3.32.74 \(\int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+(-640 x+4928 x^2-712 x^3-108 x^4-3 x^5) \log (-4 x+x^2)+(-16 x+4 x^2+(16 x^2-4 x^3) \log (-4 x+x^2)) \log (x-x^2 \log (-4 x+x^2))}{4800 x-720 x^2-108 x^3-3 x^4+(-4800 x^2+720 x^3+108 x^4+3 x^5) \log (-4 x+x^2)} \, dx\)

Optimal. Leaf size=28 \[ -8-x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{3 (20+x)} \]

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Rubi [F]  time = 5.19, 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 {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 712*x^3 - 108*x^4 - 3*x^5)*Log[-4*x + x^2
] + (-16*x + 4*x^2 + (16*x^2 - 4*x^3)*Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 - 108*x
^3 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + x^2]),x]

[Out]

-x + (2*Log[x])/15 - (2*Log[20 + x])/15 + (2*Defer[Int][1/((-4 + x)*(-1 + x*Log[(-4 + x)*x])), x])/9 + Defer[I
nt][1/(x*(-1 + x*Log[(-4 + x)*x])), x]/15 + (107*Defer[Int][1/((20 + x)*(-1 + x*Log[(-4 + x)*x])), x])/45 - (4
*Defer[Int][Log[x - x^2*Log[(-4 + x)*x]]/(20 + x)^2, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{3 (4-x) x (20+x)^2 (1-x \log ((-4+x) x))} \, dx\\ &=\frac {1}{3} \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{(4-x) x (20+x)^2 (1-x \log ((-4+x) x))} \, dx\\ &=\frac {1}{3} \int \left (-\frac {5184}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {320}{(-4+x) x (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {860 x}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {116 x^2}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {3 x^3}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))}-\frac {\left (-8+60 x+3 x^2\right ) \log ((-4+x) x)}{(20+x) (-1+x \log ((-4+x) x))}-\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\left (-8+60 x+3 x^2\right ) \log ((-4+x) x)}{(20+x) (-1+x \log ((-4+x) x))} \, dx\right )-\frac {4}{3} \int \frac {\log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2} \, dx+\frac {116}{3} \int \frac {x^2}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {320}{3} \int \frac {1}{(-4+x) x (20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {860}{3} \int \frac {x}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))} \, dx-1728 \int \frac {1}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\int \frac {x^3}{(-4+x) (20+x)^2 (-1+x \log ((-4+x) x))} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {-8+60 x+3 x^2}{x (20+x)}+\frac {-8+60 x+3 x^2}{x (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx\right )-\frac {4}{3} \int \frac {\log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2} \, dx+\frac {116}{3} \int \left (\frac {1}{36 (-4+x) (-1+x \log ((-4+x) x))}-\frac {50}{3 (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {35}{36 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx+\frac {320}{3} \int \left (\frac {1}{2304 (-4+x) (-1+x \log ((-4+x) x))}-\frac {1}{1600 x (-1+x \log ((-4+x) x))}+\frac {1}{480 (20+x)^2 (-1+x \log ((-4+x) x))}+\frac {11}{57600 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx+\frac {860}{3} \int \left (\frac {1}{144 (-4+x) (-1+x \log ((-4+x) x))}+\frac {5}{6 (20+x)^2 (-1+x \log ((-4+x) x))}-\frac {1}{144 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx-1728 \int \left (\frac {1}{576 (-4+x) (-1+x \log ((-4+x) x))}-\frac {1}{24 (20+x)^2 (-1+x \log ((-4+x) x))}-\frac {1}{576 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx+\int \left (\frac {1}{-1+x \log ((-4+x) x)}+\frac {1}{9 (-4+x) (-1+x \log ((-4+x) x))}+\frac {1000}{3 (20+x)^2 (-1+x \log ((-4+x) x))}-\frac {325}{9 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx\\ &=\frac {11}{540} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {5}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {1}{15} \int \frac {1}{x (-1+x \log ((-4+x) x))} \, dx+\frac {1}{9} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+\frac {2}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx-\frac {1}{3} \int \frac {-8+60 x+3 x^2}{x (20+x)} \, dx-\frac {1}{3} \int \frac {-8+60 x+3 x^2}{x (20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {29}{27} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {4}{3} \int \frac {\log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2} \, dx+\frac {215}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {215}{108} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-3 \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+3 \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-\frac {325}{9} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {1015}{27} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+72 \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {2150}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {1000}{3} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx-\frac {5800}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\int \frac {1}{-1+x \log ((-4+x) x)} \, dx\\ &=\frac {11}{540} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {5}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {1}{15} \int \frac {1}{x (-1+x \log ((-4+x) x))} \, dx+\frac {1}{9} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+\frac {2}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx-\frac {1}{3} \int \left (3-\frac {2}{5 x}+\frac {2}{5 (20+x)}\right ) \, dx-\frac {1}{3} \int \left (\frac {3}{-1+x \log ((-4+x) x)}-\frac {2}{5 x (-1+x \log ((-4+x) x))}+\frac {2}{5 (20+x) (-1+x \log ((-4+x) x))}\right ) \, dx+\frac {29}{27} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {4}{3} \int \frac {\log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2} \, dx+\frac {215}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {215}{108} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-3 \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+3 \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-\frac {325}{9} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {1015}{27} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+72 \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {2150}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {1000}{3} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx-\frac {5800}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\int \frac {1}{-1+x \log ((-4+x) x)} \, dx\\ &=-x+\frac {2 \log (x)}{15}-\frac {2}{15} \log (20+x)+\frac {11}{540} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {5}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {1}{15} \int \frac {1}{x (-1+x \log ((-4+x) x))} \, dx+\frac {1}{9} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+\frac {2}{15} \int \frac {1}{x (-1+x \log ((-4+x) x))} \, dx-\frac {2}{15} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {2}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {29}{27} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {4}{3} \int \frac {\log \left (x-x^2 \log ((-4+x) x)\right )}{(20+x)^2} \, dx+\frac {215}{108} \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx-\frac {215}{108} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-3 \int \frac {1}{(-4+x) (-1+x \log ((-4+x) x))} \, dx+3 \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx-\frac {325}{9} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+\frac {1015}{27} \int \frac {1}{(20+x) (-1+x \log ((-4+x) x))} \, dx+72 \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {2150}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx+\frac {1000}{3} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx-\frac {5800}{9} \int \frac {1}{(20+x)^2 (-1+x \log ((-4+x) x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 29, normalized size = 1.04 \begin {gather*} \frac {1}{3} \left (-3 x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{20+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 712*x^3 - 108*x^4 - 3*x^5)*Log[-4*x
+ x^2] + (-16*x + 4*x^2 + (16*x^2 - 4*x^3)*Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 -
108*x^3 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + x^2]),x]

