Optimal. Leaf size=45 \[ -\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \text {ArcCos}(a x)+\frac {\text {ArcSin}(a x)}{4 a^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 327, 222}
\begin {gather*} \frac {\text {ArcSin}(a x)}{4 a^2}-\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 4724
Rubi steps
\begin {align*} \int x \cos ^{-1}(a x) \, dx &=\frac {1}{2} x^2 \cos ^{-1}(a x)+\frac {1}{2} a \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \cos ^{-1}(a x)+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \cos ^{-1}(a x)+\frac {\sin ^{-1}(a x)}{4 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 42, normalized size = 0.93 \begin {gather*} \frac {-a x \sqrt {1-a^2 x^2}+2 a^2 x^2 \text {ArcCos}(a x)+\text {ArcSin}(a x)}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 40, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )}{2}-\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}+\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\) | \(40\) |
default | \(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )}{2}-\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}+\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 40, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x^{2} \arccos \left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.64, size = 37, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} a x - {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 42, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {acos}{\left (a x \right )}}{2} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a} - \frac {\operatorname {acos}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 37, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a} - \frac {\arccos \left (a x\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 38, normalized size = 0.84 \begin {gather*} \frac {\mathrm {acos}\left (a\,x\right )\,\left (2\,a^2\,x^2-1\right )}{4\,a^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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