Integrand size = 27, antiderivative size = 51 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{d x}+\frac {e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 821, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d x} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 864
Rubi steps \begin{align*} \text {integral}& = \int \frac {d-e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{d x}-e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{d x}-\frac {1}{2} e \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{d x}+\frac {\text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{d x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{d x}+\frac {e \log (x)}{\sqrt {d^2}}-\frac {e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{\sqrt {d^2}} \]
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Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\) | \(63\) |
default | \(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}}{d}-\frac {e \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{2}}+\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{2}}\) | \(228\) |
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none
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=-\frac {e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}}}{d x} \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{2} \left (d + e x\right )}\, dx \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=\frac {e^{4} x}{2 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d {\left | e \right |}} + \frac {e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{d {\left | e \right |}} - \frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{2 \, d x {\left | e \right |}} \]
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Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^2 (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^2\,\left (d+e\,x\right )} \,d x \]
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