Optimal. Leaf size=54 \[ \frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {871, 12, 272,
65, 214} \begin {gather*} \frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\int \frac {d e^2}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\text {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 )}{d e^2}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 66, normalized size = 1.22 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2}}{d+e x}+2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 88, normalized size = 1.63
method | result | size |
default | \(\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e \,d^{2} \left (x +\frac {d}{e}\right )}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d \sqrt {d^{2}}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.73, size = 63, normalized size = 1.17 \begin {gather*} \frac {x e + {\left (x e + d\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + d + \sqrt {-x^{2} e^{2} + d^{2}}}{d^{2} x e + d^{3}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 75, normalized size = 1.39 \begin {gather*} -\frac {\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 )}{d^{2}} - \frac {2}{d^{2} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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