3.67.91 \(\int \frac {2-2 x-45 x \log (4)}{-2 x^2+(-12 x-45 x^2) \log (4)+2 x \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \log \left (4-x+16 \left (x-\frac {-x+\log (x)}{24 \log (4)}\right )\right ) \]

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Rubi [F]  time = 0.21, 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 {2-2 x-45 x \log (4)}{-2 x^2+\left (-12 x-45 x^2\right ) \log (4)+2 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - 2*x - 45*x*Log[4])/(-2*x^2 + (-12*x - 45*x^2)*Log[4] + 2*x*Log[x]),x]

[Out]

(2 + 45*Log[4])*Defer[Int][(12*Log[4] + 2*x*(1 + (45*Log[4])/2) - 2*Log[x])^(-1), x] + 2*Defer[Int][1/(x*(-12*
Log[4] - 2*x*(1 + (45*Log[4])/2) + 2*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+x (-2-45 \log (4))}{-2 x^2+\left (-12 x-45 x^2\right ) \log (4)+2 x \log (x)} \, dx\\ &=\int \left (\frac {2+45 \log (4)}{12 \log (4)+2 x \left (1+\frac {45 \log (4)}{2}\right )-2 \log (x)}+\frac {2}{x \left (-12 \log (4)-2 x \left (1+\frac {45 \log (4)}{2}\right )+2 \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {1}{x \left (-12 \log (4)-2 x \left (1+\frac {45 \log (4)}{2}\right )+2 \log (x)\right )} \, dx+(2+45 \log (4)) \int \frac {1}{12 \log (4)+2 x \left (1+\frac {45 \log (4)}{2}\right )-2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 18, normalized size = 0.75 \begin {gather*} \log (2 x+12 \log (4)+45 x \log (4)-2 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 2*x - 45*x*Log[4])/(-2*x^2 + (-12*x - 45*x^2)*Log[4] + 2*x*Log[x]),x]

[Out]

Log[2*x + 12*Log[4] + 45*x*Log[4] - 2*Log[x]]

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fricas [A]  time = 0.53, size = 16, normalized size = 0.67 \begin {gather*} \log \left (-3 \, {\left (15 \, x + 4\right )} \log \relax (2) - x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x*log(2)-2*x+2)/(2*x*log(x)+2*(-45*x^2-12*x)*log(2)-2*x^2),x, algorithm="fricas")

[Out]

log(-3*(15*x + 4)*log(2) - x + log(x))

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giac [A]  time = 0.18, size = 16, normalized size = 0.67 \begin {gather*} \log \left (-45 \, x \log \relax (2) - x - 12 \, \log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x*log(2)-2*x+2)/(2*x*log(x)+2*(-45*x^2-12*x)*log(2)-2*x^2),x, algorithm="giac")

[Out]

log(-45*x*log(2) - x - 12*log(2) + log(x))

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maple [A]  time = 0.04, size = 17, normalized size = 0.71

method	result	size
norman	\(\ln \left (45 x \ln \relax (2)+12 \ln \relax (2)+x -\ln \relax (x )\right )\)	\(17\)
risch	\(\ln \left (-45 x \ln \relax (2)+\ln \relax (x )-12 \ln \relax (2)-x \right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-90*x*ln(2)-2*x+2)/(2*x*ln(x)+2*(-45*x^2-12*x)*ln(2)-2*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(45*x*ln(2)+12*ln(2)+x-ln(x))

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maxima [A]  time = 0.49, size = 17, normalized size = 0.71 \begin {gather*} \log \left (-x {\left (45 \, \log \relax (2) + 1\right )} - 12 \, \log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x*log(2)-2*x+2)/(2*x*log(x)+2*(-45*x^2-12*x)*log(2)-2*x^2),x, algorithm="maxima")

[Out]

log(-x*(45*log(2) + 1) - 12*log(2) + log(x))

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mupad [B]  time = 4.73, size = 16, normalized size = 0.67 \begin {gather*} \ln \left (x+12\,\ln \relax (2)-\ln \relax (x)+45\,x\,\ln \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 90*x*log(2) - 2)/(2*log(2)*(12*x + 45*x^2) - 2*x*log(x) + 2*x^2),x)

[Out]

log(x + 12*log(2) - log(x) + 45*x*log(2))

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sympy [A]  time = 0.14, size = 17, normalized size = 0.71 \begin {gather*} \log {\left (- 45 x \log {\relax (2 )} - x + \log {\relax (x )} - 12 \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x*ln(2)-2*x+2)/(2*x*ln(x)+2*(-45*x**2-12*x)*ln(2)-2*x**2),x)

[Out]

log(-45*x*log(2) - x + log(x) - 12*log(2))

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