3.102.44 \(\int \frac {-4 x \log (x)+8 \log (\frac {4}{\log ^2(x)})-2 \log (x) \log ^2(\frac {4}{\log ^2(x)})+(2 x \log (x)+2 \log (x) \log ^2(\frac {4}{\log ^2(x)})) \log (\frac {2}{x^2+x \log ^2(\frac {4}{\log ^2(x)})})}{x \log (x)+\log (x) \log ^2(\frac {4}{\log ^2(x)})} \, dx\)
Optimal. Leaf size=22 \[ 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 1.03, antiderivative size = 22, normalized size of antiderivative = 1.00,
number of steps used = 12, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used
= {6741, 6742, 2549} \begin {gather*} 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-4*x*Log[x] + 8*Log[4/Log[x]^2] - 2*Log[x]*Log[4/Log[x]^2]^2 + (2*x*Log[x] + 2*Log[x]*Log[4/Log[x]^2]^2)*
Log[2/(x^2 + x*Log[4/Log[x]^2]^2)])/(x*Log[x] + Log[x]*Log[4/Log[x]^2]^2),x]
[Out]
2*x*Log[2/(x*(x + Log[4/Log[x]^2]^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 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 {-4 x \log (x)+8 \log \left (\frac {4}{\log ^2(x)}\right )-2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )+\left (2 x \log (x)+2 \log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right ) \log \left (\frac {2}{x^2+x \log ^2\left (\frac {4}{\log ^2(x)}\right )}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=\int \left (-\frac {2 \left (2 x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}+2 \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )\right ) \, dx\\ &=-\left (2 \int \frac {2 x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )+\log (x) \log ^2\left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\right )+2 \int \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \left (1+\frac {x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx-2 \int \frac {4 \log \left (\frac {4}{\log ^2(x)}\right )-\log (x) \left (2 x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=-2 x+2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {x \log (x)-4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx-2 \int \left (-1+\frac {-x \log (x)+4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {-x \log (x)+4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx-2 \int \left (\frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )}-\frac {4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )-2 \int \frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )} \, dx-2 \int \left (-\frac {x}{x+\log ^2\left (\frac {4}{\log ^2(x)}\right )}+\frac {4 \log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \, dx+8 \int \frac {\log \left (\frac {4}{\log ^2(x)}\right )}{\log (x) \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )} \, dx\\ &=2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 22, normalized size = 1.00 \begin {gather*} 2 x \log \left (\frac {2}{x \left (x+\log ^2\left (\frac {4}{\log ^2(x)}\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-4*x*Log[x] + 8*Log[4/Log[x]^2] - 2*Log[x]*Log[4/Log[x]^2]^2 + (2*x*Log[x] + 2*Log[x]*Log[4/Log[x]^
2]^2)*Log[2/(x^2 + x*Log[4/Log[x]^2]^2)])/(x*Log[x] + Log[x]*Log[4/Log[x]^2]^2),x]
[Out]
2*x*Log[2/(x*(x + Log[4/Log[x]^2]^2))]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 23, normalized size = 1.05 \begin {gather*} 2 \, x \log \left (\frac {2}{x \log \left (\frac {4}{\log \relax (x)^{2}}\right )^{2} + x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2)^2+x^2))-2*log(x)*log(4/log(x)^2)^2
+8*log(4/log(x)^2)-4*x*log(x))/(log(x)*log(4/log(x)^2)^2+x*log(x)),x, algorithm="fricas")
[Out]
2*x*log(2/(x*log(4/log(x)^2)^2 + x^2))
________________________________________________________________________________________
giac [A] time = 1.65, size = 39, normalized size = 1.77 \begin {gather*} 2 \, x \log \relax (2) - 2 \, x \log \left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) \log \left (\log \relax (x)^{2}\right ) + \log \left (\log \relax (x)^{2}\right )^{2} + x\right ) - 2 \, x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2)^2+x^2))-2*log(x)*log(4/log(x)^2)^2
+8*log(4/log(x)^2)-4*x*log(x))/(log(x)*log(4/log(x)^2)^2+x*log(x)),x, algorithm="giac")
[Out]
2*x*log(2) - 2*x*log(4*log(2)^2 - 4*log(2)*log(log(x)^2) + log(log(x)^2)^2 + x) - 2*x*log(x)
________________________________________________________________________________________
maple [C] time = 0.84, size = 2139, normalized size = 97.23
|
|
|
| method |
result |
size |
|
|
|
| risch |
\(\text {Expression too large to display}\) |
\(2139\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*ln(x)*ln(4/ln(x)^2)^2+2*x*ln(x))*ln(2/(x*ln(4/ln(x)^2)^2+x^2))-2*ln(x)*ln(4/ln(x)^2)^2+8*ln(4/ln(x)^2)
-4*x*ln(x))/(ln(x)*ln(4/ln(x)^2)^2+x*ln(x)),x,method=_RETURNVERBOSE)
[Out]
-2*x*ln(-4*x+32*ln(2)*ln(ln(x))-16*ln(2)^2-16*ln(ln(x))^2+Pi^2*csgn(I*ln(x)^2)^6+16*I*Pi*ln(2)*csgn(I*ln(x))*c
sgn(I*ln(x)^2)^2-8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(ln(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)
-8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3+Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2-4*Pi^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^
3+6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4-4*Pi^2*csgn(I*ln(x))*csgn(I*ln(x)^2)^5-16*I*Pi*ln(ln(x))*csgn(I*ln(
