Optimal. Leaf size=71 \[ \frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2) \]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1586, 6742} \begin {gather*} \frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1586
Rule 6742
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {d+2 e}{48 (-2+x)}+\frac {-d-e}{18 (-1+x)}+\frac {d-e}{6 (1+x)}+\frac {-d+2 e}{12 (2+x)^2}+\frac {-19 d+26 e}{144 (2+x)}\right ) \, dx\\ &=\frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 66, normalized size = 0.93 \begin {gather*} \frac {1}{144} \left (\frac {12 (d-2 e)}{x+2}+24 (d-e) \log (-x-1)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(26 e-19 d) \log (x+2)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 93, normalized size = 1.31 \begin {gather*} -\frac {{\left ({\left (19 \, d - 26 \, e\right )} x + 38 \, d - 52 \, e\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e\right )} x + 2 \, d - 2 \, e\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e}{144 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 66, normalized size = 0.93 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 74, normalized size = 1.04 \begin {gather*} -\frac {19 d \ln \left (x +2\right )}{144}+\frac {d \ln \left (x -2\right )}{48}-\frac {d \ln \left (x -1\right )}{18}+\frac {d \ln \left (x +1\right )}{6}+\frac {13 e \ln \left (x +2\right )}{72}+\frac {e \ln \left (x -2\right )}{24}-\frac {e \ln \left (x -1\right )}{18}-\frac {e \ln \left (x +1\right )}{6}+\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 57, normalized size = 0.80 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.81, size = 64, normalized size = 0.90 \begin {gather*} \frac {\frac {d}{12}-\frac {e}{6}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 10.54, size = 1188, normalized size = 16.73
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________