Optimal. Leaf size=23 \[ -\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2098, 267, 209}
\begin {gather*} \text {ArcTan}(x)+\frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 267
Rule 2098
Rubi steps
\begin {align*} \int \frac {1-3 x+2 x^2-4 x^3+x^4}{1+3 x^2+3 x^4+x^6} \, dx &=\int \left (\frac {x}{\left (1+x^2\right )^3}-\frac {4 x}{\left (1+x^2\right )^2}+\frac {1}{1+x^2}\right ) \, dx\\ &=-\left (4 \int \frac {x}{\left (1+x^2\right )^2} \, dx\right )+\int \frac {x}{\left (1+x^2\right )^3} \, dx+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 19, normalized size = 0.83
method | result | size |
default | \(\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}}+\arctan \left (x \right )\) | \(19\) |
risch | \(\frac {2 x^{2}+\frac {7}{4}}{x^{4}+2 x^{2}+1}+\arctan \left (x \right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 24, normalized size = 1.04 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 35, normalized size = 1.52 \begin {gather*} \frac {8 \, x^{2} + 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.04, size = 20, normalized size = 0.87 \begin {gather*} \frac {8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.45, size = 19, normalized size = 0.83 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{2} + 1\right )}^{2}} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (x\right )+\frac {2\,x^2+\frac {7}{4}}{x^4+2\,x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________