3.31.54 \(\int \frac {100 x-10 x^2+(10-x) \log (\frac {10-x}{5})+(-199 x+20 x^2+(-10+x) \log (\frac {10-x}{5})) \log (4 x)}{(-100+10 x) \log ^2(4 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2+\frac {1}{10} x \log \left (2-\frac {x}{5}\right )}{\log (4 x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.67, 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 {100 x-10 x^2+(10-x) \log \left (\frac {10-x}{5}\right )+\left (-199 x+20 x^2+(-10+x) \log \left (\frac {10-x}{5}\right )\right ) \log (4 x)}{(-100+10 x) \log ^2(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100*x - 10*x^2 + (10 - x)*Log[(10 - x)/5] + (-199*x + 20*x^2 + (-10 + x)*Log[(10 - x)/5])*Log[4*x])/((-10
0 + 10*x)*Log[4*x]^2),x]

[Out]

-1/8*ExpIntegralEi[2*Log[4*x]] + x^2/Log[4*x] - Defer[Int][Log[2 - x/5]/Log[4*x]^2, x]/10 - (199*Defer[Int][x/
((-10 + x)*Log[4*x]), x])/10 + 2*Defer[Int][x^2/((-10 + x)*Log[4*x]), x] + Defer[Int][Log[2 - x/5]/Log[4*x], x
]/10

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-10 x-\log \left (2-\frac {x}{5}\right )}{10 \log ^2(4 x)}+\frac {-199 x+20 x^2-10 \log \left (2-\frac {x}{5}\right )+x \log \left (2-\frac {x}{5}\right )}{10 (-10+x) \log (4 x)}\right ) \, dx\\ &=\frac {1}{10} \int \frac {-10 x-\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)} \, dx+\frac {1}{10} \int \frac {-199 x+20 x^2-10 \log \left (2-\frac {x}{5}\right )+x \log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)} \, dx\\ &=\frac {1}{10} \int \left (-\frac {10 x}{\log ^2(4 x)}-\frac {\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)}\right ) \, dx+\frac {1}{10} \int \left (-\frac {199 x}{(-10+x) \log (4 x)}+\frac {20 x^2}{(-10+x) \log (4 x)}-\frac {10 \log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)}+\frac {x \log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)}\right ) \, dx\\ &=-\left (\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)} \, dx\right )+\frac {1}{10} \int \frac {x \log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)} \, dx+2 \int \frac {x^2}{(-10+x) \log (4 x)} \, dx-\frac {199}{10} \int \frac {x}{(-10+x) \log (4 x)} \, dx-\int \frac {x}{\log ^2(4 x)} \, dx-\int \frac {\log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)} \, dx\\ &=\frac {x^2}{\log (4 x)}+\frac {1}{10} \int \left (\frac {\log \left (2-\frac {x}{5}\right )}{\log (4 x)}+\frac {10 \log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)}\right ) \, dx-\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)} \, dx-2 \int \frac {x}{\log (4 x)} \, dx+2 \int \frac {x^2}{(-10+x) \log (4 x)} \, dx-\frac {199}{10} \int \frac {x}{(-10+x) \log (4 x)} \, dx-\int \frac {\log \left (2-\frac {x}{5}\right )}{(-10+x) \log (4 x)} \, dx\\ &=\frac {x^2}{\log (4 x)}-\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)} \, dx+\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log (4 x)} \, dx-\frac {1}{8} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (4 x)\right )+2 \int \frac {x^2}{(-10+x) \log (4 x)} \, dx-\frac {199}{10} \int \frac {x}{(-10+x) \log (4 x)} \, dx\\ &=-\frac {1}{8} \text {Ei}(2 \log (4 x))+\frac {x^2}{\log (4 x)}-\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log ^2(4 x)} \, dx+\frac {1}{10} \int \frac {\log \left (2-\frac {x}{5}\right )}{\log (4 x)} \, dx+2 \int \frac {x^2}{(-10+x) \log (4 x)} \, dx-\frac {199}{10} \int \frac {x}{(-10+x) \log (4 x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 23, normalized size = 0.96 \begin {gather*} \frac {x \left (10 x+\log \left (2-\frac {x}{5}\right )\right )}{10 \log (4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*x - 10*x^2 + (10 - x)*Log[(10 - x)/5] + (-199*x + 20*x^2 + (-10 + x)*Log[(10 - x)/5])*Log[4*x])
/((-100 + 10*x)*Log[4*x]^2),x]

