3.69.42 \(\int \frac {-41 x+x^2-120 x^3-72 x^5+(-19 x^2-24 x^4) \log (\frac {36}{x})-2 x^3 \log ^2(\frac {36}{x})+(-18 x-24 x^3-4 x^2 \log (\frac {36}{x})) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+(10 x+12 x^3) \log (\frac {36}{x})+x^2 \log ^2(\frac {36}{x})+(10+12 x^2+2 x \log (\frac {36}{x})) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ x \left (-x+\frac {x}{5+x \left (6 x+\log \left (\frac {36}{x}\right )\right )+\log (x)}\right ) \]

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Rubi [F]  time = 1.49, 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 {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-41*x + x^2 - 120*x^3 - 72*x^5 + (-19*x^2 - 24*x^4)*Log[36/x] - 2*x^3*Log[36/x]^2 + (-18*x - 24*x^3 - 4*x
^2*Log[36/x])*Log[x] - 2*x*Log[x]^2)/(25 + 60*x^2 + 36*x^4 + (10*x + 12*x^3)*Log[36/x] + x^2*Log[36/x]^2 + (10
+ 12*x^2 + 2*x*Log[36/x])*Log[x] + Log[x]^2),x]

[Out]

-x^2 - Defer[Int][x/(5 + 6*x^2 + x*Log[36/x] + Log[x])^2, x] + Defer[Int][x^2/(5 + 6*x^2 + x*Log[36/x] + Log[x
])^2, x] - 12*Defer[Int][x^3/(5 + 6*x^2 + x*Log[36/x] + Log[x])^2, x] - Defer[Int][(x^2*Log[36/x])/(5 + 6*x^2
+ x*Log[36/x] + Log[x])^2, x] + 2*Defer[Int][x/(5 + 6*x^2 + x*Log[36/x] + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-41+x-120 x^2-72 x^4-2 x^2 \log ^2\left (\frac {36}{x}\right )-6 \left (3+4 x^2\right ) \log (x)-2 \log ^2(x)-x \log \left (\frac {36}{x}\right ) \left (19+24 x^2+4 \log (x)\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx\\ &=\int \left (-2 x-\frac {x \left (1-x+12 x^2+x \log \left (\frac {36}{x}\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {2 x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)}\right ) \, dx\\ &=-x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-\int \frac {x \left (1-x+12 x^2+x \log \left (\frac {36}{x}\right )\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx\\ &=-x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-\int \left (\frac {x}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}-\frac {x^2}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {12 x^3}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}+\frac {x^2 \log \left (\frac {36}{x}\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2}\right ) \, dx\\ &=-x^2+2 \int \frac {x}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \, dx-12 \int \frac {x^3}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx-\int \frac {x}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx+\int \frac {x^2}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx-\int \frac {x^2 \log \left (\frac {36}{x}\right )}{\left (5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 29, normalized size = 1.12 \begin {gather*} -x^2+\frac {x^2}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-41*x + x^2 - 120*x^3 - 72*x^5 + (-19*x^2 - 24*x^4)*Log[36/x] - 2*x^3*Log[36/x]^2 + (-18*x - 24*x^3
- 4*x^2*Log[36/x])*Log[x] - 2*x*Log[x]^2)/(25 + 60*x^2 + 36*x^4 + (10*x + 12*x^3)*Log[36/x] + x^2*Log[36/x]^2
+ (10 + 12*x^2 + 2*x*Log[36/x])*Log[x] + Log[x]^2),x]

[Out]

-x^2 + x^2/(5 + 6*x^2 + x*Log[36/x] + Log[x])

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fricas [B]  time = 0.49, size = 59, normalized size = 2.27 \begin {gather*} -\frac {6 \, x^{4} + 2 \, x^{2} \log \relax (6) + 4 \, x^{2} + {\left (x^{3} - x^{2}\right )} \log \left (\frac {36}{x}\right )}{6 \, x^{2} + {\left (x - 1\right )} \log \left (\frac {36}{x}\right ) + 2 \, \log \relax (6) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(36/x)^2+(-24*x^4-19*x^2)*log(36/x)-72
*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x*log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x
^4+60*x^2+25),x, algorithm="fricas")

