Optimal. Leaf size=46 \[ -\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {850, 844, 217, 203, 266, 63, 208} \[ -\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 844
Rule 850
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)} \, dx &=\int \frac {d-e x}{x \sqrt {d^2-e^2 x^2}} \, dx\\ &=d \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \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 )}{e^2}\\ &=-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.00 \[ -\log \left (\sqrt {d^2-e^2 x^2}+d\right )-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 54, normalized size = 1.17 \[ 2 \, \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 48, normalized size = 1.04 \[ -\arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\relax (d) - \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 137, normalized size = 2.98 \[ -\frac {d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-\frac {e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 56, normalized size = 1.22 \[ -\frac {e {\left (\frac {d \arcsin \left (\frac {e x}{d}\right )}{e} + \frac {d \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{e}\right )}}{d} \]
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 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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