∫ 25−46x+6x2+(4−7x−x2) log(−x+2x log(5) 2 log(5) )+log(x)(21−x−5x2+4 log(−x+2x log(5) 2 log(5) )) ______________________________________________________________________________________________________________

3.24.23 \(\int \frac {25-46 x+6 x^2+(4-7 x-x^2) \log (\frac {-x+2 x \log (5)}{2 \log (5)})+\log (x) (21-x-5 x^2+4 \log (\frac {-x+2 x \log (5)}{2 \log (5)}))}{5 x^4-10 x^3 \log (x)+5 x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=36 \[ \frac {5+x^2+\frac {1}{5} (4+x) \log \left (x-\frac {x}{2 \log (5)}\right )}{x (x-\log (x))} \]

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Rubi [F] time = 3.00, 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 {25-46 x+6 x^2+\left (4-7 x-x^2\right ) \log \left (\frac {-x+2 x \log (5)}{2 \log (5)}\right )+\log (x) \left (21-x-5 x^2+4 \log \left (\frac {-x+2 x \log (5)}{2 \log (5)}\right )\right )}{5 x^4-10 x^3 \log (x)+5 x^2 \log ^2(x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(25 - 46*x + 6*x^2 + (4 - 7*x - x^2)*Log[(-x + 2*x*Log[5])/(2*Log[5])] + Log[x]*(21 - x - 5*x^2 + 4*Log[(-

x + 2*x*Log[5])/(2*Log[5])]))/(5*x^4 - 10*x^3*Log[x] + 5*x^2*Log[x]^2),x]

[Out]

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

, x] - Defer[Int][x/(x - Log[x])^2, x] - (21*Defer[Int][1/(x^2*(x - Log[x])), x])/5 + Defer[Int][1/(x*(x - Log

[x])), x]/5 - Defer[Int][(-x + Log[x])^(-1), x] - Defer[Int][Log[x*(1 - Log[25]^(-1))]/(x - Log[x])^2, x]/5 +

(4*Defer[Int][Log[x*(1 - Log[25]^(-1))]/(x^2*(x - Log[x])^2), x])/5 - (7*Defer[Int][Log[x*(1 - Log[25]^(-1))]/

(x*(x - Log[x])^2), x])/5 + (4*Defer[Int][(Log[x]*Log[x*(1 - Log[25]^(-1))])/(x^2*(x - Log[x])^2), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25-46 x+6 x^2-\log (x) \left (-21+x+5 x^2-4 \log \left (x-\frac {x}{\log (25)}\right )\right )-\left (-4+7 x+x^2\right ) \log \left (x-\frac {x}{\log (25)}\right )}{5 x^2 (x-\log (x))^2} \, dx\\ &=\frac {1}{5} \int \frac {25-46 x+6 x^2-\log (x) \left (-21+x+5 x^2-4 \log \left (x-\frac {x}{\log (25)}\right )\right )-\left (-4+7 x+x^2\right ) \log \left (x-\frac {x}{\log (25)}\right )}{x^2 (x-\log (x))^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {6}{(x-\log (x))^2}+\frac {25}{x^2 (x-\log (x))^2}-\frac {46}{x (x-\log (x))^2}-\frac {5 \log (x)}{(x-\log (x))^2}+\frac {21 \log (x)}{x^2 (x-\log (x))^2}-\frac {\log (x)}{x (x-\log (x))^2}+\frac {\left (4-7 x-x^2+4 \log (x)\right ) \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {\log (x)}{x (x-\log (x))^2} \, dx\right )+\frac {1}{5} \int \frac {\left (4-7 x-x^2+4 \log (x)\right ) \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2} \, dx+\frac {6}{5} \int \frac {1}{(x-\log (x))^2} \, dx+\frac {21}{5} \int \frac {\log (x)}{x^2 (x-\log (x))^2} \, dx+5 \int \frac {1}{x^2 (x-\log (x))^2} \, dx-\frac {46}{5} \int \frac {1}{x (x-\log (x))^2} \, dx-\int \frac {\log (x)}{(x-\log (x))^2} \, dx\\ &=-\left (\frac {1}{5} \int \left (\frac {1}{(x-\log (x))^2}-\frac {1}{x (x-\log (x))}\right ) \, dx\right )+\frac {1}{5} \int \left (-\frac {\log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{(x-\log (x))^2}+\frac {4 \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2}-\frac {7 \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x (x-\log (x))^2}+\frac {4 \log (x) \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2}\right ) \, dx+\frac {6}{5} \int \frac {1}{(x-\log (x))^2} \, dx+\frac {21}{5} \int \left (\frac {1}{x (x-\log (x))^2}-\frac {1}{x^2 (x-\log (x))}\right ) \, dx+5 \int \frac {1}{x^2 (x-\log (x))^2} \, dx-\frac {46}{5} \int \frac {1}{x (x-\log (x))^2} \, dx-\int \left (\frac {x}{(x-\log (x))^2}+\frac {1}{-x+\log (x)}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {1}{(x-\log (x))^2} \, dx\right )+\frac {1}{5} \int \frac {1}{x (x-\log (x))} \, dx-\frac {1}{5} \int \frac {\log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{(x-\log (x))^2} \, dx+\frac {4}{5} \int \frac {\log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2} \, dx+\frac {4}{5} \int \frac {\log (x) \log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x^2 (x-\log (x))^2} \, dx+\frac {6}{5} \int \frac {1}{(x-\log (x))^2} \, dx-\frac {7}{5} \int \frac {\log \left (x \left (1-\frac {1}{\log (25)}\right )\right )}{x (x-\log (x))^2} \, dx+\frac {21}{5} \int \frac {1}{x (x-\log (x))^2} \, dx-\frac {21}{5} \int \frac {1}{x^2 (x-\log (x))} \, dx+5 \int \frac {1}{x^2 (x-\log (x))^2} \, dx-\frac {46}{5} \int \frac {1}{x (x-\log (x))^2} \, dx-\int \frac {x}{(x-\log (x))^2} \, dx-\int \frac {1}{-x+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.45, size = 41, normalized size = 1.14 \begin {gather*} -\frac {-25-6 x^2+x \log (x)-(4+x) \log \left (x-\frac {x}{\log (25)}\right )}{5 x (x-\log (x))} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(25 - 46*x + 6*x^2 + (4 - 7*x - x^2)*Log[(-x + 2*x*Log[5])/(2*Log[5])] + Log[x]*(21 - x - 5*x^2 + 4*

