Optimal. Leaf size=34 \[ \frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\sin ^{-1}(a x)}{a^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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 = {807, 222}
\begin {gather*} \frac {\text {ArcSin}(a x)}{a^2}+\frac {\sqrt {1-a^2 x^2}}{a^2 (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 807
Rubi steps
\begin {align*} \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\sin ^{-1}(a x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 67, normalized size = 1.97 \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\sqrt {-a^2} \log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 65, normalized size = 1.91
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{3} \left (x +\frac {1}{a}\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 33, normalized size = 0.97 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac {\arcsin \left (a x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.41, size = 58, normalized size = 1.71 \begin {gather*} \frac {a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{3} x + a^{2}} \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 {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 52, normalized size = 1.53 \begin {gather*} \frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a {\left | a \right |}} - \frac {2}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.60, size = 57, normalized size = 1.68 \begin {gather*} \frac {1}{a^2\,\sqrt {1-a^2\,x^2}}-\frac {x}{a\,\sqrt {1-a^2\,x^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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