Optimal. Leaf size=52 \[ \frac {2 \sqrt {a^2 x^2+1}}{a^2}+\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{a^2}-\frac {2 x \sinh ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5717, 5653, 261} \[ \frac {2 \sqrt {a^2 x^2+1}}{a^2}+\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{a^2}-\frac {2 x \sinh ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{a^2}-\frac {2 \int \sinh ^{-1}(a x) \, dx}{a}\\ &=-\frac {2 x \sinh ^{-1}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{a^2}+2 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 \sqrt {1+a^2 x^2}}{a^2}-\frac {2 x \sinh ^{-1}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.92 \[ \frac {2 \sqrt {a^2 x^2+1}+\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2-2 a x \sinh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 70, normalized size = 1.35 \[ -\frac {2 \, a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 74, normalized size = 1.42 \[ \frac {\sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{a}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 64, normalized size = 1.23 \[ \frac {\arcsinh \left (a x \right )^{2} a^{2} x^{2}+\arcsinh \left (a x \right )^{2}-2 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 a^{2} x^{2}+2}{a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 48, normalized size = 0.92 \[ \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )^{2}}{a^{2}} - \frac {2 \, {\left (a x \operatorname {arsinh}\left (a x\right ) - \sqrt {a^{2} x^{2} + 1}\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 49, normalized size = 0.94 \[ \begin {cases} - \frac {2 x \operatorname {asinh}{\left (a x \right )}}{a} + \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{a^{2}} + \frac {2 \sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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