Integrand size = 82, antiderivative size = 29 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-2 x+\frac {\log \left (\frac {1}{3} (-2-x)-2 x-\frac {\log (4)}{x}\right )}{x} \] Output:
ln(-7/3*x-2*ln(2)/x-2/3)/x-2*x
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-\frac {\log (3)+\frac {x^2 \log (4096)}{\log (64)}-\log \left (-2-7 x-\frac {\log (64)}{x}\right )}{x} \] Input:
Integrate[(7*x^2 - 4*x^3 - 14*x^4 + (-3 - 6*x^2)*Log[4] + (-2*x - 7*x^2 - 3*Log[4])*Log[(-2*x - 7*x^2 - 3*Log[4])/(3*x)])/(2*x^3 + 7*x^4 + 3*x^2*Log [4]),x]
Output:
-((Log[3] + (x^2*Log[4096])/Log[64] - Log[-2 - 7*x - Log[64]/x])/x)
Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(29)=58\).
Time = 0.91 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2026, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-14 x^4-4 x^3+7 x^2+\left (-7 x^2-2 x-3 \log (4)\right ) \log \left (\frac {-7 x^2-2 x-3 \log (4)}{3 x}\right )+\left (-6 x^2-3\right ) \log (4)}{7 x^4+2 x^3+3 x^2 \log (4)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-14 x^4-4 x^3+7 x^2+\left (-7 x^2-2 x-3 \log (4)\right ) \log \left (\frac {-7 x^2-2 x-3 \log (4)}{3 x}\right )+\left (-6 x^2-3\right ) \log (4)}{x^2 \left (7 x^2+2 x+\log (64)\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {-14 x^4-4 x^3+x^2 (7-6 \log (4))-3 \log (4)}{x^2 \left (7 x^2+2 x+\log (64)\right )}-\frac {\log \left (-\frac {7 x}{3}-\frac {\log (4)}{x}-\frac {2}{3}\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {7 \log (64)-1} \log (4096) \arctan \left (\frac {7 x+1}{\sqrt {7 \log (64)-1}}\right )}{\log ^2(64)}+\frac {\left (\log (4) (6-21 \log (64))-7 \log ^2(64)\right ) \arctan \left (\frac {7 x+1}{\sqrt {7 \log (64)-1}}\right )}{\log ^2(64) \sqrt {7 \log (64)-1}}-2 x+\frac {\log \left (-\frac {7 x}{3}-\frac {\log (4)}{x}-\frac {2}{3}\right )}{x}\) |
Input:
Int[(7*x^2 - 4*x^3 - 14*x^4 + (-3 - 6*x^2)*Log[4] + (-2*x - 7*x^2 - 3*Log[ 4])*Log[(-2*x - 7*x^2 - 3*Log[4])/(3*x)])/(2*x^3 + 7*x^4 + 3*x^2*Log[4]),x ]
Output:
-2*x + (ArcTan[(1 + 7*x)/Sqrt[-1 + 7*Log[64]]]*(Log[4]*(6 - 21*Log[64]) - 7*Log[64]^2))/(Log[64]^2*Sqrt[-1 + 7*Log[64]]) + (ArcTan[(1 + 7*x)/Sqrt[-1 + 7*Log[64]]]*Sqrt[-1 + 7*Log[64]]*Log[4096])/Log[64]^2 + Log[-2/3 - (7*x )/3 - Log[4]/x]/x
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 3.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{3 x}\right )}{x}-2 x\) | \(28\) |
norman | \(\frac {-2 x^{2}+\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{3 x}\right )}{x}\) | \(30\) |
parallelrisch | \(-\frac {98 x^{2}-56 x -49 \ln \left (-\frac {7 x^{2}+6 \ln \left (2\right )+2 x}{3 x}\right )}{49 x}\) | \(36\) |
default | \(-\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{x}\right )}{x}-\frac {-\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\sqrt {-1+42 \ln \left (2\right )}\, \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{3 \ln \left (2\right )}-2 x -\frac {\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\frac {\left (1-42 \ln \left (2\right )\right ) \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{\sqrt {-1+42 \ln \left (2\right )}}}{3 \ln \left (2\right )}\) | \(136\) |
parts | \(-\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{x}\right )}{x}-\frac {-\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\sqrt {-1+42 \ln \left (2\right )}\, \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{3 \ln \left (2\right )}-2 x -\frac {\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\frac {\left (1-42 \ln \left (2\right )\right ) \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{\sqrt {-1+42 \ln \left (2\right )}}}{3 \ln \left (2\right )}\) | \(136\) |
Input:
int(((-6*ln(2)-7*x^2-2*x)*ln(1/3*(-6*ln(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*ln(2 )-14*x^4-4*x^3+7*x^2)/(6*x^2*ln(2)+7*x^4+2*x^3),x,method=_RETURNVERBOSE)
Output:
1/x*ln(1/3*(-6*ln(2)-7*x^2-2*x)/x)-2*x
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-\frac {2 \, x^{2} - \log \left (-\frac {7 \, x^{2} + 2 \, x + 6 \, \log \left (2\right )}{3 \, x}\right )}{x} \] Input:
integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^ 2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x^2*log(2)+7*x^4+2*x^3),x, algorithm="f ricas")
Output:
-(2*x^2 - log(-1/3*(7*x^2 + 2*x + 6*log(2))/x))/x
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=- 2 x + \frac {\log {\left (\frac {- \frac {7 x^{2}}{3} - \frac {2 x}{3} - 2 \log {\left (2 \right )}}{x} \right )}}{x} \] Input:
integrate(((-6*ln(2)-7*x**2-2*x)*ln(1/3*(-6*ln(2)-7*x**2-2*x)/x)+2*(-6*x** 2-3)*ln(2)-14*x**4-4*x**3+7*x**2)/(6*x**2*ln(2)+7*x**4+2*x**3),x)
Output:
-2*x + log((-7*x**2/3 - 2*x/3 - 2*log(2))/x)/x
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (21) = 42\).
