Optimal. Leaf size=84 \[ \frac {1}{16 a c^3 (1-a x)}-\frac {1}{12 a c^3 (1+a x)^3}-\frac {1}{8 a c^3 (1+a x)^2}-\frac {3}{16 a c^3 (1+a x)}+\frac {\tanh ^{-1}(a x)}{4 a c^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6275, 46, 213}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 6275
Rubi steps
\begin {align*} \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 = 0.03, size = 61, normalized size = 0.73 \begin {gather*} -\frac {-4+a x+6 a^2 x^2+3 a^3 x^3-3 (-1+a x) (1+a x)^3 \tanh ^{-1}(a x)}{12 a (-1+a x) (c+a c x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.90
method | result | size |
default | \(\frac {-\frac {1}{12 a \left (a x +1\right )^{3}}-\frac {1}{8 a \left (a x +1\right )^{2}}-\frac {3}{16 a \left (a x +1\right )}+\frac {\ln \left (a x +1\right )}{8 a}-\frac {1}{16 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{8 a}}{c^{3}}\) | \(76\) |
risch | \(\frac {-\frac {a^{2} x^{3}}{4}-\frac {x^{2} a}{2}-\frac {x}{12}+\frac {1}{3 a}}{\left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right ) c^{3}}+\frac {\ln \left (-a x -1\right )}{8 a \,c^{3}}-\frac {\ln \left (a x -1\right )}{8 a \,c^{3}}\) | \(76\) |
norman | \(\frac {\frac {3 x}{4 c}+\frac {a \,x^{2}}{2 c}-\frac {7 a^{2} x^{3}}{6 c}-\frac {5 a^{3} x^{4}}{6 c}+\frac {5 a^{4} x^{5}}{12 c}+\frac {a^{5} x^{6}}{3 c}}{\left (a x -1\right )^{2} c^{2} \left (a x +1\right )^{4}}-\frac {\ln \left (a x -1\right )}{8 a \,c^{3}}+\frac {\ln \left (a x +1\right )}{8 a \,c^{3}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 1.08 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 121, normalized size = 1.44 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 83, normalized size = 0.99 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 97, normalized size = 1.15 \begin {gather*} -\frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{3}} + \frac {1}{32 \, a c^{3} {\left (\frac {2}{a x + 1} - 1\right )}} - \frac {\frac {9 \, a^{5} c^{6}}{a x + 1} + \frac {6 \, a^{5} c^{6}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, a^{5} c^{6}}{{\left (a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 72, normalized size = 0.86 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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