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 = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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
Rubi steps
\begin {align*} \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 = 0.03, 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 = 0.73, size = 64, normalized size = 0.93
method | result | size |
risch | \(\frac {-\frac {a \,x^{2}}{8}+\frac {3 x}{8}-\frac {5}{12 a}}{\left (a x -1\right )^{3} c^{5}}+\frac {\ln \left (-a x -1\right )}{16 a \,c^{5}}-\frac {\ln \left (a x -1\right )}{16 a \,c^{5}}\) | \(57\) |
default | \(\frac {\frac {\ln \left (a x +1\right )}{16 a}-\frac {1}{6 a \left (a x -1\right )^{3}}+\frac {1}{8 \left (a x -1\right )^{2} a}-\frac {1}{8 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{16 a}}{c^{5}}\) | \(64\) |
norman | \(\frac {\frac {7 x}{8 c}-\frac {9 a \,x^{2}}{8 c}-\frac {11 a^{2} x^{3}}{24 c}+\frac {9 a^{3} x^{4}}{8 c}-\frac {5 a^{4} x^{5}}{12 c}}{\left (a x +1\right ) c^{4} \left (a x -1\right )^{4}}-\frac {\ln \left (a x -1\right )}{16 a \,c^{5}}+\frac {\ln \left (a x +1\right )}{16 a \,c^{5}}\) | \(97\) |
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 = 0.35, 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 = 0.22, size = 76, normalized size = 1.10 \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 = 0.42, 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 = 0.83, size = 64, normalized size = 0.93 \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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