Optimal. Leaf size=57 \[ -\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]
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Rubi [A] time = 0.0187291, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {5151, 264} \[ -\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5151
Rule 264
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^3} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2}+\frac{1}{2} \sqrt{-e} \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0363215, size = 54, normalized size = 0.95 \[ -\frac{\sqrt{-e} x \sqrt{d+e x^2}+d \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 d x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 67, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,{x}^{2}}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }-{\frac{1}{2\,{d}^{2}x}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ex}{2\,{d}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14299, size = 78, normalized size = 1.37 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, x^{2}} - \frac{\sqrt{-e} e x^{2} + d \sqrt{-e}}{2 \, \sqrt{e x^{2} + d} d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35543, size = 111, normalized size = 1.95 \begin{align*} -\frac{\sqrt{e x^{2} + d} \sqrt{-e} x + d \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, d x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.24586, size = 53, normalized size = 0.93 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{2 x^{2}} - \frac{\sqrt{e} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{2 d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2364, size = 142, normalized size = 2.49 \begin{align*} \frac{x e^{3}}{4 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} d} - \frac{{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} e^{\left (-1\right )}}{4 \, d x} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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