Log[(-x + 2*x*Log[5])/(2*Log[5])]))/(5*x^4 - 10*x^3*Log[x] + 5*x^2*Log[x]^2),x]

[Out]

-1/5*(-25 - 6*x^2 + x*Log[x] - (4 + x)*Log[x - x/Log[25]])/(x*(x - Log[x]))

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fricas [B] time = 0.81, size = 84, normalized size = 2.33 \begin {gather*} \frac {6 \, x^{2} - x \log \left (\frac {2 \, \log \relax (5)}{2 \, \log \relax (5) - 1}\right ) + 4 \, \log \left (\frac {2 \, x \log \relax (5) - x}{2 \, \log \relax (5)}\right ) + 25}{5 \, {\left (x^{2} - x \log \left (\frac {2 \, x \log \relax (5) - x}{2 \, \log \relax (5)}\right ) - x \log \left (\frac {2 \, \log \relax (5)}{2 \, \log \relax (5) - 1}\right )\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(1/2*(2*x*log(5)-x)/log(5))-5*x^2-x+21)*log(x)+(-x^2-7*x+4)*log(1/2*(2*x*log(5)-x)/log(5))+6*

x^2-46*x+25)/(5*x^2*log(x)^2-10*x^3*log(x)+5*x^4),x, algorithm="fricas")

[Out]

1/5*(6*x^2 - x*log(2*log(5)/(2*log(5) - 1)) + 4*log(1/2*(2*x*log(5) - x)/log(5)) + 25)/(x^2 - x*log(1/2*(2*x*l

og(5) - x)/log(5)) - x*log(2*log(5)/(2*log(5) - 1)))

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giac [A] time = 0.29, size = 67, normalized size = 1.86 \begin {gather*} \frac {6 \, x^{2} - x \log \relax (2) + x \log \left (2 \, \log \relax (5) - 1\right ) - x \log \left (\log \relax (5)\right ) + 4 \, x - 4 \, \log \relax (2) + 4 \, \log \left (2 \, \log \relax (5) - 1\right ) - 4 \, \log \left (\log \relax (5)\right ) + 25}{5 \, {\left (x^{2} - x \log \relax (x)\right )}} - \frac {4}{5 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(1/2*(2*x*log(5)-x)/log(5))-5*x^2-x+21)*log(x)+(-x^2-7*x+4)*log(1/2*(2*x*log(5)-x)/log(5))+6*

x^2-46*x+25)/(5*x^2*log(x)^2-10*x^3*log(x)+5*x^4),x, algorithm="giac")

[Out]

1/5*(6*x^2 - x*log(2) + x*log(2*log(5) - 1) - x*log(log(5)) + 4*x - 4*log(2) + 4*log(2*log(5) - 1) - 4*log(log

