Optimal. Leaf size=52 \[ \frac {2 \sqrt {1+a^2 x^2}}{a^2}-\frac {2 x \sinh ^{-1}(a x)}{a}+\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{a^2} \]
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Rubi [A]
time = 0.06, 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 = {5798, 5772,
267} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 5798
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 = 0.02, size = 48, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {1+a^2 x^2}-2 a x \sinh ^{-1}(a x)+\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.29, size = 64, normalized size = 1.23
method | result | size |
default | \(\frac {x^{2} \arcsinh \left (a x \right )^{2} a^{2}+\arcsinh \left (a x \right )^{2}-2 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +2 a^{2} x^{2}+2}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 48, normalized size = 0.92 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 70, normalized size = 1.35 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 49, normalized size = 0.94 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 74, normalized size = 1.42 \begin {gather*} \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} \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 = -0.02 \begin {gather*} \int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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