Optimal. Leaf size=59 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\sqrt{-e} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}} \]
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Rubi [A] time = 0.0329614, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {5151, 266, 63, 208} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\sqrt{-e} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 5151
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^2} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}+\sqrt{-e} \int \frac{1}{x \sqrt{d+e x^2}} \, dx\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}+\frac{1}{2} \sqrt{-e} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{\sqrt{-e}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\sqrt{-e} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}}\\ \end{align*}
Mathematica [C] time = 0.0603867, size = 86, normalized size = 1.46 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x}+\frac{i \sqrt{e} \log \left (-\frac{2 \sqrt{-e} \sqrt{d+e x^2}}{e x}+\frac{2 i \sqrt{d}}{\sqrt{e} x}\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 57, normalized size = 1. \begin{align*} -{\frac{1}{x}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }-{\sqrt{-e}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{e{x}^{2}+d} \right ) } \right ){\frac{1}{\sqrt{d}}}} \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*} \frac{-d \sqrt{-e} x \int \frac{\sqrt{e x^{2} + d}}{e^{2} x^{5} + d e x^{3} -{\left (e x^{3} + d x\right )}{\left (e x^{2} + d\right )}}\,{d x} - \arctan \left (\sqrt{-e} x, \sqrt{e x^{2} + d}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.60711, size = 339, normalized size = 5.75 \begin{align*} \left [\frac{x \sqrt{-\frac{e}{d}} \log \left (-\frac{e^{2} x^{2} + 2 \, \sqrt{e x^{2} + d} d \sqrt{-e} \sqrt{-\frac{e}{d}} + 2 \, d e}{x^{2}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, x}, -\frac{x \sqrt{\frac{e}{d}} \arctan \left (\frac{\sqrt{e x^{2} + d} d \sqrt{-e} \sqrt{\frac{e}{d}}}{e^{2} x^{2} + d e}\right ) + \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.0837, size = 60, normalized size = 1.02 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{x} + \frac{\sqrt{- e} \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x^{2}}} \right )}}{d \sqrt{- \frac{1}{d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20885, size = 73, normalized size = 1.24 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-x^{2} e^{2} - d e} e^{\left (-\frac{1}{2}\right )}}{\sqrt{d}}\right ) e^{\frac{1}{2}}}{\sqrt{d}} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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