Optimal. Leaf size=38 \[ -\frac{x e^{\text{sech}^{-1}(a x)}}{3 a^2}+\frac{x^2}{6 a}+\frac{1}{3} x^3 e^{\text{sech}^{-1}(a x)} \]
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Rubi [A] time = 0.024187, antiderivative size = 52, normalized size of antiderivative = 1.37, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6335, 30, 74} \[ -\frac{\sqrt{1-a x}}{3 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{x^2}{6 a}+\frac{1}{3} x^3 e^{\text{sech}^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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Rule 6335
Rule 30
Rule 74
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^2 \, dx &=\frac{1}{3} e^{\text{sech}^{-1}(a x)} x^3+\frac{\int x \, dx}{3 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=\frac{x^2}{6 a}+\frac{1}{3} e^{\text{sech}^{-1}(a x)} x^3-\frac{\sqrt{1-a x}}{3 a^3 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [A] time = 0.0587718, size = 48, normalized size = 1.26 \[ \frac{3 a^2 x^2+2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{6 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.173, size = 54, normalized size = 1.4 \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}-1 \right ) }{3\,{a}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{{x}^{2}}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06955, size = 51, normalized size = 1.34 \begin{align*} \frac{x^{2}}{2 \, a} + \frac{{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82112, size = 111, normalized size = 2.92 \begin{align*} \frac{3 \, a x^{2} + 2 \,{\left (a^{2} x^{3} - x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{6 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x\, dx + \int a x^{2} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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