Optimal. Leaf size=28 \[ -\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5798, 8}
\begin {gather*} \frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac {x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2}-\frac {\int 1 \, dx}{a}\\ &=-\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {x}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.09, size = 47, normalized size = 1.68
method | result | size |
default | \(\frac {\arcsinh \left (a x \right ) a^{2} x^{2}+\arcsinh \left (a x \right )-\sqrt {a^{2} x^{2}+1}\, a x}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 1.36 \begin {gather*} -\frac {a x - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 24, normalized size = 0.86 \begin {gather*} \begin {cases} - \frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{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 = 0.43, size = 38, normalized size = 1.36 \begin {gather*} -\frac {x}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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