Optimal. Leaf size=11 \[ \frac {\sin ^{-1}\left (\frac {b x}{2}\right )}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {41, 222}
\begin {gather*} \frac {\text {ArcSin}\left (\frac {b x}{2}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 222
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-b x} \sqrt {2+b x}} \, dx &=\int \frac {1}{\sqrt {4-b^2 x^2}} \, dx\\ &=\frac {\sin ^{-1}\left (\frac {b x}{2}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(11)=22\).
time = 0.04, size = 39, normalized size = 3.55 \begin {gather*} -\frac {\log \left (-\sqrt {-b^2} x+\sqrt {4-b^2 x^2}\right )}{\sqrt {-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs.
\(2(9)=18\).
time = 0.16, size = 56, normalized size = 5.09
method | result | size |
default | \(\frac {\sqrt {\left (-b x +2\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-x^{2} b^{2}+4}}\right )}{\sqrt {-b x +2}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 9, normalized size = 0.82 \begin {gather*} \frac {\arcsin \left (\frac {1}{2} \, b x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (9) = 18\).
time = 0.86, size = 31, normalized size = 2.82 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b x + 2} - 2}{b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 13.88, size = 76, normalized size = 6.91 \begin {gather*} - \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 15, normalized size = 1.36 \begin {gather*} \frac {2 \, \arcsin \left (\frac {1}{2} \, \sqrt {b x + 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 44, normalized size = 4.00 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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