out]

3*(7*x - 5)/log(x)

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giac [A]  time = 1.97, size = 11, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (7 \, x - 5\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((21*x*log(x)-21*x+15)/x/log(x)^2,x, algorithm="giac")

[Out]

3*(7*x - 5)/log(x)

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maple [A]  time = 0.01, size = 11, normalized size = 0.79

method	result	size
norman	\(\frac {-15+21 x}{\ln \relax (x )}\)	\(11\)
risch	\(\frac {-15+21 x}{\ln \relax (x )}\)	\(12\)
default	\(\frac {21 x}{\ln \relax (x )}-\frac {15}{\ln \relax (x )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((21*x*ln(x)-21*x+15)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

(-15+21*x)/ln(x)

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maxima [C]  time = 0.46, size = 20, normalized size = 1.43 \begin {gather*} -\frac {15}{\log \relax (x)} + 21 \, {\rm Ei}\left (\log \relax (x)\right ) - 21 \, \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((21*x*log(x)-21*x+15)/x/log(x)^2,x, algorithm="maxima")

[Out]

-15/log(x) + 21*Ei(log(x)) - 21*gamma(-1, -log(x))

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mupad [B]  time = 1.14, size = 11, normalized size = 0.79 \begin {gather*} \frac {3\,\left (7\,x-5\right )}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((21*x*log(x) - 21*x + 15)/(x*log(x)^2),x)

[Out]

(3*(7*x - 5))/log(x)

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sympy [A]  time = 0.08, size = 7, normalized size = 0.50 \begin {gather*} \frac {21 x - 15}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((21*x*ln(x)-21*x+15)/x/ln(x)**2,x)

[Out]

(21*x - 15)/log(x)

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