Optimal. Leaf size=43 \[ \frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4}-\frac {2 \log (1+a x)}{a^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6261, 78}
\begin {gather*} -\frac {2 \log (a x+1)}{a^4}+\frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6261
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1-a x)}{1+a x} \, dx\\ &=\int \left (\frac {2}{a^3}-\frac {2 x}{a^2}+\frac {2 x^2}{a}-x^3-\frac {2}{a^3 (1+a x)}\right ) \, dx\\ &=\frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4}-\frac {2 \log (1+a x)}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4}-\frac {2 \log (1+a x)}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 43, normalized size = 1.00
method | result | size |
risch | \(\frac {2 x}{a^{3}}-\frac {x^{2}}{a^{2}}+\frac {2 x^{3}}{3 a}-\frac {x^{4}}{4}-\frac {2 \ln \left (a x +1\right )}{a^{4}}\) | \(40\) |
default | \(-\frac {\frac {1}{4} a^{3} x^{4}-\frac {2}{3} a^{2} x^{3}+a \,x^{2}-2 x}{a^{3}}-\frac {2 \ln \left (a x +1\right )}{a^{4}}\) | \(43\) |
norman | \(\frac {\frac {x^{2}}{a^{2}}+\frac {2 x}{a^{3}}+\frac {5 x^{4}}{12}-\frac {a \,x^{5}}{4}-\frac {x^{3}}{3 a}}{a x +1}-\frac {2 \ln \left (a x +1\right )}{a^{4}}\) | \(54\) |
meijerg | \(-\frac {-\frac {a x \left (-3 a^{4} x^{4}+5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{12 \left (a x +1\right )}+5 \ln \left (a x +1\right )}{a^{4}}+\frac {-\frac {a x \left (-2 a^{2} x^{2}+6 a x +12\right )}{4 \left (a x +1\right )}+3 \ln \left (a x +1\right )}{a^{4}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 43, normalized size = 1.00 \begin {gather*} -\frac {3 \, a^{3} x^{4} - 8 \, a^{2} x^{3} + 12 \, a x^{2} - 24 \, x}{12 \, a^{3}} - \frac {2 \, \log \left (a x + 1\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 42, normalized size = 0.98 \begin {gather*} -\frac {3 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 24 \, a x + 24 \, \log \left (a x + 1\right )}{12 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 37, normalized size = 0.86 \begin {gather*} - \frac {x^{4}}{4} + \frac {2 x^{3}}{3 a} - \frac {x^{2}}{a^{2}} + \frac {2 x}{a^{3}} - \frac {2 \log {\left (a x + 1 \right )}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 66, normalized size = 1.53 \begin {gather*} \frac {{\left (a x + 1\right )}^{4} {\left (\frac {20}{a x + 1} - \frac {54}{{\left (a x + 1\right )}^{2}} + \frac {84}{{\left (a x + 1\right )}^{3}} - 3\right )}}{12 \, a^{4}} + \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 39, normalized size = 0.91 \begin {gather*} \frac {2\,x}{a^3}-\frac {2\,\ln \left (a\,x+1\right )}{a^4}-\frac {x^4}{4}+\frac {2\,x^3}{3\,a}-\frac {x^2}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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