3.17.86 \(\int \frac {-12+84 x-40 x^2+5 x^3+(-160 x+80 x^2-10 x^3) \log (x)+(4-x+(80 x-40 x^2+5 x^3) \log (x)) \log (\frac {-1+(-20 x+5 x^2) \log (x)}{-12 x^2+3 x^3})}{4-x+(80 x-40 x^2+5 x^3) \log (x)} \, dx\)

Optimal. Leaf size=34 \[ e^5-x+x \log \left (\frac {5 \left (\frac {1}{5 (4-x) x}+\log (x)\right )}{3 x}\right ) \]

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Rubi [A]  time = 2.61, antiderivative size = 33, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 4, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6741, 6688, 6742, 2549} \begin {gather*} x \log \left (\frac {5 (4-x) x \log (x)+1}{3 (4-x) x^2}\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + 84*x - 40*x^2 + 5*x^3 + (-160*x + 80*x^2 - 10*x^3)*Log[x] + (4 - x + (80*x - 40*x^2 + 5*x^3)*Log[x]
)*Log[(-1 + (-20*x + 5*x^2)*Log[x])/(-12*x^2 + 3*x^3)])/(4 - x + (80*x - 40*x^2 + 5*x^3)*Log[x]),x]

[Out]

-x + x*Log[(1 + 5*(4 - x)*x*Log[x])/(3*(4 - x)*x^2)]

Rule 2549

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x] /; ProductQ[
u]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12+84 x-40 x^2+5 x^3+\left (-160 x+80 x^2-10 x^3\right ) \log (x)+\left (4-x+\left (80 x-40 x^2+5 x^3\right ) \log (x)\right ) \log \left (\frac {-1+\left (-20 x+5 x^2\right ) \log (x)}{-12 x^2+3 x^3}\right )}{(4-x) \left (1+20 x \log (x)-5 x^2 \log (x)\right )} \, dx\\ &=\int \frac {-12+84 x-40 x^2+5 x^3-10 (-4+x)^2 x \log (x)+(-4+x) (-1+5 (-4+x) x \log (x)) \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right )}{(4-x) (1-5 (-4+x) x \log (x))} \, dx\\ &=\int \left (-\frac {12}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {84 x}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {40 x^2}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {10 (-4+x) x \log (x)}{-1-20 x \log (x)+5 x^2 \log (x)}+\log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right )\right ) \, dx\\ &=5 \int \frac {x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-10 \int \frac {(-4+x) x \log (x)}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-40 \int \frac {x^2}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+84 \int \frac {x}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+\int \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right ) \, dx\\ &=x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )+5 \int \left (\frac {16}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {64}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {4 x}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx-10 \int \left (\frac {1}{5}+\frac {1}{5 \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-40 \int \left (\frac {4}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {16}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}+\frac {x}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx+84 \int \left (\frac {1}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {4}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx-\int \frac {-8+83 x-40 x^2+5 x^3-5 (-4+x)^2 x \log (x)}{(-4+x) (-1+5 (-4+x) x \log (x))} \, dx\\ &=-2 x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \left (-1+\frac {-4+82 x-40 x^2+5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx\\ &=-x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \frac {-4+82 x-40 x^2+5 x^3}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx\\ &=-x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \left (\frac {2}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {4}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )}-\frac {20 x}{-1-20 x \log (x)+5 x^2 \log (x)}+\frac {5 x^2}{-1-20 x \log (x)+5 x^2 \log (x)}\right ) \, dx\\ &=-x+x \log \left (\frac {1+5 (4-x) x \log (x)}{3 (4-x) x^2}\right )-2 \left (2 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx\right )-4 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-12 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \left (20 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx\right )-40 \int \frac {x}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+80 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+84 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx-160 \int \frac {1}{-1-20 x \log (x)+5 x^2 \log (x)} \, dx+320 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx+336 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx-640 \int \frac {1}{(-4+x) \left (-1-20 x \log (x)+5 x^2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 29, normalized size = 0.85 \begin {gather*} -x+x \log \left (\frac {-1+5 (-4+x) x \log (x)}{3 (-4+x) x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + 84*x - 40*x^2 + 5*x^3 + (-160*x + 80*x^2 - 10*x^3)*Log[x] + (4 - x + (80*x - 40*x^2 + 5*x^3)*
Log[x])*Log[(-1 + (-20*x + 5*x^2)*Log[x])/(-12*x^2 + 3*x^3)])/(4 - x + (80*x - 40*x^2 + 5*x^3)*Log[x]),x]

[Out]

-x + x*Log[(-1 + 5*(-4 + x)*x*Log[x])/(3*(-4 + x)*x^2)]

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fricas [A]  time = 0.83, size = 33, normalized size = 0.97 \begin {gather*} x \log \left (\frac {5 \, {\left (x^{2} - 4 \, x\right )} \log \relax (x) - 1}{3 \, {\left (x^{3} - 4 \, x^{2}\right )}}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-40*x^2+80*x)*log(x)-x+4)*log(((5*x^2-20*x)*log(x)-1)/(3*x^3-12*x^2))+(-10*x^3+80*x^2-160*x)
*log(x)+5*x^3-40*x^2+84*x-12)/((5*x^3-40*x^2+80*x)*log(x)-x+4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.46, size = 35, normalized size = 1.03 \begin {gather*} x \log \left (5 \, x^{2} \log \relax (x) - 20 \, x \log \relax (x) - 1\right ) - x \log \left (3 \, x - 12\right ) - 2 \, x \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-40*x^2+80*x)*log(x)-x+4)*log(((5*x^2-20*x)*log(x)-1)/(3*x^3-12*x^2))+(-10*x^3+80*x^2-160*x)
*log(x)+5*x^3-40*x^2+84*x-12)/((5*x^3-40*x^2+80*x)*log(x)-x+4),x, algorithm="giac")

