3.62.43 \(\int \frac {-248+20 x-20 x \log (x)+x \log ^2(x)-100 \log ^2(\frac {9}{\log (5)})-10 \log ^4(\frac {9}{\log (5)})}{x \log ^2(x)} \, dx\) [6143]

Optimal. Leaf size=31 \[ x-\frac {2 \left (1+5 \left (2 x-\left (5+\log ^2\left (\frac {9}{\log (5)}\right )\right )^2\right )\right )}{\log (x)} \]

[Out]

x-2*(1+10*x-5*(ln(9/ln(5))^2+5)^2)/ln(x)

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Rubi [A]
time = 0.25, antiderivative size = 39, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 7, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6873, 6874, 2335, 2395, 2334, 2339, 30} \begin {gather*} x+\frac {2 \left (124+5 \log ^4\left (\frac {9}{\log (5)}\right )+50 \log ^2\left (\frac {9}{\log (5)}\right )\right )}{\log (x)}-\frac {20 x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-248 + 20*x - 20*x*Log[x] + x*Log[x]^2 - 100*Log[9/Log[5]]^2 - 10*Log[9/Log[5]]^4)/(x*Log[x]^2),x]

[Out]

x - (20*x)/Log[x] + (2*(124 + 50*Log[9/Log[5]]^2 + 5*Log[9/Log[5]]^4))/Log[x]

Rule 30

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

Rule 2334

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 2335

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

Rule 2339

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 2395

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 6873

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x-20 x \log (x)+x \log ^2(x)-248 \left (1+\frac {5}{124} \log ^2\left (\frac {9}{\log (5)}\right ) \left (10+\log ^2\left (\frac {9}{\log (5)}\right )\right )\right )}{x \log ^2(x)} \, dx\\ &=\int \left (1-\frac {20}{\log (x)}+\frac {2 \left (-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{x \log ^2(x)}\right ) \, dx\\ &=x+2 \int \frac {-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )}{x \log ^2(x)} \, dx-20 \int \frac {1}{\log (x)} \, dx\\ &=x-20 \text {li}(x)+2 \int \left (\frac {10}{\log ^2(x)}+\frac {-124-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )}{x \log ^2(x)}\right ) \, dx\\ &=x-20 \text {li}(x)+20 \int \frac {1}{\log ^2(x)} \, dx-\left (2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )\right ) \int \frac {1}{x \log ^2(x)} \, dx\\ &=x-\frac {20 x}{\log (x)}-20 \text {li}(x)+20 \int \frac {1}{\log (x)} \, dx-\left (2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=x-\frac {20 x}{\log (x)}+\frac {2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 35, normalized size = 1.13 \begin {gather*} x-\frac {2 \left (-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-248 + 20*x - 20*x*Log[x] + x*Log[x]^2 - 100*Log[9/Log[5]]^2 - 10*Log[9/Log[5]]^4)/(x*Log[x]^2),x]

[Out]

x - (2*(-124 + 10*x - 50*Log[9/Log[5]]^2 - 5*Log[9/Log[5]]^4))/Log[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(28)=56\).
time = 0.22, size = 110, normalized size = 3.55

method	result	size
risch	\(x +\frac {160 \ln \left (3\right )^{4}-320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )+240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}-80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}+10 \ln \left (\ln \left (5\right )\right )^{4}+400 \ln \left (3\right )^{2}-400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )+100 \ln \left (\ln \left (5\right )\right )^{2}-20 x +248}{\ln \left (x \right )}\)	\(76\)
norman	\(\frac {x \ln \left (x \right )-20 x +248+160 \ln \left (3\right )^{4}-320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )+240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}-80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}+10 \ln \left (\ln \left (5\right )\right )^{4}+400 \ln \left (3\right )^{2}-400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )+100 \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}\)	\(77\)
default	\(x +\frac {160 \ln \left (3\right )^{4}}{\ln \left (x \right )}-\frac {320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )}{\ln \left (x \right )}+\frac {240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}-\frac {80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}}{\ln \left (x \right )}+\frac {10 \ln \left (\ln \left (5\right )\right )^{4}}{\ln \left (x \right )}+\frac {400 \ln \left (3\right )^{2}}{\ln \left (x \right )}-\frac {400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )}{\ln \left (x \right )}+\frac {100 \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}-\frac {20 x}{\ln \left (x \right )}+\frac {248}{\ln \left (x \right )}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(x)^2-20*x*ln(x)-10*ln(9/ln(5))^4-100*ln(9/ln(5))^2+20*x-248)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

x+160*ln(3)^4/ln(x)-320*ln(3)^3*ln(ln(5))/ln(x)+240*ln(3)^2*ln(ln(5))^2/ln(x)-80*ln(3)*ln(ln(5))^3/ln(x)+10*ln
(ln(5))^4/ln(x)+400*ln(3)^2/ln(x)-400*ln(3)*ln(ln(5))/ln(x)+100*ln(ln(5))^2/ln(x)-20*x/ln(x)+248/ln(x)

