3.33.83 \(\int \frac {20+24 x^2+16 x^3-4 x^4+(-2 x^2-8 x^3+2 x^4+(-10-40 x+10 x^2) \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})) \log (\frac {1}{5} (x^2+5 \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})))}{(-x^2-4 x^3+x^4+(-5-20 x+5 x^2) \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})) \log ^2(\frac {1}{5} (x^2+5 \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})))} \, dx\)
Optimal. Leaf size=33 \[ \frac {2 x}{\log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{\left (2-2 \left (-4 x+x^2\right )\right )^2}\right )\right )} \]
________________________________________________________________________________________
Rubi [F] time = 36.60, 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 {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(20 + 24*x^2 + 16*x^3 - 4*x^4 + (-2*x^2 - 8*x^3 + 2*x^4 + (-10 - 40*x + 10*x^2)*Log[(25*x^2)/(4 + 32*x + 5
6*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5])/((-x^2 - 4*x^3
+ x^4 + (-5 - 20*x + 5*x^2)*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 +
32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5]^2),x]
[Out]
68*Defer[Int][1/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)
^2)])/5]^2), x] + 16*Defer[Int][x/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])/5]^2), x] + 4*Defer[Int][x^2/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2
+ 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + 88*Defer[Int][1/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x
^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + (288*(5 - 2*Sqrt[5])*Defer[Int][1/((4
- 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2
)^2)])/5]^2), x])/5 - 8*Sqrt[5]*Defer[Int][1/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + (288*(5 + 2*Sqrt[5])*Defer[Int][1/((4 + 2*
Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)
])/5]^2), x])/5 + 16*Defer[Int][x/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4
*(-1 - 4*x + x^2)^2)])/5]^2), x] + (368*(5 + 2*Sqrt[5])*Defer[Int][1/((-4 - 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*
x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x])/5 - 8*Sqrt[5]*Defe
r[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(
-1 - 4*x + x^2)^2)])/5]^2), x] + (368*(5 - 2*Sqrt[5])*Defer[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^
2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x])/5 + 34*Defer[Int][1/
((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] +
8*Defer[Int][x/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])/5]), x] + 34*Defer[Int][1/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-
1 - 4*x + x^2)^2)])/5]), x] + (144*(5 - 2*Sqrt[5])*Defer[Int][1/((4 - 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x])/5 + (144*(5 + 2*Sqrt[5])*D
efer[Int][1/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4
*(-1 - 4*x + x^2)^2)])/5]), x])/5 + 8*Defer[Int][x/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 +
5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 2*Defer[Int][x^2/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + (144*(5 + 2*Sqrt[5])*Defer[Int][1/((-4 - 2*S
qrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]
)/5]), x])/5 + (144*(5 - 2*Sqrt[5])*Defer[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x +
x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x])/5 + 10*Defer[Int][Log[(25*x^2)/(4*(-1 -
4*x + x^2)^2)]/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2
)])/5]), x] + 8*(5 - 2*Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((4 - 2*Sqrt[5] - 2*x)*(x^2 +
5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 + 2*
Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 -
4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 + 2*Sqrt[5])*Defer[Int][Log
[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((-4 - 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(
x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 - 2*Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x
+ x^2)^2)]/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])/5]), x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-20-24 x^2-16 x^3+4 x^4-2 \left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 32, normalized size = 0.97 \begin {gather*} \frac {2 x}{\log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(20 + 24*x^2 + 16*x^3 - 4*x^4 + (-2*x^2 - 8*x^3 + 2*x^4 + (-10 - 40*x + 10*x^2)*Log[(25*x^2)/(4 + 32
