Optimal. Leaf size=13 \[ \frac {\sinh ^{-1}(a x)^2}{2 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5783}
\begin {gather*} \frac {\sinh ^{-1}(a x)^2}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sinh ^{-1}(a x)^2}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sinh ^{-1}(a x)^2}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\arcsinh \left (a x \right )^{2}}{2 a}\) | \(12\) |
default | \(\frac {\arcsinh \left (a x \right )^{2}}{2 a}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 11, normalized size = 0.85 \begin {gather*} \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs.
\(2 (11) = 22\).
time = 0.37, size = 23, normalized size = 1.77 \begin {gather*} \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 10, normalized size = 0.77 \begin {gather*} \begin {cases} \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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