Optimal. Leaf size=14 \[ \frac {1}{3 (1+x)^3}+\log (1+x) \]
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Rubi [A]
time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1608, 1694, 14}
\begin {gather*} \frac {1}{3 (x+1)^3}+\log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1608
Rule 1694
Rubi steps
\begin {align*} \int \frac {3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx &=\int \frac {x \left (3+3 x+x^2\right )}{1+4 x+6 x^2+4 x^3+x^4} \, dx\\ &=\text {Subst}\left (\int \frac {-1+x^3}{x^4} \, dx,x,1+x\right )\\ &=\text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {1}{x}\right ) \, dx,x,1+x\right )\\ &=\frac {1}{3 (1+x)^3}+\log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{3 (1+x)^3}+\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 13, normalized size = 0.93
method | result | size |
default | \(\frac {1}{3 \left (1+x \right )^{3}}+\ln \left (1+x \right )\) | \(13\) |
norman | \(\frac {1}{3 \left (1+x \right )^{3}}+\ln \left (1+x \right )\) | \(13\) |
risch | \(\frac {1}{3 x^{3}+9 x^{2}+9 x +3}+\ln \left (1+x \right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 1.57 \begin {gather*} \frac {1}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 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. 38 vs.
\(2 (12) = 24\).
time = 0.40, size = 38, normalized size = 2.71 \begin {gather*} \frac {3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 1}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 1.43 \begin {gather*} \log {\left (x + 1 \right )} + \frac {1}{3 x^{3} + 9 x^{2} + 9 x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.74, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{3 \, {\left (x + 1\right )}^{3}} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (x+1\right )+\frac {1}{3\,{\left (x+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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