*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5])/((-x^2 -
4*x^3 + x^4 + (-5 - 20*x + 5*x^2)*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2
)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5]^2),x]
[Out]
(2*x)/Log[x^2/5 + Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 38, normalized size = 1.15 \begin {gather*} \frac {2 \, x}{\log \left (\frac {1}{5} \, x^{2} + \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="fricas")
[Out]
2*x/log(1/5*x^2 + log(25/4*x^2/(x^4 - 8*x^3 + 14*x^2 + 8*x + 1)))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="giac")
[Out]
sage2
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (10 x^{2}-40 x -10\right ) \ln \left (\frac {25 x^{2}}{4 x^{4}-32 x^{3}+56 x^{2}+32 x +4}\right )+2 x^{4}-8 x^{3}-2 x^{2}\right ) \ln \left (\ln \left (\frac {25 x^{2}}{4 x^{4}-32 x^{3}+56 x^{2}+32 x +4}\right )+\frac {x^{2}}{5}\right )-4 x^{4}+16 x^{3}+24 x^{2}+20}{\left (\left (5 x^{2}-20 x -5\right ) \ln \left (\frac {25 x^{2}}{4 x^{4}-32 x^{3}+56 x^{2}+32 x +4}\right )+x^{4}-4 x^{3}-x^{2}\right ) \ln \left (\ln \left (\frac {25 x^{2}}{4 x^{4}-32 x^{3}+56 x^{2}+32 x +4}\right )+\frac {x^{2}}{5}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((10*x^2-40*x-10)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*ln(ln(25*x^2/(4*x^4-32*x^3+5
6*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+x^4-4*
x^3-x^2)/ln(ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x)
[Out]
int((((10*x^2-40*x-10)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*ln(ln(25*x^2/(4*x^4-32*x^3+5
6*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+x^4-4*
x^3-x^2)/ln(ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x)
________________________________________________________________________________________
maxima [A] time = 0.73, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2 \, x}{\log \relax (5) - \log \left (x^{2} + 10 \, \log \relax (5) - 10 \, \log \relax (2) - 10 \, \log \left (x^{2} - 4 \, x - 1\right ) + 10 \, \log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="maxima")
[Out]
-2*x/(log(5) - log(x^2 + 10*log(5) - 10*log(2) - 10*log(x^2 - 4*x - 1) + 10*log(x)))
________________________________________________________________________________________
mupad [B] time = 2.90, size = 138, normalized size = 4.18 \begin {gather*} x-\frac {5\,x}{-x^4+4\,x^3+6\,x^2+5}-\frac {6\,x^3}{-x^4+4\,x^3+6\,x^2+5}-\frac {4\,x^4}{-x^4+4\,x^3+6\,x^2+5}+\frac {x^5}{-x^4+4\,x^3+6\,x^2+5}+\frac {2\,x}{\ln \left (\ln \left (x^2\right )+2\,\ln \relax (5)+\ln \left (\frac {1}{4\,x^4-32\,x^3+56\,x^2+32\,x+4}\right )+\frac {x^2}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(24*x^2 + 16*x^3 - 4*x^4 - log(log((25*x^2)/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) + x^2/5)*(log((25*x^2)/
(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4))*(40*x - 10*x^2 + 10) + 2*x^2 + 8*x^3 - 2*x^4) + 20)/(log(log((25*x^2)/(3
2*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) + x^2/5)^2*(log((25*x^2)/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4))*(20*x - 5*x
^2 + 5) + x^2 + 4*x^3 - x^4)),x)
[Out]
x - (5*x)/(6*x^2 + 4*x^3 - x^4 + 5) - (6*x^3)/(6*x^2 + 4*x^3 - x^4 + 5) - (4*x^4)/(6*x^2 + 4*x^3 - x^4 + 5) +
x^5/(6*x^2 + 4*x^3 - x^4 + 5) + (2*x)/log(log(x^2) + 2*log(5) + log(1/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) +
x^2/5)
________________________________________________________________________________________
sympy [A] time = 0.67, size = 36, normalized size = 1.09 \begin {gather*} \frac {2 x}{\log {\left (\frac {x^{2}}{5} + \log {\left (\frac {25 x^{2}}{4 x^{4} - 32 x^{3} + 56 x^{2} + 32 x + 4} \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((10*x**2-40*x-10)*ln(25*x**2/(4*x**4-32*x**3+56*x**2+32*x+4))+2*x**4-8*x**3-2*x**2)*ln(ln(25*x**2/
(4*x**4-32*x**3+56*x**2+32*x+4))+1/5*x**2)-4*x**4+16*x**3+24*x**2+20)/((5*x**2-20*x-5)*ln(25*x**2/(4*x**4-32*x
**3+56*x**2+32*x+4))+x**4-4*x**3-x**2)/ln(ln(25*x**2/(4*x**4-32*x**3+56*x**2+32*x+4))+1/5*x**2)**2,x)
[Out]
2*x/log(x**2/5 + log(25*x**2/(4*x**4 - 32*x**3 + 56*x**2 + 32*x + 4)))
________________________________________________________________________________________