Optimal. Leaf size=74 \[ -\frac {x}{c^2}-\frac {1}{4 a c^2 (1+a x)^2}+\frac {7}{4 a c^2 (1+a x)}-\frac {\log (1-a x)}{8 a c^2}+\frac {17 \log (1+a x)}{8 a c^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 74, 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 = {6292, 6285, 90}
\begin {gather*} \frac {7}{4 a c^2 (a x+1)}-\frac {1}{4 a c^2 (a x+1)^2}-\frac {\log (1-a x)}{8 a c^2}+\frac {17 \log (a x+1)}{8 a c^2}-\frac {x}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=\frac {a^4 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac {a^4 \int \frac {x^4}{(1-a x) (1+a x)^3} \, dx}{c^2}\\ &=\frac {a^4 \int \left (-\frac {1}{a^4}-\frac {1}{8 a^4 (-1+a x)}+\frac {1}{2 a^4 (1+a x)^3}-\frac {7}{4 a^4 (1+a x)^2}+\frac {17}{8 a^4 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac {x}{c^2}-\frac {1}{4 a c^2 (1+a x)^2}+\frac {7}{4 a c^2 (1+a x)}-\frac {\log (1-a x)}{8 a c^2}+\frac {17 \log (1+a x)}{8 a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.92 \begin {gather*} \frac {12+6 a x-16 a^2 x^2-8 a^3 x^3-(1+a x)^2 \log (1-a x)+17 (1+a x)^2 \log (1+a x)}{8 a (c+a c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 61, normalized size = 0.82
method | result | size |
default | \(\frac {a^{4} \left (-\frac {x}{a^{4}}-\frac {1}{4 a^{5} \left (a x +1\right )^{2}}+\frac {7}{4 a^{5} \left (a x +1\right )}+\frac {17 \ln \left (a x +1\right )}{8 a^{5}}-\frac {\ln \left (a x -1\right )}{8 a^{5}}\right )}{c^{2}}\) | \(61\) |
risch | \(-\frac {x}{c^{2}}+\frac {\frac {7 x \,c^{2}}{4}+\frac {3 c^{2}}{2 a}}{c^{4} \left (a x +1\right )^{2}}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {17 \ln \left (-a x -1\right )}{8 a \,c^{2}}\) | \(63\) |
norman | \(\frac {\frac {7 a \,x^{2}}{2 c}-\frac {a^{4} x^{5}}{c}+\frac {9 x}{4 c}-\frac {5 a^{2} x^{3}}{4 c}-\frac {7 a^{3} x^{4}}{2 c}}{\left (a x +1\right )^{3} c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {17 \ln \left (a x +1\right )}{8 a \,c^{2}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 70, normalized size = 0.95 \begin {gather*} \frac {7 \, a x + 6}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac {x}{c^{2}} + \frac {17 \, \log \left (a x + 1\right )}{8 \, a c^{2}} - \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 92, normalized size = 1.24 \begin {gather*} -\frac {8 \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 6 \, a x - 17 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 12}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 76, normalized size = 1.03 \begin {gather*} - a^{4} \left (\frac {- 7 a x - 6}{4 a^{7} c^{2} x^{2} + 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac {x}{a^{4} c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {17 \log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 101, normalized size = 1.36 \begin {gather*} -\frac {a x + 1}{a c^{2}} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c^{2}} - \frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{2}} + \frac {\frac {7 \, a^{5} c^{2}}{a x + 1} - \frac {a^{5} c^{2}}{{\left (a x + 1\right )}^{2}}}{4 \, a^{6} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 68, normalized size = 0.92 \begin {gather*} \frac {\frac {7\,x}{4}+\frac {3}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2}-\frac {x}{c^2}-\frac {\ln \left (a\,x-1\right )}{8\,a\,c^2}+\frac {17\,\ln \left (a\,x+1\right )}{8\,a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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