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.38, size = 51, normalized size = 1.65 \begin {gather*} \frac {10 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{4}}{\log \left (x\right )} + x + \frac {100 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{2}}{\log \left (x\right )} + \frac {248}{\log \left (x\right )} - 20 \, {\rm Ei}\left (\log \left (x\right )\right ) + 20 \, \Gamma \left (-1, -\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)^2-20*x*log(x)-10*log(9/log(5))^4-100*log(9/log(5))^2+20*x-248)/x/log(x)^2,x, algorithm="ma
xima")

[Out]

10*log(9/log(5))^4/log(x) + x + 100*log(9/log(5))^2/log(x) + 248/log(x) - 20*Ei(log(x)) + 20*gamma(-1, -log(x)
)

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Fricas [A]
time = 0.40, size = 36, normalized size = 1.16 \begin {gather*} \frac {10 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{4} + x \log \left (x\right ) + 100 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{2} - 20 \, x + 248}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)^2-20*x*log(x)-10*log(9/log(5))^4-100*log(9/log(5))^2+20*x-248)/x/log(x)^2,x, algorithm="fr
icas")

[Out]

(10*log(9/log(5))^4 + x*log(x) + 100*log(9/log(5))^2 - 20*x + 248)/log(x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
time = 0.04, size = 87, normalized size = 2.81 \begin {gather*} x + \frac {- 20 x - 400 \log {\left (3 \right )} \log {\left (\log {\left (5 \right )} \right )} - 320 \log {\left (3 \right )}^{3} \log {\left (\log {\left (5 \right )} \right )} - 80 \log {\left (3 \right )} \log {\left (\log {\left (5 \right )} \right )}^{3} + 10 \log {\left (\log {\left (5 \right )} \right )}^{4} + 100 \log {\left (\log {\left (5 \right )} \right )}^{2} + 240 \log {\left (3 \right )}^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 160 \log {\left (3 \right )}^{4} + 248 + 400 \log {\left (3 \right )}^{2}}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x)**2-20*x*ln(x)-10*ln(9/ln(5))**4-100*ln(9/ln(5))**2+20*x-248)/x/ln(x)**2,x)

[Out]

x + (-20*x - 400*log(3)*log(log(5)) - 320*log(3)**3*log(log(5)) - 80*log(3)*log(log(5))**3 + 10*log(log(5))**4
+ 100*log(log(5))**2 + 240*log(3)**2*log(log(5))**2 + 160*log(3)**4 + 248 + 400*log(3)**2)/log(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (28) = 56\).
time = 0.41, size = 75, normalized size = 2.42 \begin {gather*} x + \frac {2 \, {\left (80 \, \log \left (3\right )^{4} - 160 \, \log \left (3\right )^{3} \log \left (\log \left (5\right )\right ) + 120 \, \log \left (3\right )^{2} \log \left (\log \left (5\right )\right )^{2} - 40 \, \log \left (3\right ) \log \left (\log \left (5\right )\right )^{3} + 5 \, \log \left (\log \left (5\right )\right )^{4} + 200 \, \log \left (3\right )^{2} - 200 \, \log \left (3\right ) \log \left (\log \left (5\right )\right ) + 50 \, \log \left (\log \left (5\right )\right )^{2} - 10 \, x + 124\right )}}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x)^2-20*x*log(x)-10*log(9/log(5))^4-100*log(9/log(5))^2+20*x-248)/x/log(x)^2,x, algorithm="gi
ac")

[Out]

x + 2*(80*log(3)^4 - 160*log(3)^3*log(log(5)) + 120*log(3)^2*log(log(5))^2 - 40*log(3)*log(log(5))^3 + 5*log(l
og(5))^4 + 200*log(3)^2 - 200*log(3)*log(log(5)) + 50*log(log(5))^2 - 10*x + 124)/log(x)

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Mupad [B]
time = 4.31, size = 34, normalized size = 1.10 \begin {gather*} x+\frac {100\,{\ln \left (\frac {9}{\ln \left (5\right )}\right )}^2-20\,x+10\,{\ln \left (\frac {9}{\ln \left (5\right )}\right )}^4+248}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x*log(x) - x*log(x)^2 - 20*x + 100*log(9/log(5))^2 + 10*log(9/log(5))^4 + 248)/(x*log(x)^2),x)

[Out]

x + (100*log(9/log(5))^2 - 20*x + 10*log(9/log(5))^4 + 248)/log(x)

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