Optimal. Leaf size=57 \[ -\frac {1}{6} (d+4 f) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f) \tanh ^{-1}(x)-\frac {1}{6} (e+g) \log \left (1-x^2\right )+\frac {1}{6} (e+4 g) \log \left (4-x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1673, 1166, 207, 1247, 632, 31} \[ -\frac {1}{6} (d+4 f) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f) \tanh ^{-1}(x)-\frac {1}{6} (e+g) \log \left (1-x^2\right )+\frac {1}{6} (e+4 g) \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 207
Rule 632
Rule 1166
Rule 1247
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3}{4-5 x^2+x^4} \, dx &=\int \frac {d+f x^2}{4-5 x^2+x^4} \, dx+\int \frac {x \left (e+g x^2\right )}{4-5 x^2+x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+g x}{4-5 x+x^2} \, dx,x,x^2\right )-\frac {1}{3} (d+f) \int \frac {1}{-1+x^2} \, dx+\frac {1}{3} (d+4 f) \int \frac {1}{-4+x^2} \, dx\\ &=-\frac {1}{6} (d+4 f) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f) \tanh ^{-1}(x)+\frac {1}{6} (-e-g) \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )+\frac {1}{6} (e+4 g) \operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )\\ &=-\frac {1}{6} (d+4 f) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f) \tanh ^{-1}(x)-\frac {1}{6} (e+g) \log \left (1-x^2\right )+\frac {1}{6} (e+4 g) \log \left (4-x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 68, normalized size = 1.19 \[ \frac {1}{12} (-2 \log (1-x) (d+e+f+g)+\log (2-x) (d+2 e+4 f+8 g)+2 \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.48, size = 61, normalized size = 1.07 \[ -\frac {1}{12} \, {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 69, normalized size = 1.21 \[ -\frac {1}{12} \, {\left (d + 4 \, f - 8 \, g - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d + f - g - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, {\left (d + f + g + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, {\left (d + 4 \, f + 8 \, g + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 114, normalized size = 2.00 \[ -\frac {d \ln \left (x +2\right )}{12}+\frac {d \ln \left (x -2\right )}{12}-\frac {d \ln \left (x -1\right )}{6}+\frac {d \ln \left (x +1\right )}{6}+\frac {e \ln \left (x +2\right )}{6}+\frac {e \ln \left (x -2\right )}{6}-\frac {e \ln \left (x -1\right )}{6}-\frac {e \ln \left (x +1\right )}{6}-\frac {f \ln \left (x +2\right )}{3}+\frac {f \ln \left (x -2\right )}{3}-\frac {f \ln \left (x -1\right )}{6}+\frac {f \ln \left (x +1\right )}{6}+\frac {2 g \ln \left (x +2\right )}{3}+\frac {2 g \ln \left (x -2\right )}{3}-\frac {g \ln \left (x -1\right )}{6}-\frac {g \ln \left (x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 61, normalized size = 1.07 \[ -\frac {1}{12} \, {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.74, size = 75, normalized size = 1.32 \[ \ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{6}+\frac {e}{6}+\frac {f}{6}+\frac {g}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}+\frac {2\,g}{3}\right )-\ln \left (x+2\right )\,\left (\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________