Optimal. Leaf size=86 \[ \frac {1}{12 a^4 c^3 (1-a x)^3}-\frac {1}{4 a^4 c^3 (1-a x)^2}+\frac {3}{16 a^4 c^3 (1-a x)}+\frac {1}{16 a^4 c^3 (1+a x)}+\frac {\tanh ^{-1}(a x)}{8 a^4 c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6285, 90, 213}
\begin {gather*} \frac {3}{16 a^4 c^3 (1-a x)}+\frac {1}{16 a^4 c^3 (a x+1)}-\frac {1}{4 a^4 c^3 (1-a x)^2}+\frac {1}{12 a^4 c^3 (1-a x)^3}+\frac {\tanh ^{-1}(a x)}{8 a^4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 213
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^3}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac {\int \left (\frac {1}{4 a^3 (-1+a x)^4}+\frac {1}{2 a^3 (-1+a x)^3}+\frac {3}{16 a^3 (-1+a x)^2}-\frac {1}{16 a^3 (1+a x)^2}-\frac {1}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=\frac {1}{12 a^4 c^3 (1-a x)^3}-\frac {1}{4 a^4 c^3 (1-a x)^2}+\frac {3}{16 a^4 c^3 (1-a x)}+\frac {1}{16 a^4 c^3 (1+a x)}-\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{8 a^3 c^3}\\ &=\frac {1}{12 a^4 c^3 (1-a x)^3}-\frac {1}{4 a^4 c^3 (1-a x)^2}+\frac {3}{16 a^4 c^3 (1-a x)}+\frac {1}{16 a^4 c^3 (1+a x)}+\frac {\tanh ^{-1}(a x)}{8 a^4 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 64, normalized size = 0.74 \begin {gather*} \frac {-2+7 a x-6 a^2 x^2-3 a^3 x^3+3 (-1+a x)^3 (1+a x) \tanh ^{-1}(a x)}{24 a^4 c^3 (-1+a x)^3 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 76, normalized size = 0.88
method | result | size |
default | \(\frac {\frac {1}{16 a^{4} \left (a x +1\right )}+\frac {\ln \left (a x +1\right )}{16 a^{4}}-\frac {1}{12 a^{4} \left (a x -1\right )^{3}}-\frac {1}{4 a^{4} \left (a x -1\right )^{2}}-\frac {3}{16 a^{4} \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{16 a^{4}}}{c^{3}}\) | \(76\) |
risch | \(\frac {-\frac {x^{3}}{8 a}-\frac {x^{2}}{4 a^{2}}+\frac {7 x}{24 a^{3}}-\frac {1}{12 a^{4}}}{c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}-\frac {\ln \left (a x -1\right )}{16 a^{4} c^{3}}+\frac {\ln \left (-a x -1\right )}{16 a^{4} c^{3}}\) | \(81\) |
norman | \(\frac {-\frac {x^{3}}{3 a c}-\frac {a \,x^{5}}{8 c}+\frac {x}{8 a^{3} c}-\frac {x^{4}}{4 c}-\frac {a^{2} x^{6}}{12 c}}{\left (a^{2} x^{2}-1\right )^{3} c^{2}}-\frac {\ln \left (a x -1\right )}{16 a^{4} c^{3}}+\frac {\ln \left (a x +1\right )}{16 a^{4} c^{3}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 94, normalized size = 1.09 \begin {gather*} -\frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \, {\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} + \frac {\log \left (a x + 1\right )}{16 \, a^{4} c^{3}} - \frac {\log \left (a x - 1\right )}{16 \, a^{4} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 123, normalized size = 1.43 \begin {gather*} -\frac {6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 4}{48 \, {\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 88, normalized size = 1.02 \begin {gather*} \frac {- 3 a^{3} x^{3} - 6 a^{2} x^{2} + 7 a x - 2}{24 a^{8} c^{3} x^{4} - 48 a^{7} c^{3} x^{3} + 48 a^{5} c^{3} x - 24 a^{4} c^{3}} + \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{16} + \frac {\log {\left (x + \frac {1}{a} \right )}}{16}}{a^{4} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 75, normalized size = 0.87 \begin {gather*} \frac {\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} a^{4} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 77, normalized size = 0.90 \begin {gather*} \frac {\frac {1}{12\,a^4}-\frac {7\,x}{24\,a^3}+\frac {x^3}{8\,a}+\frac {x^2}{4\,a^2}}{-a^4\,c^3\,x^4+2\,a^3\,c^3\,x^3-2\,a\,c^3\,x+c^3}+\frac {\mathrm {atanh}\left (a\,x\right )}{8\,a^4\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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