3.17.50 \(\int \frac {-2 \log (x)-\log (\frac {5 x}{4})+(-x+x \log (x)) \log ^3(\frac {5 x}{4})}{x \log ^2(x) \log ^3(\frac {5 x}{4})} \, dx\)

Optimal. Leaf size=17 \[ 2+\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \]

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Rubi [F]  time = 0.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*Log[x] - Log[(5*x)/4] + (-x + x*Log[x])*Log[(5*x)/4]^3)/(x*Log[x]^2*Log[(5*x)/4]^3),x]

[Out]

x/Log[x] - 2*Defer[Int][1/(x*Log[x]*Log[(5*x)/4]^3), x] - Defer[Int][1/(x*Log[x]^2*Log[(5*x)/4]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-1+\log (x)}{\log ^2(x)}-\frac {2}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )}-\frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \frac {-1+\log (x)}{\log ^2(x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=\frac {x}{\log (x)}+\text {li}(x)-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ &=\frac {x}{\log (x)}-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} \frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*Log[x] - Log[(5*x)/4] + (-x + x*Log[x])*Log[(5*x)/4]^3)/(x*Log[x]^2*Log[(5*x)/4]^3),x]

[Out]

(x + Log[(5*x)/4]^(-2))/Log[x]

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fricas [B]  time = 0.87, size = 44, normalized size = 2.59 \begin {gather*} \frac {x \log \left (\frac {5}{4}\right )^{2} + 2 \, x \log \left (\frac {5}{4}\right ) \log \relax (x) + x \log \relax (x)^{2} + 1}{\log \left (\frac {5}{4}\right )^{2} \log \relax (x) + 2 \, \log \left (\frac {5}{4}\right ) \log \relax (x)^{2} + \log \relax (x)^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="fricas")

[Out]

(x*log(5/4)^2 + 2*x*log(5/4)*log(x) + x*log(x)^2 + 1)/(log(5/4)^2*log(x) + 2*log(5/4)*log(x)^2 + log(x)^3)

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giac [B]  time = 0.26, size = 167, normalized size = 9.82 \begin {gather*} \frac {x \log \relax (5)^{2} - 4 \, x \log \relax (5) \log \relax (2) + 4 \, x \log \relax (2)^{2} + 1}{\log \relax (5)^{2} \log \relax (x) - 4 \, \log \relax (5) \log \relax (2) \log \relax (x) + 4 \, \log \relax (2)^{2} \log \relax (x)} - \frac {2 \, \log \relax (5) - 4 \, \log \relax (2) + \log \relax (x)}{\log \relax (5)^{4} - 8 \, \log \relax (5)^{3} \log \relax (2) + 24 \, \log \relax (5)^{2} \log \relax (2)^{2} - 32 \, \log \relax (5) \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + 2 \, \log \relax (5)^{3} \log \relax (x) - 12 \, \log \relax (5)^{2} \log \relax (2) \log \relax (x) + 24 \, \log \relax (5) \log \relax (2)^{2} \log \relax (x) - 16 \, \log \relax (2)^{3} \log \relax (x) + \log \relax (5)^{2} \log \relax (x)^{2} - 4 \, \log \relax (5) \log \relax (2) \log \relax (x)^{2} + 4 \, \log \relax (2)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="giac")

[Out]

(x*log(5)^2 - 4*x*log(5)*log(2) + 4*x*log(2)^2 + 1)/(log(5)^2*log(x) - 4*log(5)*log(2)*log(x) + 4*log(2)^2*log
(x)) - (2*log(5) - 4*log(2) + log(x))/(log(5)^4 - 8*log(5)^3*log(2) + 24*log(5)^2*log(2)^2 - 32*log(5)*log(2)^
3 + 16*log(2)^4 + 2*log(5)^3*log(x) - 12*log(5)^2*log(2)*log(x) + 24*log(5)*log(2)^2*log(x) - 16*log(2)^3*log(
x) + log(5)^2*log(x)^2 - 4*log(5)*log(2)*log(x)^2 + 4*log(2)^2*log(x)^2)

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maple [B]  time = 0.22, size = 65, normalized size = 3.82

method	result	size
risch	\(\frac {4-16 x \ln \relax (2) \ln \relax (x )-16 x \ln \relax (2) \ln \relax (5)+16 x \ln \relax (2)^{2}+4 x \ln \relax (x )^{2}+4 x \ln \relax (5)^{2}+8 x \ln \relax (5) \ln \relax (x )}{\left (2 \ln \relax (5)-4 \ln \relax (2)+2 \ln \relax (x )\right )^{2} \ln \relax (x )}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*ln(x)-x)*ln(5/4*x)^3-ln(5/4*x)-2*ln(x))/x/ln(x)^2/ln(5/4*x)^3,x,method=_RETURNVERBOSE)

