Integrand size = 26, antiderivative size = 64 \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)}-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {871, 821, 272, 65, 214} \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 871
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-a^2 x^2}}{x (1-a x)}-\frac {\int \frac {-2 a^2-a^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{a^2} \\ & = -\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)}+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)}-\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a} \\ & = -\frac {2 \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {(-1+2 a x) \sqrt {1-a^2 x^2}}{x (-1+a x)}-a \log (x)+a \log \left (-1+\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}\) | \(73\) |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}}-a \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )\) | \(84\) |
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\frac {a^{2} x^{2} - a x + {\left (a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )}}{a x^{2} - x} \]
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\[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=- \int \frac {1}{a x^{3} \sqrt {- a^{2} x^{2} + 1} - x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\int { -\frac {1}{\sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 11.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\frac {a^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{x}-a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \]
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