Optimal. Leaf size=105 \[ \frac {1}{12 a^6 c^3 (1-a x)^3}-\frac {1}{2 a^6 c^3 (1-a x)^2}+\frac {23}{16 a^6 c^3 (1-a x)}+\frac {1}{16 a^6 c^3 (1+a x)}+\frac {13 \log (1-a x)}{16 a^6 c^3}+\frac {3 \log (1+a x)}{16 a^6 c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 105, 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 {23}{16 a^6 c^3 (1-a x)}+\frac {1}{16 a^6 c^3 (a x+1)}-\frac {1}{2 a^6 c^3 (1-a x)^2}+\frac {1}{12 a^6 c^3 (1-a x)^3}+\frac {13 \log (1-a x)}{16 a^6 c^3}+\frac {3 \log (a x+1)}{16 a^6 c^3} \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^5}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^5}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac {\int \left (\frac {1}{4 a^5 (-1+a x)^4}+\frac {1}{a^5 (-1+a x)^3}+\frac {23}{16 a^5 (-1+a x)^2}+\frac {13}{16 a^5 (-1+a x)}-\frac {1}{16 a^5 (1+a x)^2}+\frac {3}{16 a^5 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac {1}{12 a^6 c^3 (1-a x)^3}-\frac {1}{2 a^6 c^3 (1-a x)^2}+\frac {23}{16 a^6 c^3 (1-a x)}+\frac {1}{16 a^6 c^3 (1+a x)}+\frac {13 \log (1-a x)}{16 a^6 c^3}+\frac {3 \log (1+a x)}{16 a^6 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 87, normalized size = 0.83 \begin {gather*} \frac {-52+74 a x+36 a^2 x^2-66 a^3 x^3+39 (-1+a x)^3 (1+a x) \log (1-a x)+9 (-1+a x)^3 (1+a x) \log (1+a x)}{48 a^6 c^3 (-1+a x)^3 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.72
method | result | size |
default | \(\frac {\frac {1}{16 a^{6} \left (a x +1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a^{6}}-\frac {1}{2 a^{6} \left (a x -1\right )^{2}}-\frac {1}{12 a^{6} \left (a x -1\right )^{3}}-\frac {23}{16 a^{6} \left (a x -1\right )}+\frac {13 \ln \left (a x -1\right )}{16 a^{6}}}{c^{3}}\) | \(76\) |
risch | \(\frac {-\frac {11 x^{3}}{8 a^{3}}+\frac {3 x^{2}}{4 a^{4}}+\frac {37 x}{24 a^{5}}-\frac {13}{12 a^{6}}}{c^{3} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )}+\frac {3 \ln \left (a x +1\right )}{16 a^{6} c^{3}}+\frac {13 \ln \left (-a x +1\right )}{16 a^{6} c^{3}}\) | \(81\) |
norman | \(\frac {\frac {5 x^{3}}{3 a^{3} c}-\frac {11 x^{5}}{8 a c}-\frac {5 x}{8 c \,a^{5}}-\frac {2 x^{6}}{3 c}-\frac {5}{12 a^{6} c}+\frac {3 x^{2}}{4 a^{4} c}}{\left (a^{2} x^{2}-1\right )^{3} c^{2}}+\frac {13 \ln \left (a x -1\right )}{16 a^{6} c^{3}}+\frac {3 \ln \left (a x +1\right )}{16 a^{6} c^{3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 94, normalized size = 0.90 \begin {gather*} -\frac {33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \, {\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right )}} + \frac {3 \, \log \left (a x + 1\right )}{16 \, a^{6} c^{3}} + \frac {13 \, \log \left (a x - 1\right )}{16 \, a^{6} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 123, normalized size = 1.17 \begin {gather*} -\frac {66 \, a^{3} x^{3} - 36 \, a^{2} x^{2} - 74 \, a x - 9 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) - 39 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 52}{48 \, {\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 92, normalized size = 0.88 \begin {gather*} \frac {- 33 a^{3} x^{3} + 18 a^{2} x^{2} + 37 a x - 26}{24 a^{10} c^{3} x^{4} - 48 a^{9} c^{3} x^{3} + 48 a^{7} c^{3} x - 24 a^{6} c^{3}} + \frac {\frac {13 \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}}{a^{6} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 75, normalized size = 0.71 \begin {gather*} \frac {3 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6} c^{3}} + \frac {13 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6} c^{3}} - \frac {33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} a^{6} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 94, normalized size = 0.90 \begin {gather*} \frac {13\,\ln \left (a\,x-1\right )}{16\,a^6\,c^3}-\frac {\frac {37\,x}{24\,a^5}-\frac {13}{12\,a^6}-\frac {11\,x^3}{8\,a^3}+\frac {3\,x^2}{4\,a^4}}{-a^4\,c^3\,x^4+2\,a^3\,c^3\,x^3-2\,a\,c^3\,x+c^3}+\frac {3\,\ln \left (a\,x+1\right )}{16\,a^6\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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