Optimal. Leaf size=49 \[ -\frac {x}{a^4}-\frac {x^2}{2 a^3}-\frac {x^3}{3 a^2}-\frac {x^4}{4 a}-\frac {\log (1-a x)}{a^5} \]
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Rubi [A]
time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6285, 45}
\begin {gather*} -\frac {\log (1-a x)}{a^5}-\frac {x}{a^4}-\frac {x^2}{2 a^3}-\frac {x^3}{3 a^2}-\frac {x^4}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x^4}{1-a x} \, dx\\ &=\int \left (-\frac {1}{a^4}-\frac {x}{a^3}-\frac {x^2}{a^2}-\frac {x^3}{a}-\frac {1}{a^4 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{a^4}-\frac {x^2}{2 a^3}-\frac {x^3}{3 a^2}-\frac {x^4}{4 a}-\frac {\log (1-a x)}{a^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} -\frac {x}{a^4}-\frac {x^2}{2 a^3}-\frac {x^3}{3 a^2}-\frac {x^4}{4 a}-\frac {\log (1-a x)}{a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 42, normalized size = 0.86
method | result | size |
default | \(-\frac {\frac {1}{4} a^{3} x^{4}+\frac {1}{3} a^{2} x^{3}+\frac {1}{2} x^{2} a +x}{a^{4}}-\frac {\ln \left (a x -1\right )}{a^{5}}\) | \(42\) |
norman | \(-\frac {x}{a^{4}}-\frac {x^{2}}{2 a^{3}}-\frac {x^{3}}{3 a^{2}}-\frac {x^{4}}{4 a}-\frac {\ln \left (a x -1\right )}{a^{5}}\) | \(43\) |
risch | \(-\frac {x}{a^{4}}-\frac {x^{2}}{2 a^{3}}-\frac {x^{3}}{3 a^{2}}-\frac {x^{4}}{4 a}-\frac {\ln \left (a x -1\right )}{a^{5}}\) | \(43\) |
meijerg | \(-\frac {\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{6}+\ln \left (-a^{2} x^{2}+1\right )}{2 a^{5}}+\frac {-\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (5 a^{2} x^{2}+15\right )}{15 a^{4}}+\frac {2 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}}{2 a^{4} \sqrt {-a^{2}}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 43, normalized size = 0.88 \begin {gather*} -\frac {3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac {\log \left (a x - 1\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 42, normalized size = 0.86 \begin {gather*} -\frac {3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x + 12 \, \log \left (a x - 1\right )}{12 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 39, normalized size = 0.80 \begin {gather*} - \frac {x^{4}}{4 a} - \frac {x^{3}}{3 a^{2}} - \frac {x^{2}}{2 a^{3}} - \frac {x}{a^{4}} - \frac {\log {\left (a x - 1 \right )}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 44, normalized size = 0.90 \begin {gather*} -\frac {3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 42, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (a\,x-1\right )}{a^5}-\frac {x}{a^4}-\frac {x^4}{4\,a}-\frac {x^3}{3\,a^2}-\frac {x^2}{2\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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