3.84.17 \(\int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+(135-135 x+180 x^2-45 x^3) \log ^2(x)+(-15 x^2+5 x^3+(135 x-45 x^2) \log ^2(x)) \log (\frac {3-x}{-x^2+9 x \log ^2(x)})}{3 x^3-x^4+(-27 x^2+9 x^3) \log ^2(x)+(3 x^2-x^3+(-27 x+9 x^2) \log ^2(x)) \log (\frac {3-x}{-x^2+9 x \log ^2(x)})} \, dx\)

Optimal. Leaf size=27 \[ 5 \left (-x+\log \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )\right ) \]

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Rubi [F]  time = 5.76, 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 {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-30*x + 20*x^2 - 20*x^3 + 5*x^4 + (270 - 90*x)*Log[x] + (135 - 135*x + 180*x^2 - 45*x^3)*Log[x]^2 + (-15*
x^2 + 5*x^3 + (135*x - 45*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)])/(3*x^3 - x^4 + (-27*x^2 + 9*x^3)*
Log[x]^2 + (3*x^2 - x^3 + (-27*x + 9*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)]),x]

[Out]

-5*x + 5*Defer[Int][x/(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))]), x] - 5*Defer[Int][1/((x - 9*Log[x]^2)*(x + Log
[(-3 + x)/(x*(x - 9*Log[x]^2))])), x] + 15*Defer[Int][1/((-3 + x)*(x - 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x - 9
*Log[x]^2))])), x] + 5*Defer[Int][x/((x - 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))])), x] - 5*Defer[
Int][x^2/((x - 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))])), x] + 90*Defer[Int][Log[x]/(x*(x - 9*Log[
x]^2)*(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))])), x] - 45*Defer[Int][Log[x]^2/((-3 + x)*(x - 9*Log[x]^2)*(x + L
og[(-3 + x)/(x*(x - 9*Log[x]^2))])), x] + 45*Defer[Int][Log[x]^2/(x*(x - 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x -
9*Log[x]^2))])), x] + 45*Defer[Int][(x*Log[x]^2)/((x - 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))])),
x] + 45*Defer[Int][Log[x]^2/((-x + 9*Log[x]^2)*(x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{(3-x) x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx\\ &=\int \left (\frac {30}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}-\frac {20 x}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {20 x^2}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}-\frac {5 x^3}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {90 \log (x)}{x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {45 \left (-3+3 x-4 x^2+x^3\right ) \log ^2(x)}{(-3+x) x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )}-\frac {5 \log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )}{x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )}\right ) \, dx\\ &=-\left (5 \int \frac {x^3}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx\right )-5 \int \frac {\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )}{x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )} \, dx-20 \int \frac {x}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx+20 \int \frac {x^2}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx+30 \int \frac {1}{(-3+x) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx+45 \int \frac {\left (-3+3 x-4 x^2+x^3\right ) \log ^2(x)}{(-3+x) x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx+90 \int \frac {\log (x)}{x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 27, normalized size = 1.00 \begin {gather*} -5 x+5 \log \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30*x + 20*x^2 - 20*x^3 + 5*x^4 + (270 - 90*x)*Log[x] + (135 - 135*x + 180*x^2 - 45*x^3)*Log[x]^2 +
(-15*x^2 + 5*x^3 + (135*x - 45*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)])/(3*x^3 - x^4 + (-27*x^2 + 9
*x^3)*Log[x]^2 + (3*x^2 - x^3 + (-27*x + 9*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)]),x]

[Out]

-5*x + 5*Log[x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))]]

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fricas [A]  time = 0.59, size = 30, normalized size = 1.11 \begin {gather*} -5 \, x + 5 \, \log \left (x + \log \left (-\frac {x - 3}{9 \, x \log \relax (x)^{2} - x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2-x^2))+(-45*x^3+180*x^2-135*x+135)*l
og(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^
2-x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="fricas")

[Out]

-5*x + 5*log(x + log(-(x - 3)/(9*x*log(x)^2 - x^2)))

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giac [A]  time = 0.35, size = 28, normalized size = 1.04 \begin {gather*} -5 \, x + 5 \, \log \left (-x + \log \left (-9 \, \log \relax (x)^{2} + x\right ) - \log \left (x - 3\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2-x^2))+(-45*x^3+180*x^2-135*x+135)*l
og(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^
2-x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="giac")

[Out]

-5*x + 5*log(-x + log(-9*log(x)^2 + x) - log(x - 3) + log(x))