[Out]

x*log(5*x^2*log(x) - 20*x*log(x) - 1) - x*log(3*x - 12) - 2*x*log(x) - x

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maple [C]  time = 0.18, size = 444, normalized size = 13.06

method	result	size
risch	\(x \ln \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )-x \ln \left (x -4\right )-2 x \ln \relax (x )+\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -4}\right ) \mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )^{2}}{2}-i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x^{2} \left (x -4\right )}\right )}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-4 x \ln \relax (x )-\frac {1}{5}\right )}{x -4}\right )^{3}}{2}+x \ln \relax (5)-x \ln \relax (3)-x\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^3-40*x^2+80*x)*ln(x)-x+4)*ln(((5*x^2-20*x)*ln(x)-1)/(3*x^3-12*x^2))+(-10*x^3+80*x^2-160*x)*ln(x)+5*
x^3-40*x^2+84*x-12)/((5*x^3-40*x^2+80*x)*ln(x)-x+4),x,method=_RETURNVERBOSE)

[Out]

x*ln(x^2*ln(x)-4*x*ln(x)-1/5)-x*ln(x-4)-2*x*ln(x)+1/2*I*Pi*x*csgn(I*x^2)^3+1/2*I*Pi*x*csgn(I/(x-4))*csgn(I/(x-
4)*(x^2*ln(x)-4*x*ln(x)-1/5))^2-1/2*I*Pi*x*csgn(I/(x-4))*csgn(I*(x^2*ln(x)-4*x*ln(x)-1/5))*csgn(I/(x-4)*(x^2*l
n(x)-4*x*ln(x)-1/5))+1/2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+1/2*I*Pi*x*csgn(I/x^2)*csgn(I/x^2/(x-4)*(x^2*ln(x)-4*x
*ln(x)-1/5))^2+1/2*I*Pi*x*csgn(I*(x^2*ln(x)-4*x*ln(x)-1/5))*csgn(I/(x-4)*(x^2*ln(x)-4*x*ln(x)-1/5))^2+1/2*I*Pi
*x*csgn(I/(x-4)*(x^2*ln(x)-4*x*ln(x)-1/5))*csgn(I/x^2/(x-4)*(x^2*ln(x)-4*x*ln(x)-1/5))^2-I*Pi*x*csgn(I*x)*csgn
(I*x^2)^2-1/2*I*Pi*x*csgn(I/x^2/(x-4)*(x^2*ln(x)-4*x*ln(x)-1/5))^3-1/2*I*Pi*x*csgn(I/x^2)*csgn(I/(x-4)*(x^2*ln
(x)-4*x*ln(x)-1/5))*csgn(I/x^2/(x-4)*(x^2*ln(x)-4*x*ln(x)-1/5))-1/2*I*Pi*x*csgn(I/(x-4)*(x^2*ln(x)-4*x*ln(x)-1
/5))^3+x*ln(5)-x*ln(3)-x

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maxima [A]  time = 0.66, size = 36, normalized size = 1.06 \begin {gather*} -x {\left (\log \relax (3) + 1\right )} + x \log \left (5 \, {\left (x^{2} - 4 \, x\right )} \log \relax (x) - 1\right ) - x \log \left (x - 4\right ) - 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3-40*x^2+80*x)*log(x)-x+4)*log(((5*x^2-20*x)*log(x)-1)/(3*x^3-12*x^2))+(-10*x^3+80*x^2-160*x)
*log(x)+5*x^3-40*x^2+84*x-12)/((5*x^3-40*x^2+80*x)*log(x)-x+4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.30, size = 33, normalized size = 0.97 \begin {gather*} x\,\left (\ln \left (\frac {\ln \relax (x)\,\left (20\,x-5\,x^2\right )+1}{12\,x^2-3\,x^3}\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((84*x + log((log(x)*(20*x - 5*x^2) + 1)/(12*x^2 - 3*x^3))*(log(x)*(80*x - 40*x^2 + 5*x^3) - x + 4) - 40*x^
2 + 5*x^3 - log(x)*(160*x - 80*x^2 + 10*x^3) - 12)/(log(x)*(80*x - 40*x^2 + 5*x^3) - x + 4),x)

[Out]

x*(log((log(x)*(20*x - 5*x^2) + 1)/(12*x^2 - 3*x^3)) - 1)

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sympy [B]  time = 1.32, size = 56, normalized size = 1.65 \begin {gather*} - x + \left (x - \frac {2}{3}\right ) \log {\left (\frac {\left (5 x^{2} - 20 x\right ) \log {\relax (x )} - 1}{3 x^{3} - 12 x^{2}} \right )} - \frac {2 \log {\relax (x )}}{3} + \frac {2 \log {\left (\log {\relax (x )} - \frac {1}{5 x^{2} - 20 x} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**3-40*x**2+80*x)*ln(x)-x+4)*ln(((5*x**2-20*x)*ln(x)-1)/(3*x**3-12*x**2))+(-10*x**3+80*x**2-16
0*x)*ln(x)+5*x**3-40*x**2+84*x-12)/((5*x**3-40*x**2+80*x)*ln(x)-x+4),x)

[Out]

-x + (x - 2/3)*log(((5*x**2 - 20*x)*log(x) - 1)/(3*x**3 - 12*x**2)) - 2*log(x)/3 + 2*log(log(x) - 1/(5*x**2 -
20*x))/3

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