Integrand size = 75, antiderivative size = 33 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=(4+5 x) \left (2+\frac {e^x}{x}+x-\frac {\log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}\right ) \] Output:
(4+5*x)*(exp(x)/x+2-ln(3/4+1/4*x^2)/x+x)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.27 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=5 e^x+\frac {4 e^x}{x}+14 x+5 x^2+\frac {8 \arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \left (15-4 i \sqrt {3}\right ) \log \left (\sqrt {3}+i x\right )-\frac {1}{3} \left (15+4 i \sqrt {3}\right ) \log \left (i \sqrt {3}+x\right )-\frac {4 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x} \] Input:
Integrate[(34*x^2 + 20*x^3 + 14*x^4 + 10*x^5 + E^x*(-12 + 12*x + 11*x^2 + 4*x^3 + 5*x^4) + (12 + 4*x^2)*Log[(3 + x^2)/4])/(3*x^2 + x^4),x]
Output:
5*E^x + (4*E^x)/x + 14*x + 5*x^2 + (8*ArcTan[x/Sqrt[3]])/Sqrt[3] - ((15 - (4*I)*Sqrt[3])*Log[Sqrt[3] + I*x])/3 - ((15 + (4*I)*Sqrt[3])*Log[I*Sqrt[3] + x])/3 - (4*Log[3/4 + x^2/4])/x
Time = 0.97 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2026, 7276, 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.
\(\displaystyle \int \frac {10 x^5+14 x^4+20 x^3+34 x^2+\left (4 x^2+12\right ) \log \left (\frac {1}{4} \left (x^2+3\right )\right )+e^x \left (5 x^4+4 x^3+11 x^2+12 x-12\right )}{x^4+3 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {10 x^5+14 x^4+20 x^3+34 x^2+\left (4 x^2+12\right ) \log \left (\frac {1}{4} \left (x^2+3\right )\right )+e^x \left (5 x^4+4 x^3+11 x^2+12 x-12\right )}{x^2 \left (x^2+3\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {e^x \left (5 x^2+4 x-4\right )}{x^2}+\frac {2 \left (5 x^5+7 x^4+10 x^3+17 x^2+2 x^2 \log \left (\frac {1}{4} \left (x^2+3\right )\right )+6 \log \left (\frac {1}{4} \left (x^2+3\right )\right )\right )}{x^2 \left (x^2+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 x^2-5 \log \left (x^2+3\right )-\frac {4 \log \left (\frac {x^2}{4}+\frac {3}{4}\right )}{x}+14 x+5 e^x+\frac {4 e^x}{x}\) |
Input:
Int[(34*x^2 + 20*x^3 + 14*x^4 + 10*x^5 + E^x*(-12 + 12*x + 11*x^2 + 4*x^3 + 5*x^4) + (12 + 4*x^2)*Log[(3 + x^2)/4])/(3*x^2 + x^4),x]
Output:
5*E^x + (4*E^x)/x + 14*x + 5*x^2 - (4*Log[3/4 + x^2/4])/x - 5*Log[3 + x^2]
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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.98 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {4 \,{\mathrm e}^{x}}{x}+5 \,{\mathrm e}^{x}-\frac {4 \ln \left (x^{2}+3\right )}{x}+\frac {8 \ln \left (2\right )}{x}+5 x^{2}+14 x -5 \ln \left (x^{2}+3\right )\) | \(47\) |
parts | \(\frac {4 \,{\mathrm e}^{x}}{x}+5 \,{\mathrm e}^{x}-\frac {4 \ln \left (x^{2}+3\right )}{x}+\frac {8 \ln \left (2\right )}{x}+5 x^{2}+14 x -5 \ln \left (x^{2}+3\right )\) | \(47\) |
parallelrisch | \(-\frac {-30 x^{3}+30 \ln \left (x^{2}+3\right ) x -84 x^{2}-30 \,{\mathrm e}^{x} x +45 x -24 \,{\mathrm e}^{x}+24 \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )}{6 x}\) | \(48\) |