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maple [C]  time = 0.52, size = 332, normalized size = 12.30

method	result	size
risch	\(-5 x +5 \ln \left (\ln \left (x -9 \ln \relax (x )^{2}\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{x \left (9 \ln \relax (x )^{2}-x \right )}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{x \left (9 \ln \relax (x )^{2}-x \right )}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{9 \ln \relax (x )^{2}-x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{9 \ln \relax (x )^{2}-x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right ) \mathrm {csgn}\left (i \left (x -3\right )\right )-\pi \mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right )^{2} \mathrm {csgn}\left (i \left (x -3\right )\right )+\pi \,\mathrm {csgn}\left (\frac {i \left (x -3\right )}{9 \ln \relax (x )^{2}-x}\right ) \mathrm {csgn}\left (\frac {i \left (x -3\right )}{x \left (9 \ln \relax (x )^{2}-x \right )}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (x -3\right )}{x \left (9 \ln \relax (x )^{2}-x \right )}\right )^{3}+2 i x -2 i \ln \relax (x )+2 i \ln \left (x -3\right )\right )}{2}\right )\)	\(332\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-45*x^2+135*x)*ln(x)^2+5*x^3-15*x^2)*ln((3-x)/(9*x*ln(x)^2-x^2))+(-45*x^3+180*x^2-135*x+135)*ln(x)^2+(-
90*x+270)*ln(x)+5*x^4-20*x^3+20*x^2-30*x)/(((9*x^2-27*x)*ln(x)^2-x^3+3*x^2)*ln((3-x)/(9*x*ln(x)^2-x^2))+(9*x^3
-27*x^2)*ln(x)^2-x^4+3*x^3),x,method=_RETURNVERBOSE)

[Out]

-5*x+5*ln(ln(x-9*ln(x)^2)+1/2*I*(Pi*csgn(I/x)*csgn(I*(x-3)/(9*ln(x)^2-x))*csgn(I*(x-3)/x/(9*ln(x)^2-x))-Pi*csg
n(I/x)*csgn(I*(x-3)/x/(9*ln(x)^2-x))^2+Pi*csgn(I/(9*ln(x)^2-x))*csgn(I*(x-3)/(9*ln(x)^2-x))^2+Pi*csgn(I/(9*ln(
x)^2-x))*csgn(I*(x-3)/(9*ln(x)^2-x))*csgn(I*(x-3))-Pi*csgn(I*(x-3)/(9*ln(x)^2-x))^3-Pi*csgn(I*(x-3)/(9*ln(x)^2
-x))^2*csgn(I*(x-3))+Pi*csgn(I*(x-3)/(9*ln(x)^2-x))*csgn(I*(x-3)/x/(9*ln(x)^2-x))^2-Pi*csgn(I*(x-3)/x/(9*ln(x)
^2-x))^3+2*I*x-2*I*ln(x)+2*I*ln(x-3)))

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maxima [A]  time = 0.41, size = 28, normalized size = 1.04 \begin {gather*} -5 \, x + 5 \, \log \left (-x + \log \left (-9 \, \log \relax (x)^{2} + x\right ) - \log \left (x - 3\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2-x^2))+(-45*x^3+180*x^2-135*x+135)*l
og(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^
2-x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="maxima")

[Out]

-5*x + 5*log(-x + log(-9*log(x)^2 + x) - log(x - 3) + log(x))

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mupad [B]  time = 5.81, size = 30, normalized size = 1.11 \begin {gather*} 5\,\ln \left (x+\ln \left (-\frac {x-3}{9\,x\,{\ln \relax (x)}^2-x^2}\right )\right )-5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x + log(x)^2*(135*x - 180*x^2 + 45*x^3 - 135) + log(x)*(90*x - 270) - 20*x^2 + 20*x^3 - 5*x^4 - log(-(
x - 3)/(9*x*log(x)^2 - x^2))*(log(x)^2*(135*x - 45*x^2) - 15*x^2 + 5*x^3))/(log(x)^2*(27*x^2 - 9*x^3) + log(-(
x - 3)/(9*x*log(x)^2 - x^2))*(log(x)^2*(27*x - 9*x^2) - 3*x^2 + x^3) - 3*x^3 + x^4),x)

[Out]

5*log(x + log(-(x - 3)/(9*x*log(x)^2 - x^2))) - 5*x

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sympy [A]  time = 1.09, size = 24, normalized size = 0.89 \begin {gather*} - 5 x + 5 \log {\left (x + \log {\left (\frac {3 - x}{- x^{2} + 9 x \log {\relax (x )}^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-45*x**2+135*x)*ln(x)**2+5*x**3-15*x**2)*ln((3-x)/(9*x*ln(x)**2-x**2))+(-45*x**3+180*x**2-135*x+1
35)*ln(x)**2+(-90*x+270)*ln(x)+5*x**4-20*x**3+20*x**2-30*x)/(((9*x**2-27*x)*ln(x)**2-x**3+3*x**2)*ln((3-x)/(9*
x*ln(x)**2-x**2))+(9*x**3-27*x**2)*ln(x)**2-x**4+3*x**3),x)

[Out]

-5*x + 5*log(x + log((3 - x)/(-x**2 + 9*x*log(x)**2)))

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