Integrand size = 24, antiderivative size = 46 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2}}{e}+\frac {d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {679, 223, 209} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {\sqrt {d^2-e^2 x^2}}{e} \]
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Rule 209
Rule 223
Rule 679
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2}}{e}+d \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {\sqrt {d^2-e^2 x^2}}{e}+d \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = \frac {\sqrt {d^2-e^2 x^2}}{e}+\frac {d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2}}{e}-\frac {d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \]
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Time = 0.37 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e}+\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}\) | \(49\) |
default | \(\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}}{e}\) | \(78\) |
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {2 \, d \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - \sqrt {-e^{2} x^{2} + d^{2}}}{e} \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {d \arcsin \left (\frac {e x}{d}\right )}{e} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {d \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} \]
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Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]
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