Integrand size = 17, antiderivative size = 48 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{12} \log \left (9-x^4\right ) \]
-1/3*ln(x)+1/12*ln(-x^4+9)+1/3*arctan(1/3*x*3^(1/2))*3^(1/2)-1/3*arctanh(1 /3*x*3^(1/2))*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{12} \left (4 \sqrt {3} \arctan \left (\frac {x}{\sqrt {3}}\right )-4 \log (x)+2 \sqrt {3} \log \left (3-\sqrt {3} x\right )-2 \sqrt {3} \log \left (3+\sqrt {3} x\right )+\log \left (9-x^4\right )\right ) \]
(4*Sqrt[3]*ArcTan[x/Sqrt[3]] - 4*Log[x] + 2*Sqrt[3]*Log[3 - Sqrt[3]*x] - 2 *Sqrt[3]*Log[3 + Sqrt[3]*x] + Log[9 - x^4])/12
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2026, 2370, 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.
\(\displaystyle \int \frac {2 x^3+3}{x^5-9 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^3+3}{x \left (x^4-9\right )}dx\) |
\(\Big \downarrow \) 2370 |
\(\displaystyle \int \left (\frac {3}{\left (x^4-9\right ) x}+\frac {2 x^2}{x^4-9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{12} \log \left (9-x^4\right )-\frac {\log (x)}{3}\) |
3.2.14.3.1 Defintions of rubi rules used
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[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\ln \left (x^{2}+3\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (x^{2}-3\right )}{12}-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(46\) |
risch | \(\frac {\ln \left (x^{2}+3\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (x -\sqrt {3}\right )}{12}+\frac {\sqrt {3}\, \ln \left (x -\sqrt {3}\right )}{6}+\frac {\ln \left (x +\sqrt {3}\right )}{12}-\frac {\sqrt {3}\, \ln \left (x +\sqrt {3}\right )}{6}-\frac {\ln \left (x \right )}{3}\) | \(68\) |
meijerg | \(-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (3\right )}{6}-\frac {i \pi }{12}+\frac {\ln \left (1-\frac {x^{4}}{9}\right )}{12}+\frac {x^{3} \sqrt {3}\, \left (\ln \left (1-\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )-\ln \left (1+\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )+2 \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )\right )}{6 \left (x^{4}\right )^{\frac {3}{4}}}\) | \(79\) |
1/12*ln(x^2+3)+1/3*arctan(1/3*x*3^(1/2))*3^(1/2)-1/3*ln(x)+1/12*ln(x^2-3)- 1/3*arctanh(1/3*x*3^(1/2))*3^(1/2)
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \left (x\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log((x^2 - 2*sqrt(3)*x + 3 )/(x^2 - 3)) + 1/12*log(x^2 + 3) + 1/12*log(x^2 - 3) - 1/3*log(x)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 6.38 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=- \frac {\log {\left (x \right )}}{3} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {943 \sqrt {3} i}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{2}}{481} - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{4}}{481} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{4}}{481} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{2}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {943 \sqrt {3} i}{5772} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{4}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {943 \sqrt {3}}{5772} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{4}}{481} - \frac {943 \sqrt {3}}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} \]
-log(x)/3 + (1/12 + sqrt(3)*I/6)*log(x + 17413/11544 - 943*sqrt(3)*I/5772 + 1368*(1/12 + sqrt(3)*I/6)**3/481 + 4158*(1/12 + sqrt(3)*I/6)**2/481 - 10 8000*(1/12 + sqrt(3)*I/6)**4/481) + (1/12 - sqrt(3)*I/6)*log(x + 17413/115 44 - 108000*(1/12 - sqrt(3)*I/6)**4/481 + 4158*(1/12 - sqrt(3)*I/6)**2/481 + 1368*(1/12 - sqrt(3)*I/6)**3/481 + 943*sqrt(3)*I/5772) + (1/12 - sqrt(3 )/6)*log(x - 108000*(1/12 - sqrt(3)/6)**4/481 + 1368*(1/12 - sqrt(3)/6)**3 /481 + 943*sqrt(3)/5772 + 4158*(1/12 - sqrt(3)/6)**2/481 + 17413/11544) + (1/12 + sqrt(3)/6)*log(x - 108000*(1/12 + sqrt(3)/6)**4/481 - 943*sqrt(3)/ 5772 + 1368*(1/12 + sqrt(3)/6)**3/481 + 4158*(1/12 + sqrt(3)/6)**2/481 + 1 7413/11544)
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.12 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \left (x\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log((x - sqrt(3))/(x + sqr t(3))) + 1/12*log(x^2 + 3) + 1/12*log(x^2 - 3) - 1/3*log(x)
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.33 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left ({\left | x^{2} - 3 \right |}\right ) - \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/6*sqrt(3)*log(abs(2*x - 2*sqrt(3))/a bs(2*x + 2*sqrt(3))) + 1/12*log(x^2 + 3) + 1/12*log(abs(x^2 - 3)) - 1/3*lo g(abs(x))
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.52 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\ln \left (x-\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{12}\right )-\ln \left (x+\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{12}\right )-\frac {\ln \left (x\right )}{3}-\ln \left (x-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]