x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(ln(x))*csgn(I*ln(x)^2)^3)-2*I*Pi*x*csgn(I/(4*x-32*ln(2)*ln(ln(x))+16*ln(2)^2+1
6*ln(ln(x))^2-Pi^2*csgn(I*ln(x)^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csgn(I*ln(x))^
2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-Pi^2*csgn(I*
ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4
*Pi^2*csgn(I*ln(x))*csgn(I*ln(x)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(ln(x))*csgn(
I*ln(x)^2)^3)/x)^2-I*Pi*x*csgn(I/x)*csgn(I/(4*x-32*ln(2)*ln(ln(x))+16*ln(2)^2+16*ln(ln(x))^2-Pi^2*csgn(I*ln(x)
^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln(
x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi
^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*csgn(I*ln(x))*csgn(I*ln(x
)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(ln(x))*csgn(I*ln(x)^2)^3))*csgn(I/(4*x-32*l
n(2)*ln(ln(x))+16*ln(2)^2+16*ln(ln(x))^2-Pi^2*csgn(I*ln(x)^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+
8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csg
n(I*ln(x)^2)^3-Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*l
n(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*csgn(I*ln(x))*csgn(I*ln(x)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^
2)^2-8*I*Pi*ln(ln(x))*csgn(I*ln(x)^2)^3)/x)+I*Pi*x*csgn(I/x)*csgn(I/(4*x-32*ln(2)*ln(ln(x))+16*ln(2)^2+16*ln(l
n(x))^2-Pi^2*csgn(I*ln(x)^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn
(I*ln(x)^2)-8*I*Pi*ln(ln(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-Pi^2*csgn(I*ln(x))
^4*csgn(I*ln(x)^2)^2+4*Pi^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*
csgn(I*ln(x))*csgn(I*ln(x)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(ln(x))*csgn(I*ln(x
)^2)^3)/x)^2-I*Pi*x*csgn(I/(4*x-32*ln(2)*ln(ln(x))+16*ln(2)^2+16*ln(ln(x))^2-Pi^2*csgn(I*ln(x)^2)^6-16*I*Pi*ln
(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln(x))*csgn(I*ln(x)
)^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi^2*csgn(I*ln(x))
^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*csgn(I*ln(x))*csgn(I*ln(x)^2)^5+16*I*Pi*l
n(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(ln(x))*csgn(I*ln(x)^2)^3))*csgn(I/(4*x-32*ln(2)*ln(ln(x))+1
6*ln(2)^2+16*ln(ln(x))^2-Pi^2*csgn(I*ln(x)^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csg
n(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-P
i^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*l
n(x)^2)^4+4*Pi^2*csgn(I*ln(x))*csgn(I*ln(x)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(l
n(x))*csgn(I*ln(x)^2)^3)/x)^2-I*Pi*x*csgn(I/(4*x-32*ln(2)*ln(ln(x))+16*ln(2)^2+16*ln(ln(x))^2-Pi^2*csgn(I*ln(x
)^2)^6-16*I*Pi*ln(2)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2+8*I*Pi*ln(2)*csgn(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(ln
(x))*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*csgn(I*ln(x)^2)^3-Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*P
i^2*csgn(I*ln(x))^3*csgn(I*ln(x)^2)^3-6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*csgn(I*ln(x))*csgn(I*ln(
x)^2)^5+16*I*Pi*ln(ln(x))*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-8*I*Pi*ln(ln(x))*csgn(I*ln(x)^2)^3)/x)^3+2*I*Pi*x+6*
x*ln(2)-2*x*ln(x)
________________________________________________________________________________________
maxima [A] time = 0.48, size = 37, normalized size = 1.68 \begin {gather*} 2 \, x \log \relax (2) - 2 \, x \log \left (4 \, \log \relax (2)^{2} - 8 \, \log \relax (2) \log \left (\log \relax (x)\right ) + 4 \, \log \left (\log \relax (x)\right )^{2} + x\right ) - 2 \, x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*log(x)*log(4/log(x)^2)^2+2*x*log(x))*log(2/(x*log(4/log(x)^2)^2+x^2))-2*log(x)*log(4/log(x)^2)^2
+8*log(4/log(x)^2)-4*x*log(x))/(log(x)*log(4/log(x)^2)^2+x*log(x)),x, algorithm="maxima")
[Out]
2*x*log(2) - 2*x*log(4*log(2)^2 - 8*log(2)*log(log(x)) + 4*log(log(x))^2 + x) - 2*x*log(x)
________________________________________________________________________________________
mupad [B] time = 8.48, size = 24, normalized size = 1.09 \begin {gather*} 2\,x\,\left (\ln \left (\frac {1}{x^2+x\,{\ln \left (\frac {4}{{\ln \relax (x)}^2}\right )}^2}\right )+\ln \relax (2)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((8*log(4/log(x)^2) - 2*log(4/log(x)^2)^2*log(x) - 4*x*log(x) + log(2/(x*log(4/log(x)^2)^2 + x^2))*(2*log(4
/log(x)^2)^2*log(x) + 2*x*log(x)))/(log(4/log(x)^2)^2*log(x) + x*log(x)),x)
[Out]
2*x*(log(1/(x*log(4/log(x)^2)^2 + x^2)) + log(2))
________________________________________________________________________________________
sympy [A] time = 0.88, size = 20, normalized size = 0.91 \begin {gather*} 2 x \log {\left (\frac {2}{x^{2} + x \log {\left (\frac {4}{\log {\relax (x )}^{2}} \right )}^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*ln(x)*ln(4/ln(x)**2)**2+2*x*ln(x))*ln(2/(x*ln(4/ln(x)**2)**2+x**2))-2*ln(x)*ln(4/ln(x)**2)**2+8*
ln(4/ln(x)**2)-4*x*ln(x))/(ln(x)*ln(4/ln(x)**2)**2+x*ln(x)),x)
[Out]
2*x*log(2/(x**2 + x*log(4/log(x)**2)**2))
________________________________________________________________________________________