Optimal. Leaf size=53 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]
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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 = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {661, 205} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-d+e x} \sqrt {d^2-e^2 x^2}} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{2 d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {-d+e x}}\right )\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {-d+e x}}\right )}{\sqrt {d} e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 1.00 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\right )}{\sqrt {d} e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 147, normalized size = 2.77 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{d}} \log \left (-\frac {e^{2} x^{2} + 2 \, d e x - 2 \, \sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d} d \sqrt {-\frac {1}{d}} - 3 \, d^{2}}{e^{2} x^{2} - 2 \, d e x + d^{2}}\right )}{2 \, e}, \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d} \sqrt {d}}{e^{2} x^{2} - d^{2}}\right )}{\sqrt {d} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 63, normalized size = 1.19 \[ \frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-e x -d}\, \sqrt {2}}{2 \sqrt {d}}\right )}{\sqrt {e x -d}\, \sqrt {-e x -d}\, \sqrt {d}\, e} \]
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 \[ \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} \sqrt {e x - d}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {d^2-e^2\,x^2}\,\sqrt {e\,x-d}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \sqrt {- d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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