[Out]

-(6*x^4 + 2*x^2*log(6) + 4*x^2 + (x^3 - x^2)*log(36/x))/(6*x^2 + (x - 1)*log(36/x) + 2*log(6) + 5)

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giac [A]  time = 0.21, size = 31, normalized size = 1.19 \begin {gather*} -x^{2} + \frac {x^{2}}{6 \, x^{2} + 2 \, x \log \relax (6) - x \log \relax (x) + \log \relax (x) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(36/x)^2+(-24*x^4-19*x^2)*log(36/x)-72
*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x*log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x
^4+60*x^2+25),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 40, normalized size = 1.54

method	result	size
risch	\(-x^{2}+\frac {2 x^{2}}{10+4 x \ln \relax (2)+4 x \ln \relax (3)+12 x^{2}-2 x \ln \relax (x )+2 \ln \relax (x )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)^2+(-4*x^2*ln(36/x)-24*x^3-18*x)*ln(x)-2*x^3*ln(36/x)^2+(-24*x^4-19*x^2)*ln(36/x)-72*x^5-120*x^
3+x^2-41*x)/(ln(x)^2+(2*x*ln(36/x)+12*x^2+10)*ln(x)+x^2*ln(36/x)^2+(12*x^3+10*x)*ln(36/x)+36*x^4+60*x^2+25),x,
method=_RETURNVERBOSE)

[Out]

-x^2+2*x^2/(10+4*x*ln(2)+4*x*ln(3)+12*x^2-2*x*ln(x)+2*ln(x))

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maxima [B]  time = 0.48, size = 60, normalized size = 2.31 \begin {gather*} -\frac {6 \, x^{4} + 2 \, x^{3} {\left (\log \relax (3) + \log \relax (2)\right )} + 4 \, x^{2} - {\left (x^{3} - x^{2}\right )} \log \relax (x)}{6 \, x^{2} + 2 \, x {\left (\log \relax (3) + \log \relax (2)\right )} - {\left (x - 1\right )} \log \relax (x) + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(36/x)^2+(-24*x^4-19*x^2)*log(36/x)-72
*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x*log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x
^4+60*x^2+25),x, algorithm="maxima")

[Out]

-(6*x^4 + 2*x^3*(log(3) + log(2)) + 4*x^2 - (x^3 - x^2)*log(x))/(6*x^2 + 2*x*(log(3) + log(2)) - (x - 1)*log(x
) + 5)

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mupad [B]  time = 4.63, size = 41, normalized size = 1.58 \begin {gather*} -\frac {x^2\,\left (\ln \relax (x)+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+4\right )}{\ln \relax (x)+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(41*x + 2*x*log(x)^2 + 2*x^3*log(36/x)^2 + log(x)*(18*x + 24*x^3 + 4*x^2*log(36/x)) + log(36/x)*(19*x^2 +
24*x^4) - x^2 + 120*x^3 + 72*x^5)/(x^2*log(36/x)^2 + log(x)^2 + log(x)*(2*x*log(36/x) + 12*x^2 + 10) + log(36
/x)*(10*x + 12*x^3) + 60*x^2 + 36*x^4 + 25),x)

[Out]

-(x^2*(log(x) + x*log(36/x) + 6*x^2 + 4))/(log(x) + x*log(36/x) + 6*x^2 + 5)

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sympy [A]  time = 0.60, size = 27, normalized size = 1.04 \begin {gather*} - x^{2} - \frac {x^{2}}{- 6 x^{2} - 2 x \log {\relax (6 )} + \left (x - 1\right ) \log {\relax (x )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x)**2+(-4*x**2*ln(36/x)-24*x**3-18*x)*ln(x)-2*x**3*ln(36/x)**2+(-24*x**4-19*x**2)*ln(36/x)-
72*x**5-120*x**3+x**2-41*x)/(ln(x)**2+(2*x*ln(36/x)+12*x**2+10)*ln(x)+x**2*ln(36/x)**2+(12*x**3+10*x)*ln(36/x)
+36*x**4+60*x**2+25),x)

[Out]

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

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