Optimal. Leaf size=87 \[ -\frac {\sinh ^{-1}(a x)^3}{6 a^3}-\frac {\sinh ^{-1}(a x)}{4 a^3}+\frac {x \sqrt {a^2 x^2+1}}{4 a^2}+\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {x^2 \sinh ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.15, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5758, 5675, 5661, 321, 215} \[ \frac {x \sqrt {a^2 x^2+1}}{4 a^2}+\frac {x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}-\frac {\sinh ^{-1}(a x)}{4 a^3}-\frac {x^2 \sinh ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^2 \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx &=\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}-\frac {\int x \sinh ^{-1}(a x) \, dx}{a}\\ &=-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}+\frac {1}{2} \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}-\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {x \sqrt {1+a^2 x^2}}{4 a^2}-\frac {\sinh ^{-1}(a x)}{4 a^3}-\frac {x^2 \sinh ^{-1}(a x)}{2 a}+\frac {x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 a^2}-\frac {\sinh ^{-1}(a x)^3}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.83 \[ \frac {3 a x \sqrt {a^2 x^2+1}+6 a x \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2-3 \left (2 a^2 x^2+1\right ) \sinh ^{-1}(a x)-2 \sinh ^{-1}(a x)^3}{12 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 102, normalized size = 1.17 \[ \frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{12 \, a^{3}} \]
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 \[ \int \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 69, normalized size = 0.79 \[ -\frac {-6 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x +6 \arcsinh \left (a x \right ) x^{2} a^{2}+2 \arcsinh \left (a x \right )^{3}-3 \sqrt {a^{2} x^{2}+1}\, x a +3 \arcsinh \left (a x \right )}{12 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {arsinh}\left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {asinh}\left (a\,x\right )}^2}{\sqrt {a^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 78, normalized size = 0.90 \[ \begin {cases} - \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a^{2}} - \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{6 a^{3}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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