[Out]

(-3*x + (4*Log[x - x^2*Log[(-4 + x)*x]])/(20 + x))/3

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fricas [A]  time = 0.62, size = 34, normalized size = 1.21 \begin {gather*} -\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x^{2} \log \left (x^{2} - 4 \, x\right ) + x\right )}{3 \, {\left (x + 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x)+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2
-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-10
8*x^3-720*x^2+4800*x),x, algorithm="fricas")

[Out]

-1/3*(3*x^2 + 60*x - 4*log(-x^2*log(x^2 - 4*x) + x))/(x + 20)

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giac [A]  time = 0.42, size = 34, normalized size = 1.21 \begin {gather*} -x + \frac {4 \, \log \left (-x \log \left (x^{2} - 4 \, x\right ) + 1\right )}{3 \, {\left (x + 20\right )}} + \frac {4 \, \log \relax (x)}{3 \, {\left (x + 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x)+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2
-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-10
8*x^3-720*x^2+4800*x),x, algorithm="giac")

[Out]

-x + 4/3*log(-x*log(x^2 - 4*x) + 1)/(x + 20) + 4/3*log(x)/(x + 20)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-4 x^{3}+16 x^{2}\right ) \ln \left (x^{2}-4 x \right )+4 x^{2}-16 x \right ) \ln \left (-x^{2} \ln \left (x^{2}-4 x \right )+x \right )+\left (-3 x^{5}-108 x^{4}-712 x^{3}+4928 x^{2}-640 x \right ) \ln \left (x^{2}-4 x \right )+3 x^{4}+116 x^{3}+860 x^{2}-5184 x +320}{\left (3 x^{5}+108 x^{4}+720 x^{3}-4800 x^{2}\right ) \ln \left (x^{2}-4 x \right )-3 x^{4}-108 x^{3}-720 x^{2}+4800 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^3+16*x^2)*ln(x^2-4*x)+4*x^2-16*x)*ln(-x^2*ln(x^2-4*x)+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*l
n(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*ln(x^2-4*x)-3*x^4-108*x^3-720*x
^2+4800*x),x)

[Out]

int((((-4*x^3+16*x^2)*ln(x^2-4*x)+4*x^2-16*x)*ln(-x^2*ln(x^2-4*x)+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*l
n(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*ln(x^2-4*x)-3*x^4-108*x^3-720*x
^2+4800*x),x)

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maxima [A]  time = 0.51, size = 37, normalized size = 1.32 \begin {gather*} -\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x \log \left (x - 4\right ) - x \log \relax (x) + 1\right ) - 4 \, \log \relax (x)}{3 \, {\left (x + 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x)+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2
-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-10
8*x^3-720*x^2+4800*x),x, algorithm="maxima")

[Out]

-1/3*(3*x^2 + 60*x - 4*log(-x*log(x - 4) - x*log(x) + 1) - 4*log(x))/(x + 20)

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mupad [B]  time = 2.21, size = 29, normalized size = 1.04 \begin {gather*} \frac {4\,\ln \left (x-x^2\,\ln \left (x^2-4\,x\right )\right )}{3\,\left (x+20\right )}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - x^2*log(x^2 - 4*x))*(log(x^2 - 4*x)*(16*x^2 - 4*x^3) - 16*x + 4*x^2) - 5184*x + 860*x^2 + 116*x^
3 + 3*x^4 - log(x^2 - 4*x)*(640*x - 4928*x^2 + 712*x^3 + 108*x^4 + 3*x^5) + 320)/(720*x^2 - log(x^2 - 4*x)*(72
0*x^3 - 4800*x^2 + 108*x^4 + 3*x^5) - 4800*x + 108*x^3 + 3*x^4),x)

[Out]

(4*log(x - x^2*log(x^2 - 4*x)))/(3*(x + 20)) - x

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sympy [A]  time = 0.65, size = 22, normalized size = 0.79 \begin {gather*} - x + \frac {4 \log {\left (- x^{2} \log {\left (x^{2} - 4 x \right )} + x \right )}}{3 x + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**3+16*x**2)*ln(x**2-4*x)+4*x**2-16*x)*ln(-x**2*ln(x**2-4*x)+x)+(-3*x**5-108*x**4-712*x**3+49
28*x**2-640*x)*ln(x**2-4*x)+3*x**4+116*x**3+860*x**2-5184*x+320)/((3*x**5+108*x**4+720*x**3-4800*x**2)*ln(x**2
-4*x)-3*x**4-108*x**3-720*x**2+4800*x),x)

[Out]

-x + 4*log(-x**2*log(x**2 - 4*x) + x)/(3*x + 60)

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