Optimal. Leaf size=54 \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]
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Rubi [A] time = 0.0443948, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {857, 12, 266, 63, 208} \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 857
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}+\frac{\int \frac{d e^2}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}+\frac{\int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}-\frac{\operatorname{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 )}{d e^2}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0394156, size = 52, normalized size = 0.96 \[ \frac{\frac{\sqrt{d^2-e^2 x^2}}{d+e x}-\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 88, normalized size = 1.6 \begin{align*} -{\frac{1}{d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{e{d}^{2}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64702, size = 131, normalized size = 2.43 \begin{align*} \frac{e x +{\left (e x + d\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + d + \sqrt{-e^{2} x^{2} + d^{2}}}{d^{2} e x + d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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