Optimal. Leaf size=34 \[ -\frac {\sqrt {1+a^2 x^4}}{2 a}+\frac {1}{2} x^2 \sinh ^{-1}\left (a x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6847, 5772, 267}
\begin {gather*} \frac {1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac {\sqrt {a^2 x^4+1}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 6847
Rubi steps
\begin {align*} \int x \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \sinh ^{-1}(a x) \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac {1}{2} a \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^4}}{2 a}+\frac {1}{2} x^2 \sinh ^{-1}\left (a x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+a^2 x^4}}{2 a}+\frac {1}{2} x^2 \sinh ^{-1}\left (a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 31, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {a \,x^{2} \arcsinh \left (a \,x^{2}\right )-\sqrt {a^{2} x^{4}+1}}{2 a}\) | \(31\) |
default | \(\frac {a \,x^{2} \arcsinh \left (a \,x^{2}\right )-\sqrt {a^{2} x^{4}+1}}{2 a}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 0.88 \begin {gather*} \frac {a x^{2} \operatorname {arsinh}\left (a x^{2}\right ) - \sqrt {a^{2} x^{4} + 1}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 42, normalized size = 1.24 \begin {gather*} \frac {a x^{2} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \sqrt {a^{2} x^{4} + 1}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 0.79 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {asinh}{\left (a x^{2} \right )}}{2} - \frac {\sqrt {a^{2} x^{4} + 1}}{2 a} & \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 = 0.38, size = 40, normalized size = 1.18 \begin {gather*} \frac {1}{2} \, x^{2} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \frac {\sqrt {a^{2} x^{4} + 1}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 28, normalized size = 0.82 \begin {gather*} \frac {x^2\,\mathrm {asinh}\left (a\,x^2\right )}{2}-\frac {\sqrt {a^2\,x^4+1}}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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