Optimal. Leaf size=10 \[ -\frac {1}{3} \tanh ^{-1}\left (2+x^3\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 2.10, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 630, 31}
\begin {gather*} \frac {1}{6} \log \left (x^3+1\right )-\frac {1}{6} \log \left (x^3+3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 1366
Rubi steps
\begin {align*} \int \frac {x^2}{3+4 x^3+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^3\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{3+x} \, dx,x,x^3\right )\\ &=\frac {1}{6} \log \left (1+x^3\right )-\frac {1}{6} \log \left (3+x^3\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
time = 0.00, size = 21, normalized size = 2.10 \begin {gather*} \frac {1}{6} \log \left (1+x^3\right )-\frac {1}{6} \log \left (3+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(17\) vs.
\(2(8)=16\).
time = 0.02, size = 18, normalized size = 1.80
method | result | size |
default | \(\frac {\ln \left (x^{3}+1\right )}{6}-\frac {\ln \left (x^{3}+3\right )}{6}\) | \(18\) |
risch | \(\frac {\ln \left (x^{3}+1\right )}{6}-\frac {\ln \left (x^{3}+3\right )}{6}\) | \(18\) |
norman | \(\frac {\ln \left (1+x \right )}{6}-\frac {\ln \left (x^{3}+3\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{6}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs.
\(2 (8) = 16\).
time = 0.31, size = 17, normalized size = 1.70 \begin {gather*} -\frac {1}{6} \, \log \left (x^{3} + 3\right ) + \frac {1}{6} \, \log \left (x^{3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs.
\(2 (8) = 16\).
time = 0.34, size = 17, normalized size = 1.70 \begin {gather*} -\frac {1}{6} \, \log \left (x^{3} + 3\right ) + \frac {1}{6} \, \log \left (x^{3} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 1.50 \begin {gather*} \frac {\log {\left (x^{3} + 1 \right )}}{6} - \frac {\log {\left (x^{3} + 3 \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs.
\(2 (8) = 16\).
time = 4.27, size = 19, normalized size = 1.90 \begin {gather*} -\frac {1}{6} \, \log \left ({\left | x^{3} + 3 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 16, normalized size = 1.60 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {9}{2\,\left (8\,x^3+6\right )}+\frac {5}{4}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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