Integrand size = 25, antiderivative size = 12 \[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=-\frac {e^{\text {sech}^{-1}(a x)} x}{a} \]
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Leaf count is larger than twice the leaf count of optimal. \(26\) vs. \(2(12)=24\).
Time = 0.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 266, 6476, 1972, 75} \[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=-\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{a x+1}}} \]
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Rule 75
Rule 266
Rule 1972
Rule 6476
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{-1+a^2 x^2}-\frac {a e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2}\right ) \, dx \\ & = -\left (a \int \frac {e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2} \, dx\right )+\int \frac {x}{-1+a^2 x^2} \, dx \\ & = \frac {\log \left (1-a^2 x^2\right )}{2 a^2}+\int \frac {x \sqrt {\frac {1}{1+a x}}}{\sqrt {1-a x}} \, dx-\int \frac {x}{-1+a^2 x^2} \, dx \\ & = \left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = -\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{1+a x}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(28\) vs. \(2(12)=24\).
Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.33 \[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a^2} \]
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Time = 0.73 (sec) , antiderivative size = 36, normalized size of antiderivative = 3.00
method | result | size |
gosper | \(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\) | \(36\) |
default | \(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\) | \(36\) |
risch | \(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\) | \(36\) |
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none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.92 \[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=-\frac {x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{a} \]
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\[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=- a \int \frac {x^{2} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{a^{2} x^{2} - 1}\, dx \]
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\[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=\int { -\frac {{\left (a x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1} \,d x } \]
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\[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=\int { -\frac {{\left (a x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1} \,d x } \]
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Time = 5.78 (sec) , antiderivative size = 76, normalized size of antiderivative = 6.33 \[ \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx=\frac {\ln \left (\frac {1}{x}\right )}{a^2}-\frac {\ln \left (a+\frac {1}{x}\right )}{2\,a^2}-\frac {\ln \left (\frac {1}{x}-a\right )}{2\,a^2}+\frac {\ln \left (a^2\,x^2-1\right )}{2\,a^2}-\frac {x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{a} \]
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