[Out]

(x*(10*x + Log[2 - x/5]))/(10*Log[4*x])

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 22, normalized size = 0.92 \begin {gather*} \frac {10 \, x^{2} + x \log \left (-\frac {1}{5} \, x + 2\right )}{10 \, \log \left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-10)*log(2-1/5*x)+20*x^2-199*x)*log(4*x)+(10-x)*log(2-1/5*x)-10*x^2+100*x)/(10*x-100)/log(4*x)^2
,x, algorithm="fricas")

[Out]

1/10*(10*x^2 + x*log(-1/5*x + 2))/log(4*x)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 35, normalized size = 1.46 \begin {gather*} \frac {x \log \left (-x + 10\right )}{10 \, \log \left (4 \, x\right )} + \frac {10 \, x^{2} - x \log \relax (5)}{10 \, \log \left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-10)*log(2-1/5*x)+20*x^2-199*x)*log(4*x)+(10-x)*log(2-1/5*x)-10*x^2+100*x)/(10*x-100)/log(4*x)^2
,x, algorithm="giac")

[Out]

1/10*x*log(-x + 10)/log(4*x) + 1/10*(10*x^2 - x*log(5))/log(4*x)

________________________________________________________________________________________

maple [A]  time = 0.49, size = 27, normalized size = 1.12

method	result	size
risch	\(\frac {x \ln \left (2-\frac {x}{5}\right )}{10 \ln \left (4 x \right )}+\frac {x^{2}}{\ln \left (4 x \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-10)*ln(2-1/5*x)+20*x^2-199*x)*ln(4*x)+(10-x)*ln(2-1/5*x)-10*x^2+100*x)/(10*x-100)/ln(4*x)^2,x,method=
_RETURNVERBOSE)

[Out]

1/10*x/ln(4*x)*ln(2-1/5*x)+x^2/ln(4*x)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 30, normalized size = 1.25 \begin {gather*} \frac {10 \, x^{2} - x \log \relax (5) + x \log \left (-x + 10\right )}{10 \, {\left (2 \, \log \relax (2) + \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-10)*log(2-1/5*x)+20*x^2-199*x)*log(4*x)+(10-x)*log(2-1/5*x)-10*x^2+100*x)/(10*x-100)/log(4*x)^2
,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.04, size = 19, normalized size = 0.79 \begin {gather*} \frac {x\,\left (10\,x+\ln \left (2-\frac {x}{5}\right )\right )}{10\,\ln \left (4\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x - 10*x^2 - log(2 - x/5)*(x - 10) + log(4*x)*(20*x^2 - 199*x + log(2 - x/5)*(x - 10)))/(log(4*x)^2*(
10*x - 100)),x)

[Out]

(x*(10*x + log(2 - x/5)))/(10*log(4*x))

________________________________________________________________________________________

sympy [A]  time = 0.37, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^{2}}{\log {\left (4 x \right )}} + \frac {x \log {\left (2 - \frac {x}{5} \right )}}{10 \log {\left (4 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-10)*ln(2-1/5*x)+20*x**2-199*x)*ln(4*x)+(10-x)*ln(2-1/5*x)-10*x**2+100*x)/(10*x-100)/ln(4*x)**2,
x)

[Out]

x**2/log(4*x) + x*log(2 - x/5)/(10*log(4*x))

________________________________________________________________________________________