Optimal. Leaf size=19 \[ \frac {x}{2 \left (1+x^2\right )}+\frac {\arctan (x)}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 205, 209}
\begin {gather*} \frac {\arctan (x)}{2}+\frac {x}{2 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 205
Rule 209
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{2 \left (1+x^2\right )}+\frac {\arctan (x)}{2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{2} \left (\frac {x}{1+x^2}+\arctan (x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 16, normalized size = 0.84
method | result | size |
default | \(\frac {x}{2 x^{2}+2}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
risch | \(\frac {x}{2 x^{2}+2}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
parallelrisch | \(-\frac {i \ln \left (x -i\right ) x^{2}-i \ln \left (i+x \right ) x^{2}+i \ln \left (x -i\right )-i \ln \left (i+x \right )-2 x}{4 \left (x^{2}+1\right )}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.43, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.56, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 12, normalized size = 0.63 \begin {gather*} \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.46, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 16, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{2}+\frac {x}{2\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.58 \begin {gather*} -\frac {1}{2 x^{2}+2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________