3.95.58 \(\int \frac {-5 \log (256)+5 \log (256) \log (x)+(-14+10 x+5 \log (4)) \log ^2(x)}{5 \log ^2(x)} \, dx\) [9458]

Optimal. Leaf size=16 \[ x \left (-\frac {14}{5}+x+\log (4)+\frac {\log (256)}{\log (x)}\right ) \]

[Out]

(8*ln(2)/ln(x)+x+2*ln(2)-14/5)*x

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 23, normalized size of antiderivative = 1.44, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {12, 6874, 2334, 2335} \begin {gather*} x^2+\frac {x \log (256)}{\log (x)}-\frac {2}{5} x (7-5 \log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5*Log[256] + 5*Log[256]*Log[x] + (-14 + 10*x + 5*Log[4])*Log[x]^2)/(5*Log[x]^2),x]

[Out]

x^2 - (2*x*(7 - 5*Log[2]))/5 + (x*Log[256])/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 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 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-5 \log (256)+5 \log (256) \log (x)+(-14+10 x+5 \log (4)) \log ^2(x)}{\log ^2(x)} \, dx\\ &=\frac {1}{5} \int \left (10 x-14 \left (1-\frac {5 \log (2)}{7}\right )-\frac {5 \log (256)}{\log ^2(x)}+\frac {5 \log (256)}{\log (x)}\right ) \, dx\\ &=x^2-\frac {2}{5} x (7-5 \log (2))-\log (256) \int \frac {1}{\log ^2(x)} \, dx+\log (256) \int \frac {1}{\log (x)} \, dx\\ &=x^2-\frac {2}{5} x (7-5 \log (2))+\frac {x \log (256)}{\log (x)}+\log (256) \text {li}(x)-\log (256) \int \frac {1}{\log (x)} \, dx\\ &=x^2-\frac {2}{5} x (7-5 \log (2))+\frac {x \log (256)}{\log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 21, normalized size = 1.31 \begin {gather*} -\frac {14 x}{5}+x^2+x \log (4)+\frac {x \log (256)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*Log[256] + 5*Log[256]*Log[x] + (-14 + 10*x + 5*Log[4])*Log[x]^2)/(5*Log[x]^2),x]

[Out]

(-14*x)/5 + x^2 + x*Log[4] + (x*Log[256])/Log[x]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.64, size = 43, normalized size = 2.69

method	result	size
risch	\(2 x \ln \left (2\right )+x^{2}-\frac {14 x}{5}+\frac {8 x \ln \left (2\right )}{\ln \left (x \right )}\)	\(22\)
norman	\(\frac {x^{2} \ln \left (x \right )+\left (2 \ln \left (2\right )-\frac {14}{5}\right ) x \ln \left (x \right )+8 x \ln \left (2\right )}{\ln \left (x \right )}\)	\(28\)
default	\(2 x \ln \left (2\right )+x^{2}-\frac {14 x}{5}-8 \ln \left (2\right ) \expIntegral \left (1, -\ln \left (x \right )\right )-8 \ln \left (2\right ) \left (-\frac {x}{\ln \left (x \right )}-\expIntegral \left (1, -\ln \left (x \right )\right )\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((10*ln(2)+10*x-14)*ln(x)^2+40*ln(2)*ln(x)-40*ln(2))/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

2*x*ln(2)+x^2-14/5*x-8*ln(2)*Ei(1,-ln(x))-8*ln(2)*(-x/ln(x)-Ei(1,-ln(x)))

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.29, size = 29, normalized size = 1.81 \begin {gather*} x^{2} + 2 \, x \log \left (2\right ) + 8 \, {\rm Ei}\left (\log \left (x\right )\right ) \log \left (2\right ) - 8 \, \Gamma \left (-1, -\log \left (x\right )\right ) \log \left (2\right ) - \frac {14}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((10*log(2)+10*x-14)*log(x)^2+40*log(2)*log(x)-40*log(2))/log(x)^2,x, algorithm="maxima")

[Out]

x^2 + 2*x*log(2) + 8*Ei(log(x))*log(2) - 8*gamma(-1, -log(x))*log(2) - 14/5*x

________________________________________________________________________________________

Fricas [A]
time = 0.49, size = 29, normalized size = 1.81 \begin {gather*} \frac {40 \, x \log \left (2\right ) + {\left (5 \, x^{2} + 10 \, x \log \left (2\right ) - 14 \, x\right )} \log \left (x\right )}{5 \, \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((10*log(2)+10*x-14)*log(x)^2+40*log(2)*log(x)-40*log(2))/log(x)^2,x, algorithm="fricas")

[Out]

1/5*(40*x*log(2) + (5*x^2 + 10*x*log(2) - 14*x)*log(x))/log(x)

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 22, normalized size = 1.38 \begin {gather*} x^{2} + x \left (- \frac {14}{5} + 2 \log {\left (2 \right )}\right ) + \frac {8 x \log {\left (2 \right )}}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((10*ln(2)+10*x-14)*ln(x)**2+40*ln(2)*ln(x)-40*ln(2))/ln(x)**2,x)

[Out]

x**2 + x*(-14/5 + 2*log(2)) + 8*x*log(2)/log(x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 21, normalized size = 1.31 \begin {gather*} x^{2} + 2 \, x \log \left (2\right ) - \frac {14}{5} \, x + \frac {8 \, x \log \left (2\right )}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((10*log(2)+10*x-14)*log(x)^2+40*log(2)*log(x)-40*log(2))/log(x)^2,x, algorithm="giac")

[Out]

x^2 + 2*x*log(2) - 14/5*x + 8*x*log(2)/log(x)

________________________________________________________________________________________

Mupad [B]
time = 5.45, size = 22, normalized size = 1.38 \begin {gather*} \frac {x\,\left (5\,x+10\,\ln \left (2\right )-14\right )}{5}+\frac {8\,x\,\ln \left (2\right )}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(x)^2*(10*x + 10*log(2) - 14))/5 - 8*log(2) + 8*log(2)*log(x))/log(x)^2,x)

[Out]

(x*(5*x + 10*log(2) - 14))/5 + (8*x*log(2))/log(x)

________________________________________________________________________________________