risch | \(-\frac {4 \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )}{x}-\frac {-5 x^{3}+5 \ln \left (x^{2}+3\right ) x -14 x^{2}-5 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}}{x}\) | \(49\) |
orering | \(-\frac {\left (625 x^{10}+500 x^{9}+1850 x^{8}+18320 x^{7}+40117 x^{6}+105640 x^{5}+379268 x^{4}+294250 x^{3}+1393974 x^{2}+280890 x +1437534\right ) \left (\left (4 x^{2}+12\right ) \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+10 x^{5}+14 x^{4}+20 x^{3}+34 x^{2}\right )}{20 \left (75 x^{8}+280 x^{7}+706 x^{6}+2420 x^{5}+2275 x^{4}+6082 x^{3}+1242 x^{2}+4482 x +2718\right ) \left (x^{4}+3 x^{2}\right )}+\frac {\left (625 x^{11}+2125 x^{10}+6200 x^{9}+9550 x^{8}+19079 x^{7}+42718 x^{6}-17604 x^{5}+356872 x^{4}-153432 x^{3}+1413261 x^{2}-185652 x +1623186\right ) \left (\frac {8 x \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\frac {\left (4 x^{2}+12\right ) x}{\frac {3}{2}+\frac {x^{2}}{2}}+\left (20 x^{3}+12 x^{2}+22 x +12\right ) {\mathrm e}^{x}+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+50 x^{4}+56 x^{3}+60 x^{2}+68 x}{x^{4}+3 x^{2}}-\frac {\left (\left (4 x^{2}+12\right ) \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+10 x^{5}+14 x^{4}+20 x^{3}+34 x^{2}\right ) \left (4 x^{3}+6 x \right )}{\left (x^{4}+3 x^{2}\right )^{2}}\right )}{1500 x^{8}+5600 x^{7}+14120 x^{6}+48400 x^{5}+45500 x^{4}+121640 x^{3}+24840 x^{2}+89640 x +54360}-\frac {\left (625 x^{8}+1500 x^{7}+4325 x^{6}-11896 x^{4}-31895 x^{3}-109496 x^{2}-59235 x -180354\right ) \left (x^{2}+3\right ) x \left (\frac {8 \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\frac {8 x^{2}}{\frac {3}{4}+\frac {x^{2}}{4}}+\frac {4 x^{2}+12}{\frac {3}{2}+\frac {x^{2}}{2}}-\frac {\left (4 x^{2}+12\right ) x^{2}}{4 \left (\frac {3}{4}+\frac {x^{2}}{4}\right )^{2}}+\left (60 x^{2}+24 x +22\right ) {\mathrm e}^{x}+2 \left (20 x^{3}+12 x^{2}+22 x +12\right ) {\mathrm e}^{x}+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+200 x^{3}+168 x^{2}+120 x +68}{x^{4}+3 x^{2}}-\frac {2 \left (8 x \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\frac {\left (4 x^{2}+12\right ) x}{\frac {3}{2}+\frac {x^{2}}{2}}+\left (20 x^{3}+12 x^{2}+22 x +12\right ) {\mathrm e}^{x}+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+50 x^{4}+56 x^{3}+60 x^{2}+68 x \right ) \left (4 x^{3}+6 x \right )}{\left (x^{4}+3 x^{2}\right )^{2}}+\frac {2 \left (\left (4 x^{2}+12\right ) \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+10 x^{5}+14 x^{4}+20 x^{3}+34 x^{2}\right ) \left (4 x^{3}+6 x \right )^{2}}{\left (x^{4}+3 x^{2}\right )^{3}}-\frac {\left (\left (4 x^{2}+12\right ) \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )+\left (5 x^{4}+4 x^{3}+11 x^{2}+12 x -12\right ) {\mathrm e}^{x}+10 x^{5}+14 x^{4}+20 x^{3}+34 x^{2}\right ) \left (12 x^{2}+6\right )}{\left (x^{4}+3 x^{2}\right )^{2}}\right )}{20 \left (75 x^{8}+280 x^{7}+706 x^{6}+2420 x^{5}+2275 x^{4}+6082 x^{3}+1242 x^{2}+4482 x +2718\right )}\) | \(958\) |