Time = 0.60 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.86 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {1}{6} \, {\left (\frac {2 \, {\left (21 \, \log \left (2\right ) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \left (2\right ) - 1}}\right )}{\sqrt {42 \, \log \left (2\right ) - 1} \log \left (2\right )^{2}} - \frac {\log \left (7 \, x^{2} + 2 \, x + 6 \, \log \left (2\right )\right )}{\log \left (2\right )^{2}} + \frac {2 \, \log \left (x\right )}{\log \left (2\right )^{2}} + \frac {6}{x \log \left (2\right )}\right )} \log \left (2\right ) - \frac {{\left (21 \, \log \left (2\right ) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \left (2\right ) - 1}}\right )}{3 \, \sqrt {42 \, \log \left (2\right ) - 1} \log \left (2\right )} - \frac {12 \, x^{2} \log \left (2\right ) + 6 \, {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - {\left (x + 6 \, \log \left (2\right )\right )} \log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \left (2\right )\right ) + 2 \, {\left (x + 3 \, \log \left (2\right )\right )} \log \left (x\right )}{6 \, x \log \left (2\right )} \] Input:
integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^ 2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x^2*log(2)+7*x^4+2*x^3),x, algorithm="m axima")
Output:
1/6*(2*(21*log(2) - 1)*arctan((7*x + 1)/sqrt(42*log(2) - 1))/(sqrt(42*log( 2) - 1)*log(2)^2) - log(7*x^2 + 2*x + 6*log(2))/log(2)^2 + 2*log(x)/log(2) ^2 + 6/(x*log(2)))*log(2) - 1/3*(21*log(2) - 1)*arctan((7*x + 1)/sqrt(42*l og(2) - 1))/(sqrt(42*log(2) - 1)*log(2)) - 1/6*(12*x^2*log(2) + 6*(log(3) + 1)*log(2) - (x + 6*log(2))*log(-7*x^2 - 2*x - 6*log(2)) + 2*(x + 3*log(2 ))*log(x))/(x*log(2))
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-2 \, x + \frac {\log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \left (2\right )\right )}{x} - \frac {\log \left (3 \, x\right )}{x} \] Input:
integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^ 2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x^2*log(2)+7*x^4+2*x^3),x, algorithm="g iac")
Output:
-2*x + log(-7*x^2 - 2*x - 6*log(2))/x - log(3*x)/x
Time = 3.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {\ln \left (-\frac {\frac {7\,x^2}{3}+\frac {2\,x}{3}+\ln \left (4\right )}{x}\right )}{x}-2\,x \] Input:
int(-(2*log(2)*(6*x^2 + 3) + log(-((2*x)/3 + 2*log(2) + (7*x^2)/3)/x)*(2*x + 6*log(2) + 7*x^2) - 7*x^2 + 4*x^3 + 14*x^4)/(6*x^2*log(2) + 2*x^3 + 7*x ^4),x)
Output:
log(-((2*x)/3 + log(4) + (7*x^2)/3)/x)/x - 2*x
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.83 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {-\mathrm {log}\left (6 \,\mathrm {log}\left (2\right )+7 x^{2}+2 x \right ) x +6 \,\mathrm {log}\left (\frac {-6 \,\mathrm {log}\left (2\right )-7 x^{2}-2 x}{3 x}\right ) \mathrm {log}\left (2\right )+\mathrm {log}\left (\frac {-6 \,\mathrm {log}\left (2\right )-7 x^{2}-2 x}{3 x}\right ) x +\mathrm {log}\left (x \right ) x -12 \,\mathrm {log}\left (2\right ) x^{2}}{6 \,\mathrm {log}\left (2\right ) x} \] Input:
int(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*l og(2)-14*x^4-4*x^3+7*x^2)/(6*x^2*log(2)+7*x^4+2*x^3),x)
Output:
( - log(6*log(2) + 7*x**2 + 2*x)*x + 6*log(( - 6*log(2) - 7*x**2 - 2*x)/(3 *x))*log(2) + log(( - 6*log(2) - 7*x**2 - 2*x)/(3*x))*x + log(x)*x - 12*lo g(2)*x**2)/(6*log(2)*x)