Integrand size = 16, antiderivative size = 26 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=3 c x-\frac {1}{2} a c x^2-\frac {4 c \log (1+a x)}{a} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6302, 6264, 45} \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} a c x^2-\frac {4 c \log (a x+1)}{a}+3 c x \]
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Rule 45
Rule 6264
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} (c-a c x) \, dx \\ & = -\left (c \int \frac {(1-a x)^2}{1+a x} \, dx\right ) \\ & = -\left (c \int \left (-3+a x+\frac {4}{1+a x}\right ) \, dx\right ) \\ & = 3 c x-\frac {1}{2} a c x^2-\frac {4 c \log (1+a x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=3 c x-\frac {1}{2} a c x^2-\frac {4 c \log (1+a x)}{a} \]
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Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
default | \(c \left (-\frac {a \,x^{2}}{2}+3 x -\frac {4 \ln \left (a x +1\right )}{a}\right )\) | \(24\) |
norman | \(3 c x -\frac {a c \,x^{2}}{2}-\frac {4 c \ln \left (a x +1\right )}{a}\) | \(25\) |
risch | \(3 c x -\frac {a c \,x^{2}}{2}-\frac {4 c \ln \left (a x +1\right )}{a}\) | \(25\) |
parallelrisch | \(-\frac {a^{2} c \,x^{2}-6 a c x +8 c \ln \left (a x +1\right )}{2 a}\) | \(29\) |
meijerg | \(-\frac {c \left (-\frac {a x \left (-3 a x +6\right )}{6}+\ln \left (a x +1\right )\right )}{a}+\frac {2 c \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {c \ln \left (a x +1\right )}{a}\) | \(55\) |
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none
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {a^{2} c x^{2} - 6 \, a c x + 8 \, c \log \left (a x + 1\right )}{2 \, a} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=- \frac {a c x^{2}}{2} + 3 c x - \frac {4 c \log {\left (a x + 1 \right )}}{a} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {1}{2} \, a c x^{2} + 3 \, c x - \frac {4 \, c \log \left (a x + 1\right )}{a} \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} - \frac {a^{3} c x^{2} - 6 \, a^{2} c x}{2 \, a^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx=-\frac {c\,\left (8\,\ln \left (a\,x+1\right )-6\,a\,x+a^2\,x^2\right )}{2\,a} \]
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