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 = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 264
Rule 5151
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.62, size = 44, normalized size = 0.77 \[ -\frac {\sqrt {e x^{2} + d} \sqrt {-e} x + d \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 105, normalized size = 1.84 \[ \frac {x e^{3}}{4 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} d} - \frac {{\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} e^{\left (-1\right )}}{4 \, d x} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 67, normalized size = 1.18 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{2 x^{2}}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d^{2} x}+\frac {\sqrt {-e}\, e x \sqrt {e \,x^{2}+d}}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 58, normalized size = 1.02 \[ -\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, x^{2}} - \frac {\sqrt {-e} e x^{2} + d \sqrt {-e}}{2 \, \sqrt {e x^{2} + d} d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.60, size = 53, normalized size = 0.93 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{2 x^{2}} - \frac {\sqrt {e} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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