3.73.98 \(\int \frac {1+(-3+2 x) \log (3 e^2)}{x+(-4-3 x+x^2) \log (3 e^2)} \, dx\)
Optimal. Leaf size=22 \[ \log \left (4+3 x-x^2-\frac {x}{\log \left (3 e^2\right )}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.86,
number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used =
{1587} \begin {gather*} \log \left (x-\left (-x^2+3 x+4\right ) (2+\log (3))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(1 + (-3 + 2*x)*Log[3*E^2])/(x + (-4 - 3*x + x^2)*Log[3*E^2]),x]
[Out]
Log[x - (4 + 3*x - x^2)*(2 + Log[3])]
Rule 1587
Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log \left (x-\left (4+3 x-x^2\right ) (2+\log (3))\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 16, normalized size = 0.73 \begin {gather*} \log \left (x+\left (-4-3 x+x^2\right ) (2+\log (3))\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 + (-3 + 2*x)*Log[3*E^2])/(x + (-4 - 3*x + x^2)*Log[3*E^2]),x]
[Out]
Log[x + (-4 - 3*x + x^2)*(2 + Log[3])]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 22, normalized size = 1.00 \begin {gather*} \log \left (2 \, x^{2} + {\left (x^{2} - 3 \, x - 4\right )} \log \relax (3) - 5 \, x - 8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x-3)*log(3*exp(2))+1)/((x^2-3*x-4)*log(3*exp(2))+x),x, algorithm="fricas")
[Out]
log(2*x^2 + (x^2 - 3*x - 4)*log(3) - 5*x - 8)
________________________________________________________________________________________
giac [A] time = 0.17, size = 24, normalized size = 1.09 \begin {gather*} \log \left ({\left | {\left (x^{2} - 3 \, x\right )} \log \left (3 \, e^{2}\right ) + x - 4 \, \log \left (3 \, e^{2}\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x-3)*log(3*exp(2))+1)/((x^2-3*x-4)*log(3*exp(2))+x),x, algorithm="giac")
[Out]
log(abs((x^2 - 3*x)*log(3*e^2) + x - 4*log(3*e^2)))
________________________________________________________________________________________
maple [A] time = 0.26, size = 18, normalized size = 0.82
|
|
|
| method |
result |
size |
|
|
|
| derivativedivides |
\(\ln \left (\left (x^{2}-3 x -4\right ) \ln \left (3 \,{\mathrm e}^{2}\right )+x \right )\) |
\(18\) |
| risch |
\(\ln \left (\left (2+\ln \relax (3)\right ) x^{2}+\left (-5-3 \ln \relax (3)\right ) x -4 \ln \relax (3)-8\right )\) |
\(24\) |
| norman |
\(\ln \left (x^{2} \ln \relax (3)-3 x \ln \relax (3)+2 x^{2}-4 \ln \relax (3)-5 x -8\right )\) |
\(27\) |
| default |
\(\ln \left (\ln \left (3 \,{\mathrm e}^{2}\right ) x^{2}-3 \ln \left (3 \,{\mathrm e}^{2}\right ) x -4 \ln \left (3 \,{\mathrm e}^{2}\right )+x \right )\) |
\(28\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*x-3)*ln(3*exp(2))+1)/((x^2-3*x-4)*ln(3*exp(2))+x),x,method=_RETURNVERBOSE)
[Out]
ln((x^2-3*x-4)*ln(3*exp(2))+x)
________________________________________________________________________________________
maxima [A] time = 0.35, size = 17, normalized size = 0.77 \begin {gather*} \log \left ({\left (x^{2} - 3 \, x - 4\right )} \log \left (3 \, e^{2}\right ) + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x-3)*log(3*exp(2))+1)/((x^2-3*x-4)*log(3*exp(2))+x),x, algorithm="maxima")
[Out]
log((x^2 - 3*x - 4)*log(3*e^2) + x)
________________________________________________________________________________________
mupad [B] time = 6.07, size = 1027, normalized size = 46.68 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((log(3*exp(2))*(2*x - 3) + 1)/(x - log(3*exp(2))*(3*x - x^2 + 4)),x)
[Out]
(log(x*(log(9) + 4) - log(27) + ((log(27) - x*(2*log(3) + 4) + 5)*(128*log(3) + 89*log(9) + 40*log(27) + 32*lo
