Optimal. Leaf size=84 \[ -\frac {1}{16 a c^3 (1-a x)}+\frac {3}{16 a c^3 (a x+1)}+\frac {1}{8 a c^3 (a x+1)^2}+\frac {1}{12 a c^3 (a x+1)^3}-\frac {\tanh ^{-1}(a x)}{4 a c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6140, 44, 207} \[ -\frac {1}{16 a c^3 (1-a x)}+\frac {3}{16 a c^3 (a x+1)}+\frac {1}{8 a c^3 (a x+1)^2}+\frac {1}{12 a c^3 (a x+1)^3}-\frac {\tanh ^{-1}(a x)}{4 a c^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\\ &=-\frac {\int \frac {1}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=-\frac {\int \left (\frac {1}{16 (-1+a x)^2}+\frac {1}{4 (1+a x)^4}+\frac {1}{4 (1+a x)^3}+\frac {3}{16 (1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=-\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}+\frac {1}{8 a c^3 (1+a x)^2}+\frac {3}{16 a c^3 (1+a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^3}\\ &=-\frac {1}{16 a c^3 (1-a x)}+\frac {1}{12 a c^3 (1+a x)^3}+\frac {1}{8 a c^3 (1+a x)^2}+\frac {3}{16 a c^3 (1+a x)}-\frac {\tanh ^{-1}(a x)}{4 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 61, normalized size = 0.73 \[ \frac {3 a^3 x^3+6 a^2 x^2+a x-3 (a x-1) (a x+1)^3 \tanh ^{-1}(a x)-4}{12 a (a x-1) (a c x+c)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 121, normalized size = 1.44 \[ \frac {6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 2 \, a x - 3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x - 1\right ) - 8}{24 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 74, normalized size = 0.88 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{3}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{3}} + \frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \, {\left (a x + 1\right )}^{3} {\left (a x - 1\right )} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 90, normalized size = 1.07 \[ \frac {1}{16 a \,c^{3} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 c^{3} a}+\frac {1}{12 a \,c^{3} \left (a x +1\right )^{3}}+\frac {1}{8 a \,c^{3} \left (a x +1\right )^{2}}+\frac {3}{16 a \,c^{3} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 91, normalized size = 1.08 \[ \frac {3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + a x - 4}{12 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{3}} + \frac {\log \left (a x - 1\right )}{8 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 73, normalized size = 0.87 \[ -\frac {\frac {x}{12}+\frac {a\,x^2}{2}-\frac {1}{3\,a}+\frac {a^2\,x^3}{4}}{-a^4\,c^3\,x^4-2\,a^3\,c^3\,x^3+2\,a\,c^3\,x+c^3}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 85, normalized size = 1.01 \[ - \frac {- 3 a^{3} x^{3} - 6 a^{2} x^{2} - a x + 4}{12 a^{5} c^{3} x^{4} + 24 a^{4} c^{3} x^{3} - 24 a^{2} c^{3} x - 12 a c^{3}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{8} + \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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