Optimal. Leaf size=44 \[ -\frac {x \sqrt {1+a^2 x^2}}{4 a}+\frac {\sinh ^{-1}(a x)}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x) \]
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Rubi [A]
time = 0.01, antiderivative size = 44, 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 = {5776, 327, 221}
\begin {gather*} -\frac {x \sqrt {a^2 x^2+1}}{4 a}+\frac {\sinh ^{-1}(a x)}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rubi steps
\begin {align*} \int x \sinh ^{-1}(a x) \, dx &=\frac {1}{2} x^2 \sinh ^{-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 \sinh ^{-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 {\sinh ^{-1}(a x)}{4 a^2}+\frac {1}{2} x^2 \sinh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.91 \begin {gather*} \frac {-a x \sqrt {1+a^2 x^2}+\left (1+2 a^2 x^2\right ) \sinh ^{-1}(a x)}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 39, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \arcsinh \left (a x \right )}{2}-\frac {a x \sqrt {a^{2} x^{2}+1}}{4}+\frac {\arcsinh \left (a x \right )}{4}}{a^{2}}\) | \(39\) |
default | \(\frac {\frac {a^{2} x^{2} \arcsinh \left (a x \right )}{2}-\frac {a x \sqrt {a^{2} x^{2}+1}}{4}+\frac {\arcsinh \left (a x \right )}{4}}{a^{2}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 39, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arsinh}\left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\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 = 0.36, size = 48, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {a^{2} x^{2} + 1} a x - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 37, normalized size = 0.84 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2} - \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a} + \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 68, normalized size = 1.55 \begin {gather*} \frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} + \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2} {\left | a \right |}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.65, size = 36, normalized size = 0.82 \begin {gather*} x\,\mathrm {asinh}\left (a\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,a^2\,x}\right )-\frac {x\,\sqrt {a^2\,x^2+1}}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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