[Out]

(4-16*x*ln(2)*ln(x)-16*x*ln(2)*ln(5)+16*x*ln(2)^2+4*x*ln(x)^2+4*x*ln(5)^2+8*x*ln(5)*ln(x))/(2*ln(5)-4*ln(2)+2*
ln(x))^2/ln(x)

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maxima [C]  time = 0.64, size = 186, normalized size = 10.94 \begin {gather*} -\frac {3 \, \log \relax (5) - 6 \, \log \relax (2) + 2 \, \log \relax (x)}{\log \relax (5)^{4} - 8 \, \log \relax (5)^{3} \log \relax (2) + 24 \, \log \relax (5)^{2} \log \relax (2)^{2} - 32 \, \log \relax (5) \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + {\left (\log \relax (5)^{2} - 4 \, \log \relax (5) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (\log \relax (5)^{3} - 6 \, \log \relax (5)^{2} \log \relax (2) + 12 \, \log \relax (5) \log \relax (2)^{2} - 8 \, \log \relax (2)^{3}\right )} \log \relax (x)} + \frac {\log \relax (5) - 2 \, \log \relax (2) + 2 \, \log \relax (x)}{{\left (\log \relax (5)^{2} - 4 \, \log \relax (5) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2} + {\left (\log \relax (5)^{3} - 6 \, \log \relax (5)^{2} \log \relax (2) + 12 \, \log \relax (5) \log \relax (2)^{2} - 8 \, \log \relax (2)^{3}\right )} \log \relax (x)} + {\rm Ei}\left (\log \relax (x)\right ) - \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="maxima")

[Out]

-(3*log(5) - 6*log(2) + 2*log(x))/(log(5)^4 - 8*log(5)^3*log(2) + 24*log(5)^2*log(2)^2 - 32*log(5)*log(2)^3 +
16*log(2)^4 + (log(5)^2 - 4*log(5)*log(2) + 4*log(2)^2)*log(x)^2 + 2*(log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)
*log(2)^2 - 8*log(2)^3)*log(x)) + (log(5) - 2*log(2) + 2*log(x))/((log(5)^2 - 4*log(5)*log(2) + 4*log(2)^2)*lo
g(x)^2 + (log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)*log(2)^2 - 8*log(2)^3)*log(x)) + Ei(log(x)) - gamma(-1, -lo
g(x))

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mupad [B]  time = 1.18, size = 21, normalized size = 1.24 \begin {gather*} \frac {x\,{\ln \left (\frac {5\,x}{4}\right )}^2+1}{{\ln \left (\frac {5\,x}{4}\right )}^2\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((5*x)/4) + 2*log(x) + log((5*x)/4)^3*(x - x*log(x)))/(x*log((5*x)/4)^3*log(x)^2),x)

[Out]

(x*log((5*x)/4)^2 + 1)/(log((5*x)/4)^2*log(x))

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sympy [B]  time = 0.43, size = 94, normalized size = 5.53 \begin {gather*} \frac {x \log {\relax (x )}^{2} - 4 x \log {\relax (2 )} \log {\relax (5 )} + 4 x \log {\relax (2 )}^{2} + x \log {\relax (5 )}^{2} + \left (- 4 x \log {\relax (2 )} + 2 x \log {\relax (5 )}\right ) \log {\relax (x )} + 1}{\log {\relax (x )}^{3} + \left (- 4 \log {\relax (2 )} + 2 \log {\relax (5 )}\right ) \log {\relax (x )}^{2} + \left (- 4 \log {\relax (2 )} \log {\relax (5 )} + 4 \log {\relax (2 )}^{2} + \log {\relax (5 )}^{2}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*ln(x)-x)*ln(5/4*x)**3-ln(5/4*x)-2*ln(x))/x/ln(x)**2/ln(5/4*x)**3,x)

[Out]

(x*log(x)**2 - 4*x*log(2)*log(5) + 4*x*log(2)**2 + x*log(5)**2 + (-4*x*log(2) + 2*x*log(5))*log(x) + 1)/(log(x
)**3 + (-4*log(2) + 2*log(5))*log(x)**2 + (-4*log(2)*log(5) + 4*log(2)**2 + log(5)**2)*log(x))

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