Optimal. Leaf size=25 \[ x \sinh ^{-1}(a x)-\frac {\sqrt {a^2 x^2+1}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5653, 261} \[ x \sinh ^{-1}(a x)-\frac {\sqrt {a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rubi steps
\begin {align*} \int \sinh ^{-1}(a x) \, dx &=x \sinh ^{-1}(a x)-a \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{a}+x \sinh ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ x \sinh ^{-1}(a x)-\frac {\sqrt {a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 37, normalized size = 1.48 \[ \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 35, normalized size = 1.40 \[ x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 1.04 \[ \frac {a x \arcsinh \left (a x \right )-\sqrt {a^{2} x^{2}+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 25, normalized size = 1.00 \[ \frac {a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 23, normalized size = 0.92 \[ x\,\mathrm {asinh}\left (a\,x\right )-\frac {\sqrt {a^2\,x^2+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.80 \[ \begin {cases} x \operatorname {asinh}{\left (a x \right )} - \frac {\sqrt {a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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