Integrand size = 26, antiderivative size = 51 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x}-a \arcsin (a x)-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {864, 827, 858, 222, 272, 65, 214} \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (1-a x)}{x}-a \arcsin (a x) \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 827
Rule 858
Rule 864
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{x^2} \, dx \\ & = -\frac {(1-a x) \sqrt {1-a^2 x^2}}{x}-\frac {1}{2} \int \frac {-2 a+2 a^2 x}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {(1-a x) \sqrt {1-a^2 x^2}}{x}+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-a^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {(1-a x) \sqrt {1-a^2 x^2}}{x}-a \sin ^{-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 {(1-a x) \sqrt {1-a^2 x^2}}{x}-a \sin ^{-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 {(1-a x) \sqrt {1-a^2 x^2}}{x}-a \sin ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.43 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=\frac {(-1+a x) \sqrt {1-a^2 x^2}}{x}-2 a \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )-a \log (x)+a \log \left (-1+\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.71
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a \sqrt {-a^{2} x^{2}+1}-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(87\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )+a \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-a \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\) | \(253\) |
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=\frac {2 \, a x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + a x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + a x + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )}}{x} \]
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Result contains complex when optimal does not.
Time = 2.68 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.33 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=a \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.33 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=-a \arcsin \left (a x\right ) - a \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \sqrt {-a^{2} x^{2} + 1} a - \frac {\sqrt {-a^{2} x^{2} + 1}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (45) = 90\).
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.45 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=\frac {a^{4} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \sqrt {-a^{2} x^{2} + 1} a - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, x {\left | a \right |}} \]
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Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^2 (1-a x)} \, dx=a\,\sqrt {1-a^2\,x^2}-\frac {\sqrt {1-a^2\,x^2}}{x}-\frac {a^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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