Input:
int(((4*x^2+12)*ln(3/4+1/4*x^2)+(5*x^4+4*x^3+11*x^2+12*x-12)*exp(x)+10*x^5 +14*x^4+20*x^3+34*x^2)/(x^4+3*x^2),x,method=_RETURNVERBOSE)
Output:
4*exp(x)/x+5*exp(x)-4/x*ln(x^2+3)+8*ln(2)/x+5*x^2+14*x-5*ln(x^2+3)
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=\frac {5 \, x^{3} + 14 \, x^{2} + {\left (5 \, x + 4\right )} e^{x} - {\left (5 \, x + 4\right )} \log \left (\frac {1}{4} \, x^{2} + \frac {3}{4}\right )}{x} \] Input:
integrate(((4*x^2+12)*log(3/4+1/4*x^2)+(5*x^4+4*x^3+11*x^2+12*x-12)*exp(x) +10*x^5+14*x^4+20*x^3+34*x^2)/(x^4+3*x^2),x, algorithm="fricas")
Output:
(5*x^3 + 14*x^2 + (5*x + 4)*e^x - (5*x + 4)*log(1/4*x^2 + 3/4))/x
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=5 x^{2} + 14 x - 5 \log {\left (x^{2} + 3 \right )} + \frac {\left (5 x + 4\right ) e^{x}}{x} - \frac {4 \log {\left (\frac {x^{2}}{4} + \frac {3}{4} \right )}}{x} \] Input:
integrate(((4*x**2+12)*ln(3/4+1/4*x**2)+(5*x**4+4*x**3+11*x**2+12*x-12)*ex p(x)+10*x**5+14*x**4+20*x**3+34*x**2)/(x**4+3*x**2),x)
Output:
5*x**2 + 14*x - 5*log(x**2 + 3) + (5*x + 4)*exp(x)/x - 4*log(x**2/4 + 3/4) /x
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=5 \, x^{2} + 14 \, x + \frac {{\left (5 \, x + 4\right )} e^{x} + 8 \, \log \left (2\right ) - 4 \, \log \left (x^{2} + 3\right )}{x} - 5 \, \log \left (x^{2} + 3\right ) \] Input:
integrate(((4*x^2+12)*log(3/4+1/4*x^2)+(5*x^4+4*x^3+11*x^2+12*x-12)*exp(x) +10*x^5+14*x^4+20*x^3+34*x^2)/(x^4+3*x^2),x, algorithm="maxima")
Output:
5*x^2 + 14*x + ((5*x + 4)*e^x + 8*log(2) - 4*log(x^2 + 3))/x - 5*log(x^2 + 3)
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=\frac {5 \, x^{3} + 14 \, x^{2} + 5 \, x e^{x} - 5 \, x \log \left (x^{2} + 3\right ) + 4 \, e^{x} - 4 \, \log \left (\frac {1}{4} \, x^{2} + \frac {3}{4}\right )}{x} \] Input:
integrate(((4*x^2+12)*log(3/4+1/4*x^2)+(5*x^4+4*x^3+11*x^2+12*x-12)*exp(x) +10*x^5+14*x^4+20*x^3+34*x^2)/(x^4+3*x^2),x, algorithm="giac")
Output:
(5*x^3 + 14*x^2 + 5*x*e^x - 5*x*log(x^2 + 3) + 4*e^x - 4*log(1/4*x^2 + 3/4 ))/x
Time = 7.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=14\,x-5\,\ln \left (x^2+3\right )-\frac {4\,\ln \left (\frac {x^2}{4}+\frac {3}{4}\right )}{x}+5\,x^2+\frac {{\mathrm {e}}^x\,\left (5\,x+4\right )}{x} \] Input:
int((exp(x)*(12*x + 11*x^2 + 4*x^3 + 5*x^4 - 12) + 34*x^2 + 20*x^3 + 14*x^ 4 + 10*x^5 + log(x^2/4 + 3/4)*(4*x^2 + 12))/(3*x^2 + x^4),x)
Output:
14*x - 5*log(x^2 + 3) - (4*log(x^2/4 + 3/4))/x + 5*x^2 + (exp(x)*(5*x + 4) )/x
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{3 x^2+x^4} \, dx=\frac {5 e^{x} x +4 e^{x}-5 \,\mathrm {log}\left (x^{2}+3\right ) x -4 \,\mathrm {log}\left (\frac {x^{2}}{4}+\frac {3}{4}\right )+5 x^{3}+14 x^{2}}{x} \] Input:
int(((4*x^2+12)*log(3/4+1/4*x^2)+(5*x^4+4*x^3+11*x^2+12*x-12)*exp(x)+10*x^ 5+14*x^4+20*x^3+34*x^2)/(x^4+3*x^2),x)
Output:
(5*e**x*x + 4*e**x - 5*log(x**2 + 3)*x - 4*log((x**2 + 3)/4) + 5*x**3 + 14 *x**2)/x