g(81) + 32*log(3)*log(9) + 10*log(9)*log(27) + 16*log(3)*log(81) + 8*log(9)*log(81) + log(9)*log(27)^2 + 4*log
(27)^2 - 10*log(3)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) + 5*log(9)*(
32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) - 2*log(3)*log(27)*(32*log(3) +
10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) + log(9)*log(27)*(32*log(3) + 10*log(27) + 8
*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) + 4*log(3)*log(9)*log(81) + 356))/(2*(153*log(3) + 20*log(
27) + 16*log(81) + 10*log(3)*log(27) + 16*log(3)*log(81) + log(3)*log(27)^2 + 4*log(3)^2*log(81) + 32*log(3)^2
+ 2*log(27)^2 + 178)) - 5)*(128*log(3) + 89*log(9) + 40*log(27) + 32*log(81) + 32*log(3)*log(9) + 10*log(9)*l
og(27) + 16*log(3)*log(81) + 8*log(9)*log(81) + log(9)*log(27)^2 + 4*log(27)^2 - 10*log(3)*(32*log(3) + 10*log
(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) + 5*log(9)*(32*log(3) + 10*log(27) + 8*log(81) + 4
*log(3)*log(81) + log(27)^2 + 89)^(1/2) - 2*log(3)*log(27)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(
81) + log(27)^2 + 89)^(1/2) + log(9)*log(27)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^
2 + 89)^(1/2) + 4*log(3)*log(9)*log(81) + 356))/(2*(153*log(3) + 20*log(27) + 16*log(81) + 10*log(3)*log(27) +
16*log(3)*log(81) + log(3)*log(27)^2 + 4*log(3)^2*log(81) + 32*log(3)^2 + 2*log(27)^2 + 178)) + (log(x*(log(9
) + 4) - log(27) + ((log(27) - x*(2*log(3) + 4) + 5)*(128*log(3) + 89*log(9) + 40*log(27) + 32*log(81) + 32*lo
g(3)*log(9) + 10*log(9)*log(27) + 16*log(3)*log(81) + 8*log(9)*log(81) + log(9)*log(27)^2 + 4*log(27)^2 + 10*l
og(3)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) - 5*log(9)*(32*log(3) + 1
0*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) + 2*log(3)*log(27)*(32*log(3) + 10*log(27) +
8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) - log(9)*log(27)*(32*log(3) + 10*log(27) + 8*log(81) + 4*
log(3)*log(81) + log(27)^2 + 89)^(1/2) + 4*log(3)*log(9)*log(81) + 356))/(2*(153*log(3) + 20*log(27) + 16*log(
81) + 10*log(3)*log(27) + 16*log(3)*log(81) + log(3)*log(27)^2 + 4*log(3)^2*log(81) + 32*log(3)^2 + 2*log(27)^
2 + 178)) - 5)*(128*log(3) + 89*log(9) + 40*log(27) + 32*log(81) + 32*log(3)*log(9) + 10*log(9)*log(27) + 16*l
og(3)*log(81) + 8*log(9)*log(81) + log(9)*log(27)^2 + 4*log(27)^2 + 10*log(3)*(32*log(3) + 10*log(27) + 8*log(
81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2) - 5*log(9)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(8
1) + log(27)^2 + 89)^(1/2) + 2*log(3)*log(27)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)
^2 + 89)^(1/2) - log(9)*log(27)*(32*log(3) + 10*log(27) + 8*log(81) + 4*log(3)*log(81) + log(27)^2 + 89)^(1/2)
+ 4*log(3)*log(9)*log(81) + 356))/(2*(153*log(3) + 20*log(27) + 16*log(81) + 10*log(3)*log(27) + 16*log(3)*lo
g(81) + log(3)*log(27)^2 + 4*log(3)^2*log(81) + 32*log(3)^2 + 2*log(27)^2 + 178))
________________________________________________________________________________________
sympy [A] time = 0.29, size = 26, normalized size = 1.18 \begin {gather*} \log {\left (x^{2} \left (\log {\relax (3 )} + 2\right ) + x \left (-5 - 3 \log {\relax (3 )}\right ) - 8 - 4 \log {\relax (3 )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x-3)*ln(3*exp(2))+1)/((x**2-3*x-4)*ln(3*exp(2))+x),x)
[Out]
log(x**2*(log(3) + 2) + x*(-5 - 3*log(3)) - 8 - 4*log(3))
________________________________________________________________________________________