Optimal. Leaf size=117 \[ \frac{(1-a x) (a x+1)^3}{4 a^4}+\frac{\sqrt{\frac{1-a x}{a x+1}} \left (4-3 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^3}{6 a^4}+\frac{\left (3-8 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^2}{6 a^4}-\frac{x}{a^3} \]
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Rubi [A] time = 0.546433, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6337, 1804, 1811, 1814, 12, 261} \[ \frac{(1-a x) (a x+1)^3}{4 a^4}+\frac{\sqrt{\frac{1-a x}{a x+1}} \left (4-3 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^3}{6 a^4}+\frac{\left (3-8 \sqrt{\frac{1-a x}{a x+1}}\right ) (a x+1)^2}{6 a^4}-\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1804
Rule 1811
Rule 1814
Rule 12
Rule 261
Rubi steps
\begin{align*} \int e^{2 \text{sech}^{-1}(a x)} x^3 \, dx &=\int x^3 \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{(-1+x) x (1+x)^5}{\left (1+x^2\right )^5} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^4}\\ &=\frac{(1-a x) (1+a x)^3}{4 a^4}-\frac{\operatorname{Subst}\left (\int \frac{24 x+32 x^2-32 x^3-32 x^4-8 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^4}\\ &=\frac{(1-a x) (1+a x)^3}{4 a^4}-\frac{\operatorname{Subst}\left (\int \frac{x \left (24+32 x-32 x^2-32 x^3-8 x^4\right )}{\left (1+x^2\right )^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^4}\\ &=\frac{(1-a x) (1+a x)^3}{4 a^4}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}+\frac{\operatorname{Subst}\left (\int \frac{-64-48 x+192 x^2+48 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{12 a^4}\\ &=\frac{(1-a x) (1+a x)^3}{4 a^4}+\frac{(1+a x)^2 \left (3-8 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}-\frac{\operatorname{Subst}\left (\int -\frac{192 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{48 a^4}\\ &=\frac{(1-a x) (1+a x)^3}{4 a^4}+\frac{(1+a x)^2 \left (3-8 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}+\frac{4 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^4}\\ &=-\frac{x}{a^3}+\frac{(1-a x) (1+a x)^3}{4 a^4}+\frac{(1+a x)^2 \left (3-8 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x)^3 \left (4-3 \sqrt{\frac{1-a x}{1+a x}}\right )}{6 a^4}\\ \end{align*}
Mathematica [A] time = 0.0679191, size = 52, normalized size = 0.44 \[ \frac{x^2}{a^2}+\frac{2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{3 a^4}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 72, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{{x}^{2}}{2}} \right ) }+{\frac{2\,x \left ({a}^{2}{x}^{2}-1 \right ) }{3\,{a}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{{x}^{2}}{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04463, size = 57, normalized size = 0.49 \begin{align*} -\frac{1}{4} \, x^{4} + \frac{x^{2}}{a^{2}} + \frac{2 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9465, size = 131, normalized size = 1.12 \begin{align*} -\frac{3 \, a^{3} x^{4} - 12 \, a x^{2} - 8 \,{\left (a^{2} x^{3} - x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{12 \, a^{3}} \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 2 x\, dx + \int - a^{2} x^{3}\, dx + \int 2 a x^{2} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a^{2}} \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^{3}{\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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