Optimal. Leaf size=25 \[ -\frac {\sqrt {1+a^2 x^2}}{a}+x \sinh ^{-1}(a x) \]
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Rubi [A]
time = 0.00, 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 = {5772, 267}
\begin {gather*} x \sinh ^{-1}(a x)-\frac {\sqrt {a^2 x^2+1}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
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 \begin {gather*} -\frac {\sqrt {1+a^2 x^2}}{a}+x \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 26, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {a x \arcsinh \left (a x \right )-\sqrt {a^{2} x^{2}+1}}{a}\) | \(26\) |
default | \(\frac {a x \arcsinh \left (a x \right )-\sqrt {a^{2} x^{2}+1}}{a}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 37, normalized size = 1.48 \begin {gather*} \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 35, normalized size = 1.40 \begin {gather*} x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a} \end {gather*}
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 \begin {gather*} x\,\mathrm {asinh}\left (a\,x\right )-\frac {\sqrt {a^2\,x^2+1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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