Optimal. Leaf size=68 \[ \frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (x+1)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (x+1)+\frac{1}{144} \log (x+2) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.057872, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1586, 2074} \[ \frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (x+1)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (x+1)+\frac{1}{144} \log (x+2) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1586
Rule 2074
Rubi steps
\begin{align*} \int \frac{2+x}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{1}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac{1}{36 (-2+x)^2}-\frac{35}{432 (-2+x)}+\frac{1}{12 (-1+x)^2}+\frac{1}{18 (-1+x)}+\frac{1}{36 (1+x)^2}+\frac{1}{54 (1+x)}+\frac{1}{144 (2+x)}\right ) \, dx\\ &=\frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (1+x)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (1+x)+\frac{1}{144} \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0288866, size = 60, normalized size = 0.88 \[ \frac{1}{432} \left (\frac{12 \left (-5 x^2+6 x+5\right )}{x^3-2 x^2-x+2}+24 \log (1-x)-35 \log (2-x)+8 \log (x+1)+3 \log (x+2)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 47, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( 2+x \right ) }{144}}-{\frac{1}{36+36\,x}}+{\frac{\ln \left ( 1+x \right ) }{54}}-{\frac{1}{36\,x-72}}-{\frac{35\,\ln \left ( x-2 \right ) }{432}}-{\frac{1}{12\,x-12}}+{\frac{\ln \left ( x-1 \right ) }{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.95761, size = 70, normalized size = 1.03 \begin{align*} -\frac{5 \, x^{2} - 6 \, x - 5}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} + \frac{1}{144} \, \log \left (x + 2\right ) + \frac{1}{54} \, \log \left (x + 1\right ) + \frac{1}{18} \, \log \left (x - 1\right ) - \frac{35}{432} \, \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.73935, size = 271, normalized size = 3.99 \begin{align*} -\frac{60 \, x^{2} - 3 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 2\right ) - 8 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 1\right ) - 24 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 1\right ) + 35 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 2\right ) - 72 \, x - 60}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.279434, size = 53, normalized size = 0.78 \begin{align*} - \frac{5 x^{2} - 6 x - 5}{36 x^{3} - 72 x^{2} - 36 x + 72} - \frac{35 \log{\left (x - 2 \right )}}{432} + \frac{\log{\left (x - 1 \right )}}{18} + \frac{\log{\left (x + 1 \right )}}{54} + \frac{\log{\left (x + 2 \right )}}{144} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10012, size = 76, normalized size = 1.12 \begin{align*} -\frac{5 \, x^{2} - 6 \, x - 5}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} + \frac{1}{144} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{54} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{18} \, \log \left ({\left | x - 1 \right |}\right ) - \frac{35}{432} \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]