Optimal. Leaf size=63 \[ \frac {3 x}{a^4 c}+\frac {x^2}{a^3 c}+\frac {x^3}{3 a^2 c}+\frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c} \]
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Rubi [A]
time = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 45}
\begin {gather*} \frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c}+\frac {3 x}{a^4 c}+\frac {x^2}{a^3 c}+\frac {x^3}{3 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^4}{(1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {3}{a^4}+\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {1}{a^4 (-1+a x)^2}+\frac {4}{a^4 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {3 x}{a^4 c}+\frac {x^2}{a^3 c}+\frac {x^3}{3 a^2 c}+\frac {1}{a^5 c (1-a x)}+\frac {4 \log (1-a x)}{a^5 c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 49, normalized size = 0.78 \begin {gather*} \frac {9 a x+3 a^2 x^2+a^3 x^3+\frac {3}{1-a x}+12 \log (1-a x)}{3 a^5 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 50, normalized size = 0.79
method | result | size |
default | \(\frac {\frac {\frac {1}{3} a^{2} x^{3}+x^{2} a +3 x}{a^{4}}+\frac {4 \ln \left (a x -1\right )}{a^{5}}-\frac {1}{a^{5} \left (a x -1\right )}}{c}\) | \(50\) |
risch | \(\frac {x^{3}}{3 a^{2} c}+\frac {x^{2}}{a^{3} c}+\frac {3 x}{a^{4} c}-\frac {1}{a^{5} c \left (a x -1\right )}+\frac {4 \ln \left (a x -1\right )}{c \,a^{5}}\) | \(61\) |
norman | \(\frac {\frac {x^{4}}{a c}+\frac {x^{5}}{3 c}-\frac {4 x}{a^{4} c}+\frac {8 x^{3}}{3 a^{2} c}-\frac {2}{c \,a^{5}}}{a^{2} x^{2}-1}+\frac {4 \ln \left (a x -1\right )}{c \,a^{5}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 57, normalized size = 0.90 \begin {gather*} -\frac {1}{a^{6} c x - a^{5} c} + \frac {a^{2} x^{3} + 3 \, a x^{2} + 9 \, x}{3 \, a^{4} c} + \frac {4 \, \log \left (a x - 1\right )}{a^{5} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 59, normalized size = 0.94 \begin {gather*} \frac {a^{4} x^{4} + 2 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 9 \, a x + 12 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 3}{3 \, {\left (a^{6} c x - a^{5} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 53, normalized size = 0.84 \begin {gather*} - \frac {1}{a^{6} c x - a^{5} c} + \frac {x^{3}}{3 a^{2} c} + \frac {x^{2}}{a^{3} c} + \frac {3 x}{a^{4} c} + \frac {4 \log {\left (a x - 1 \right )}}{a^{5} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 70, normalized size = 1.11 \begin {gather*} \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{5} c} - \frac {1}{{\left (a x - 1\right )} a^{5} c} + \frac {a^{4} c^{2} x^{3} + 3 \, a^{3} c^{2} x^{2} + 9 \, a^{2} c^{2} x}{3 \, a^{6} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 64, normalized size = 1.02 \begin {gather*} \frac {1}{a\,\left (a^4\,c-a^5\,c\,x\right )}+\frac {3\,x}{a^4\,c}+\frac {x^3}{3\,a^2\,c}+\frac {x^2}{a^3\,c}+\frac {4\,\ln \left (a\,x-1\right )}{a^5\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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