Optimal. Leaf size=69 \[ -\frac {1}{6 a c^5 (1-a x)^3}-\frac {1}{8 a c^5 (1-a x)^2}-\frac {1}{8 a c^5 (1-a x)}-\frac {\tanh ^{-1}(a x)}{8 a c^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6264, 46,
213} \begin {gather*} -\frac {1}{8 a c^5 (1-a x)}-\frac {1}{8 a c^5 (1-a x)^2}-\frac {1}{6 a c^5 (1-a x)^3}-\frac {\tanh ^{-1}(a x)}{8 a c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 6264
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx\\ &=-\frac {\int \frac {1}{(1-a x)^4 (1+a x)} \, dx}{c^5}\\ &=-\frac {\int \left (\frac {1}{2 (-1+a x)^4}-\frac {1}{4 (-1+a x)^3}+\frac {1}{8 (-1+a x)^2}-\frac {1}{8 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^5}\\ &=-\frac {1}{6 a c^5 (1-a x)^3}-\frac {1}{8 a c^5 (1-a x)^2}-\frac {1}{8 a c^5 (1-a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{8 c^5}\\ &=-\frac {1}{6 a c^5 (1-a x)^3}-\frac {1}{8 a c^5 (1-a x)^2}-\frac {1}{8 a c^5 (1-a x)}-\frac {\tanh ^{-1}(a x)}{8 a c^5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 44, normalized size = 0.64 \begin {gather*} \frac {10-9 a x+3 a^2 x^2-3 (-1+a x)^3 \tanh ^{-1}(a x)}{24 a c^5 (-1+a x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 64, normalized size = 0.93
method | result | size |
risch | \(\frac {\frac {a \,x^{2}}{8}-\frac {3 x}{8}+\frac {5}{12 a}}{\left (a x -1\right )^{3} c^{5}}+\frac {\ln \left (-a x +1\right )}{16 c^{5} a}-\frac {\ln \left (a x +1\right )}{16 c^{5} a}\) | \(57\) |
default | \(\frac {-\frac {\ln \left (a x +1\right )}{16 a}+\frac {1}{6 a \left (a x -1\right )^{3}}-\frac {1}{8 a \left (a x -1\right )^{2}}+\frac {1}{8 a \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{16 a}}{c^{5}}\) | \(64\) |
norman | \(\frac {-\frac {7 x}{8 c}+\frac {2 a \,x^{2}}{c}-\frac {37 a^{2} x^{3}}{24 c}+\frac {5 a^{3} x^{4}}{12 c}}{c^{4} \left (a x -1\right )^{4}}+\frac {\ln \left (a x -1\right )}{16 c^{5} a}-\frac {\ln \left (a x +1\right )}{16 c^{5} a}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 84, normalized size = 1.22 \begin {gather*} \frac {3 \, a^{2} x^{2} - 9 \, a x + 10}{24 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {\log \left (a x + 1\right )}{16 \, a c^{5}} + \frac {\log \left (a x - 1\right )}{16 \, a c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 113, normalized size = 1.64 \begin {gather*} \frac {6 \, a^{2} x^{2} - 18 \, a x - 3 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 20}{48 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 78, normalized size = 1.13 \begin {gather*} - \frac {- 3 a^{2} x^{2} + 9 a x - 10}{24 a^{4} c^{5} x^{3} - 72 a^{3} c^{5} x^{2} + 72 a^{2} c^{5} x - 24 a c^{5}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{16} + \frac {\log {\left (x + \frac {1}{a} \right )}}{16}}{a c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 89, normalized size = 1.29 \begin {gather*} -\frac {\log \left ({\left | -\frac {2 \, c}{a c x - c} - 1 \right |}\right )}{16 \, a c^{5}} + \frac {\frac {3 \, a^{2} c^{2}}{a c x - c} - \frac {3 \, a^{2} c^{3}}{{\left (a c x - c\right )}^{2}} + \frac {4 \, a^{2} c^{4}}{{\left (a c x - c\right )}^{3}}}{24 \, a^{3} c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 65, normalized size = 0.94 \begin {gather*} -\frac {\frac {a\,x^2}{8}-\frac {3\,x}{8}+\frac {5}{12\,a}}{-a^3\,c^5\,x^3+3\,a^2\,c^5\,x^2-3\,a\,c^5\,x+c^5}-\frac {\mathrm {atanh}\left (a\,x\right )}{8\,a\,c^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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