3.60.50 \(\int \frac {50-130 x-25 \log (x^2)-100 \log ^2(x^2)}{(25-130 x+169 x^2) \log ^2(x^2)} \, dx\)

Optimal. Leaf size=32 \[ \frac {4 x+\frac {x}{\log \left (x^2\right )}}{x \left (2+\frac {-5+3 x}{5 x}\right )} \]

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Rubi [F]  time = 0.26, 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 {50-130 x-25 \log \left (x^2\right )-100 \log ^2\left (x^2\right )}{\left (25-130 x+169 x^2\right ) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50 - 130*x - 25*Log[x^2] - 100*Log[x^2]^2)/((25 - 130*x + 169*x^2)*Log[x^2]^2),x]

[Out]

-100/(13*(5 - 13*x)) - 10*Defer[Int][1/((-5 + 13*x)*Log[x^2]^2), x] - 25*Defer[Int][1/((-5 + 13*x)^2*Log[x^2])
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50-130 x-25 \log \left (x^2\right )-100 \log ^2\left (x^2\right )}{(-5+13 x)^2 \log ^2\left (x^2\right )} \, dx\\ &=\int \frac {5 \left (10-26 x-5 \log \left (x^2\right )-20 \log ^2\left (x^2\right )\right )}{(5-13 x)^2 \log ^2\left (x^2\right )} \, dx\\ &=5 \int \frac {10-26 x-5 \log \left (x^2\right )-20 \log ^2\left (x^2\right )}{(5-13 x)^2 \log ^2\left (x^2\right )} \, dx\\ &=5 \int \left (-\frac {20}{(-5+13 x)^2}-\frac {2}{(-5+13 x) \log ^2\left (x^2\right )}-\frac {5}{(-5+13 x)^2 \log \left (x^2\right )}\right ) \, dx\\ &=-\frac {100}{13 (5-13 x)}-10 \int \frac {1}{(-5+13 x) \log ^2\left (x^2\right )} \, dx-25 \int \frac {1}{(-5+13 x)^2 \log \left (x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 30, normalized size = 0.94 \begin {gather*} -5 \left (-\frac {20}{13 (-5+13 x)}-\frac {x}{(-5+13 x) \log \left (x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 - 130*x - 25*Log[x^2] - 100*Log[x^2]^2)/((25 - 130*x + 169*x^2)*Log[x^2]^2),x]

[Out]

-5*(-20/(13*(-5 + 13*x)) - x/((-5 + 13*x)*Log[x^2]))

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fricas [A]  time = 0.62, size = 25, normalized size = 0.78 \begin {gather*} \frac {5 \, {\left (13 \, x + 20 \, \log \left (x^{2}\right )\right )}}{13 \, {\left (13 \, x - 5\right )} \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x^2)^2-25*log(x^2)-130*x+50)/(169*x^2-130*x+25)/log(x^2)^2,x, algorithm="fricas")

[Out]

5/13*(13*x + 20*log(x^2))/((13*x - 5)*log(x^2))

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giac [A]  time = 0.16, size = 29, normalized size = 0.91 \begin {gather*} \frac {5 \, x}{13 \, x \log \left (x^{2}\right ) - 5 \, \log \left (x^{2}\right )} + \frac {100}{13 \, {\left (13 \, x - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x^2)^2-25*log(x^2)-130*x+50)/(169*x^2-130*x+25)/log(x^2)^2,x, algorithm="giac")

[Out]

5*x/(13*x*log(x^2) - 5*log(x^2)) + 100/13/(13*x - 5)

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maple [A]  time = 0.57, size = 26, normalized size = 0.81

method	result	size
norman	\(\frac {20 x \ln \left (x^{2}\right )+5 x}{\left (13 x -5\right ) \ln \left (x^{2}\right )}\)	\(26\)
risch	\(\frac {100}{13 \left (13 x -5\right )}+\frac {5 x}{\left (13 x -5\right ) \ln \left (x^{2}\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-100*ln(x^2)^2-25*ln(x^2)-130*x+50)/(169*x^2-130*x+25)/ln(x^2)^2,x,method=_RETURNVERBOSE)

[Out]

(20*x*ln(x^2)+5*x)/(13*x-5)/ln(x^2)

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maxima [A]  time = 0.40, size = 21, normalized size = 0.66 \begin {gather*} \frac {5 \, {\left (13 \, x + 40 \, \log \relax (x)\right )}}{26 \, {\left (13 \, x - 5\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(x^2)^2-25*log(x^2)-130*x+50)/(169*x^2-130*x+25)/log(x^2)^2,x, algorithm="maxima")

[Out]

5/26*(13*x + 40*log(x))/((13*x - 5)*log(x))

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mupad [B]  time = 4.25, size = 24, normalized size = 0.75 \begin {gather*} \frac {5\,x\,\left (4\,\ln \left (x^2\right )+1\right )}{\ln \left (x^2\right )\,\left (13\,x-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(130*x + 25*log(x^2) + 100*log(x^2)^2 - 50)/(log(x^2)^2*(169*x^2 - 130*x + 25)),x)

[Out]

(5*x*(4*log(x^2) + 1))/(log(x^2)*(13*x - 5))

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sympy [A]  time = 0.14, size = 19, normalized size = 0.59 \begin {gather*} \frac {5 x}{\left (13 x - 5\right ) \log {\left (x^{2} \right )}} + \frac {100}{169 x - 65} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*ln(x**2)**2-25*ln(x**2)-130*x+50)/(169*x**2-130*x+25)/ln(x**2)**2,x)

[Out]

5*x/((13*x - 5)*log(x**2)) + 100/(169*x - 65)

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