Optimal. Leaf size=79 \[ -\frac {x}{a^4 c^2}+\frac {1}{4 a^5 c^2 (1-a x)^2}-\frac {7}{4 a^5 c^2 (1-a x)}-\frac {17 \log (1-a x)}{8 a^5 c^2}+\frac {\log (1+a x)}{8 a^5 c^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 79, 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, 90}
\begin {gather*} -\frac {7}{4 a^5 c^2 (1-a x)}+\frac {1}{4 a^5 c^2 (1-a x)^2}-\frac {17 \log (1-a x)}{8 a^5 c^2}+\frac {\log (a x+1)}{8 a^5 c^2}-\frac {x}{a^4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^4}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (-\frac {1}{a^4}-\frac {1}{2 a^4 (-1+a x)^3}-\frac {7}{4 a^4 (-1+a x)^2}-\frac {17}{8 a^4 (-1+a x)}+\frac {1}{8 a^4 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac {x}{a^4 c^2}+\frac {1}{4 a^5 c^2 (1-a x)^2}-\frac {7}{4 a^5 c^2 (1-a x)}-\frac {17 \log (1-a x)}{8 a^5 c^2}+\frac {\log (1+a x)}{8 a^5 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.86 \begin {gather*} \frac {-\frac {x}{a^4}+\frac {1}{4 a^5 (1-a x)^2}-\frac {7}{4 a^5 (1-a x)}-\frac {17 \log (1-a x)}{8 a^5}+\frac {\log (1+a x)}{8 a^5}}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 58, normalized size = 0.73
method | result | size |
default | \(\frac {-\frac {x}{a^{4}}+\frac {\ln \left (a x +1\right )}{8 a^{5}}+\frac {7}{4 a^{5} \left (a x -1\right )}-\frac {17 \ln \left (a x -1\right )}{8 a^{5}}+\frac {1}{4 a^{5} \left (a x -1\right )^{2}}}{c^{2}}\) | \(58\) |
risch | \(-\frac {x}{a^{4} c^{2}}+\frac {\frac {7 c^{2} x}{4}-\frac {3 c^{2}}{2 a}}{a^{4} c^{4} \left (a x -1\right )^{2}}-\frac {17 \ln \left (a x -1\right )}{8 c^{2} a^{5}}+\frac {\ln \left (-a x -1\right )}{8 a^{5} c^{2}}\) | \(69\) |
norman | \(\frac {-\frac {x^{5}}{c}-\frac {9 x}{4 a^{4} c}+\frac {15 x^{3}}{4 a^{2} c}-\frac {3}{2 c \,a^{5}}+\frac {2 x^{2}}{a^{3} c}}{\left (a^{2} x^{2}-1\right )^{2} c}-\frac {17 \ln \left (a x -1\right )}{8 c^{2} a^{5}}+\frac {\ln \left (a x +1\right )}{8 a^{5} c^{2}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 75, normalized size = 0.95 \begin {gather*} \frac {7 \, a x - 6}{4 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right )}} - \frac {x}{a^{4} c^{2}} + \frac {\log \left (a x + 1\right )}{8 \, a^{5} c^{2}} - \frac {17 \, \log \left (a x - 1\right )}{8 \, a^{5} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 95, normalized size = 1.20 \begin {gather*} -\frac {8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 12}{8 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 71, normalized size = 0.90 \begin {gather*} - \frac {- 7 a x + 6}{4 a^{7} c^{2} x^{2} - 8 a^{6} c^{2} x + 4 a^{5} c^{2}} - \frac {x}{a^{4} c^{2}} - \frac {\frac {17 \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 61, normalized size = 0.77 \begin {gather*} -\frac {x}{a^{4} c^{2}} + \frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{5} c^{2}} - \frac {17 \, \log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{5} c^{2}} + \frac {7 \, a x - 6}{4 \, {\left (a x - 1\right )}^{2} a^{5} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 77, normalized size = 0.97 \begin {gather*} \frac {\frac {7\,x}{4}-\frac {3}{2\,a}}{a^6\,c^2\,x^2-2\,a^5\,c^2\,x+a^4\,c^2}-\frac {x}{a^4\,c^2}-\frac {17\,\ln \left (a\,x-1\right )}{8\,a^5\,c^2}+\frac {\ln \left (a\,x+1\right )}{8\,a^5\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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