Optimal. Leaf size=85 \[ -\frac{(a x+1)^2}{2 a^2}+\frac{\left (2 \sqrt{\frac{1-a x}{a x+1}}+1\right ) (a x+1)}{a^2}+\frac{2 \log (a x+1)}{a^2}+\frac{4 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}{a^2} \]
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Rubi [A] time = 0.434226, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6337, 1647, 1593, 801, 260} \[ -\frac{(a x+1)^2}{2 a^2}+\frac{\left (2 \sqrt{\frac{1-a x}{a x+1}}+1\right ) (a x+1)}{a^2}+\frac{2 \log (a x+1)}{a^2}+\frac{4 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1647
Rule 1593
Rule 801
Rule 260
Rubi steps
\begin{align*} \int e^{2 \text{sech}^{-1}(a x)} x \, dx &=\int x \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)^3}{(-1+x) \left (1+x^2\right )^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}-\frac{\operatorname{Subst}\left (\int \frac{-12 x-4 x^2}{(-1+x) \left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}-\frac{\operatorname{Subst}\left (\int \frac{(-12-4 x) x}{(-1+x) \left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}+\frac{(1+a x) \left (1+2 \sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{8+8 x}{(-1+x) \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}+\frac{(1+a x) \left (1+2 \sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \left (\frac{8}{-1+x}-\frac{8 x}{1+x^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}+\frac{(1+a x) \left (1+2 \sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}+\frac{4 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}-\frac{4 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ &=-\frac{(1+a x)^2}{2 a^2}+\frac{(1+a x) \left (1+2 \sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}+\frac{2 \log (1+a x)}{a^2}+\frac{4 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0600083, size = 89, normalized size = 1.05 \[ \frac{-a^2 x^2+4 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-4 \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )+8 \log (x)}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.208, size = 89, normalized size = 1.1 \begin{align*} -{\frac{{x}^{2}}{2}}+2\,{\frac{\ln \left ( x \right ) }{{a}^{2}}}+2\,{\frac{x \left ( \sqrt{-{a}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) }{a\sqrt{-{a}^{2}{x}^{2}+1}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}} \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*} \int x{\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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Fricas [A] time = 2.10643, size = 281, normalized size = 3.31 \begin{align*} -\frac{a^{2} x^{2} - 4 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 2 \, \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - 2 \, \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) - 4 \, \log \left (x\right )}{2 \, a^{2}} \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{\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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