Optimal. Leaf size=53 \[ -\frac {\sqrt {e} \sqrt {d+e x^2}}{2 d x}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6221, 264} \[ -\frac {\sqrt {e} \sqrt {d+e x^2}}{2 d x}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6221
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^3} \, dx &=-\frac {\tanh ^{-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 {\tanh ^{-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 = 50, normalized size = 0.94 \[ -\frac {\sqrt {e} x \sqrt {d+e x^2}+d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 d x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 54, normalized size = 1.02 \[ -\frac {2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{4 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 71, normalized size = 1.34 \[ \frac {e}{{\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - d} - \frac {\log \left (-\frac {\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} + 1}{\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}} - 1}\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 1.13 \[ -\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{2 x^{2}}-\frac {\sqrt {e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d^{2} x}+\frac {e^{\frac {3}{2}} x \sqrt {e \,x^{2}+d}}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 51, normalized size = 0.96 \[ -\frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, x^{2}} - \frac {e^{\frac {3}{2}} x^{2} + d \sqrt {e}}{2 \, \sqrt {e x^{2} + d} d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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