3.24.23 \(\int \frac {25-46 x+6 x^2+(4-7 x-x^2) \log (\frac {-x+2 x \log (5)}{2 \log (5)})+\log (x) (21-x-5 x^2+4 \log (\frac {-x+2 x \log (5)}{2 \log (5)}))}{5 x^4-10 x^3 \log (x)+5 x^2 \log ^2(x)} \, dx\) [2323]

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

[Out]

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

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Rubi [F]
time = 1.86, 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 = 0.11, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(32)=64\).
time = 0.72, size = 92, normalized size = 2.56

method	result	size
risch	\(-\frac {4}{5 x}+\frac {50-2 x \ln \left (2\right )-2 x \ln \left (\ln \left (5\right )\right )+2 x \ln \left (2 \ln \left (5\right )-1\right )+12 x^{2}-8 \ln \left (2\right )-8 \ln \left (\ln \left (5\right )\right )+8 \ln \left (2 \ln \left (5\right )-1\right )+8 x}{10 x \left (x -\ln \left (x \right )\right )}\)	\(69\)
default	\(\frac {-25-6 x^{2}-4 \ln \left (x \right )}{5 x \left (\ln \left (x \right )-x \right )}+\frac {\ln \left (2 \ln \left (5\right )-1\right ) \left (-x -4\right )}{5 x \left (\ln \left (x \right )-x \right )}-\frac {\ln \left (2\right ) \left (-x -4\right )}{5 x \left (\ln \left (x \right )-x \right )}-\frac {\ln \left (\ln \left (5\right )\right ) \left (-x -4\right )}{5 x \left (\ln \left (x \right )-x \right )}\)	\(92\)

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]

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

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Maxima [A]
time = 0.50, size = 60, normalized size = 1.67 \begin {gather*} \frac {6 \, x^{2} - x {\left (\log \left (2\right ) - \log \left (2 \, \log \left (5\right ) - 1\right ) + \log \left (\log \left (5\right )\right )\right )} - 4 \, \log \left (2\right ) + 4 \, \log \left (x\right ) + 4 \, \log \left (2 \, \log \left (5\right ) - 1\right ) - 4 \, \log \left (\log \left (5\right )\right ) + 25}{5 \, {\left (x^{2} - x \log \left (x\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="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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).
time = 0.38, size = 84, normalized size = 2.33 \begin {gather*} \frac {6 \, x^{2} - x \log \left (\frac {2 \, \log \left (5\right )}{2 \, \log \left (5\right ) - 1}\right ) + 4 \, \log \left (\frac {2 \, x \log \left (5\right ) - x}{2 \, \log \left (5\right )}\right ) + 25}{5 \, {\left (x^{2} - x \log \left (\frac {2 \, x \log \left (5\right ) - x}{2 \, \log \left (5\right )}\right ) - x \log \left (\frac {2 \, \log \left (5\right )}{2 \, \log \left (5\right ) - 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
time = 0.09, size = 70, normalized size = 1.94 \begin {gather*} \frac {- 6 x^{2} - 4 x - x \log {\left (-1 + 2 \log {\left (5 \right )} \right )} + x \log {\left (\log {\left (5 \right )} \right )} + x \log {\left (2 \right )} - 25 - 4 \log {\left (-1 + 2 \log {\left (5 \right )} \right )} + 4 \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}}{- 5 x^{2} + 5 x \log {\left (x \right )}} - \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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Giac [A]
time = 0.42, size = 67, normalized size = 1.86 \begin {gather*} \frac {6 \, x^{2} - x \log \left (2\right ) + x \log \left (2 \, \log \left (5\right ) - 1\right ) - x \log \left (\log \left (5\right )\right ) + 4 \, x - 4 \, \log \left (2\right ) + 4 \, \log \left (2 \, \log \left (5\right ) - 1\right ) - 4 \, \log \left (\log \left (5\right )\right ) + 25}{5 \, {\left (x^{2} - x \log \left (x\right )\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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Mupad [B]
time = 1.61, size = 49, normalized size = 1.36 \begin {gather*} \frac {4\,\ln \left (\ln \left (5\right )-\frac {1}{2}\right )-4\,\ln \left (\ln \left (5\right )\right )+4\,\ln \left (x\right )+x\,\ln \left (\ln \left (5\right )-\frac {1}{2}\right )-x\,\ln \left (\ln \left (5\right )\right )+6\,x^2+25}{5\,x\,\left (x-\ln \left (x\right )\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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