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

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maple [B] time = 0.28, size = 69, normalized size = 1.92


method	result	size

risch	\(-\frac {4}{5 x}+\frac {50-2 x \ln \left (\ln \relax (5)\right )+2 x \ln \left (2 \ln \relax (5)-1\right )-2 x \ln \relax (2)+12 x^{2}-8 \ln \left (\ln \relax (5)\right )+8 \ln \left (2 \ln \relax (5)-1\right )-8 \ln \relax (2)+8 x}{10 x \left (x -\ln \relax (x )\right )}\)	\(69\)
default	\(\frac {25+6 x \ln \relax (x )+4 \ln \relax (x )}{5 x \left (x -\ln \relax (x )\right )}+\frac {x \ln \left (2 \ln \relax (5)-1\right )+4 \ln \left (2 \ln \relax (5)-1\right )}{5 x \left (x -\ln \relax (x )\right )}+\frac {-x \ln \relax (2)-4 \ln \relax (2)}{5 x \left (x -\ln \relax (x )\right )}+\frac {-x \ln \left (\ln \relax (5)\right )-4 \ln \left (\ln \relax (5)\right )}{5 x \left (x -\ln \relax (x )\right )}\)	\(106\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(1/2*(2*x*ln(5)-x)/ln(5))-5*x^2-x+21)*ln(x)+(-x^2-7*x+4)*ln(1/2*(2*x*ln(5)-x)/ln(5))+6*x^2-46*x+25)/

(5*x^2*ln(x)^2-10*x^3*ln(x)+5*x^4),x,method=_RETURNVERBOSE)

[Out]

-4/5/x+1/10*(50-2*x*ln(ln(5))+2*x*ln(2*ln(5)-1)-2*x*ln(2)+12*x^2-8*ln(ln(5))+8*ln(2*ln(5)-1)-8*ln(2)+8*x)/x/(x

-ln(x))

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maxima [A] time = 0.56, size = 60, normalized size = 1.67 \begin {gather*} \frac {6 \, x^{2} - x {\left (\log \relax (2) - \log \left (2 \, \log \relax (5) - 1\right ) + \log \left (\log \relax (5)\right )\right )} - 4 \, \log \relax (2) + 4 \, \log \relax (x) + 4 \, \log \left (2 \, \log \relax (5) - 1\right ) - 4 \, \log \left (\log \relax (5)\right ) + 25}{5 \, {\left (x^{2} - x \log \relax (x)\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(1/2*(2*x*log(5)-x)/log(5))-5*x^2-x+21)*log(x)+(-x^2-7*x+4)*log(1/2*(2*x*log(5)-x)/log(5))+6*

x^2-46*x+25)/(5*x^2*log(x)^2-10*x^3*log(x)+5*x^4),x, algorithm="maxima")

[Out]

1/5*(6*x^2 - x*(log(2) - log(2*log(5) - 1) + log(log(5))) - 4*log(2) + 4*log(x) + 4*log(2*log(5) - 1) - 4*log(

log(5)) + 25)/(x^2 - x*log(x))

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mupad [B] time = 1.61, size = 49, normalized size = 1.36 \begin {gather*} \frac {4\,\ln \left (\ln \relax (5)-\frac {1}{2}\right )-4\,\ln \left (\ln \relax (5)\right )+4\,\ln \relax (x)+x\,\ln \left (\ln \relax (5)-\frac {1}{2}\right )-x\,\ln \left (\ln \relax (5)\right )+6\,x^2+25}{5\,x\,\left (x-\ln \relax (x)\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(46*x + log(-(x/2 - x*log(5))/log(5))*(7*x + x^2 - 4) + log(x)*(x - 4*log(-(x/2 - x*log(5))/log(5)) + 5*x

^2 - 21) - 6*x^2 - 25)/(5*x^2*log(x)^2 - 10*x^3*log(x) + 5*x^4),x)

[Out]

(4*log(log(5) - 1/2) - 4*log(log(5)) + 4*log(x) + x*log(log(5) - 1/2) - x*log(log(5)) + 6*x^2 + 25)/(5*x*(x -

log(x)))

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sympy [B] time = 0.37, size = 70, normalized size = 1.94 \begin {gather*} \frac {- 6 x^{2} - 4 x - x \log {\left (-1 + 2 \log {\relax (5 )} \right )} + x \log {\left (\log {\relax (5 )} \right )} + x \log {\relax (2 )} - 25 - 4 \log {\left (-1 + 2 \log {\relax (5 )} \right )} + 4 \log {\left (\log {\relax (5 )} \right )} + 4 \log {\relax (2 )}}{- 5 x^{2} + 5 x \log {\relax (x )}} - \frac {4}{5 x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(1/2*(2*x*ln(5)-x)/ln(5))-5*x**2-x+21)*ln(x)+(-x**2-7*x+4)*ln(1/2*(2*x*ln(5)-x)/ln(5))+6*x**2-

46*x+25)/(5*x**2*ln(x)**2-10*x**3*ln(x)+5*x**4),x)

[Out]

(-6*x**2 - 4*x - x*log(-1 + 2*log(5)) + x*log(log(5)) + x*log(2) - 25 - 4*log(-1 + 2*log(5)) + 4*log(log(5)) +

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

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