Optimal. Leaf size=169 \[ \frac{(a x+1)^3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}{12 a^3}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}{6 a^3}+\frac{(a x+1) \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{2 a^3}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a^3} \]
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Rubi [A] time = 0.46505, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6337, 821, 12, 729, 723, 203} \[ \frac{(a x+1)^3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^4}{12 a^3}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3}{6 a^3}+\frac{(a x+1) \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{2 a^3}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 821
Rule 12
Rule 729
Rule 723
Rule 203
Rubi steps
\begin{align*} \int e^{2 \text{sech}^{-1}(a x)} x^2 \, dx &=\int x^2 \left (\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}\right )^2 \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{x (1+x)^4}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ &=\frac{(1+a x)^3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}{12 a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{4 (1+x)^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{3 a^3}\\ &=\frac{(1+a x)^3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}{12 a^3}-\frac{8 \operatorname{Subst}\left (\int \frac{(1+x)^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{3 a^3}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}{6 a^3}+\frac{(1+a x)^3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}{12 a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{(1+x)^2}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ &=\frac{(1+a x) \left (1-\sqrt{\frac{1-a x}{1+a x}}\right ) \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^3}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}{6 a^3}+\frac{(1+a x)^3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}{12 a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ &=\frac{(1+a x) \left (1-\sqrt{\frac{1-a x}{1+a x}}\right ) \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^3}-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}{6 a^3}+\frac{(1+a x)^3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^4}{12 a^3}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{a^3}\\ \end{align*}
Mathematica [C] time = 0.0703845, size = 86, normalized size = 0.51 \[ \sqrt{\frac{1-a x}{a x+1}} \left (\frac{x}{a^2}+\frac{x^2}{a}\right )+\frac{2 x}{a^2}+\frac{i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{a^3}-\frac{x^3}{3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.182, size = 97, normalized size = 0.6 \begin{align*} -{\frac{{x}^{3}}{3}}+2\,{\frac{x}{{a}^{2}}}+{\frac{x{\it csgn} \left ( a \right ) }{{a}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( x\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) a+\arctan \left ({x{\it csgn} \left ( a \right ) a{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, x}{a^{2}} + \frac{2 \,{\left (\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}}\right )}}{a^{2}} - \int x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11647, size = 192, normalized size = 1.14 \begin{align*} -\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 6 \, a x + 3 \, \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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