Optimal. Leaf size=18 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {272, 45}
\begin {gather*} \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{1+x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}-\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 15, normalized size = 0.83
method | result | size |
default | \(\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(15\) |
norman | \(\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(15\) |
meijerg | \(\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(15\) |
risch | \(\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.62, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.12, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.67 \begin {gather*} \frac {x^{2}}{2} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 14, normalized size = 0.78 \begin {gather*} \frac {x^2}{2}-\frac {\ln \left (x^2+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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