Optimal. Leaf size=57 \[ -\frac{4}{a \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{4 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a}-x \]
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Rubi [A] time = 0.173586, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6332, 1647, 12, 801, 203} \[ -\frac{4}{a \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}+\frac{4 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a}-x \]
Antiderivative was successfully verified.
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Rule 6332
Rule 1647
Rule 12
Rule 801
Rule 203
Rubi steps
\begin{align*} \int e^{2 \text{sech}^{-1}(a x)} \, dx &=\int \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)^2}{(-1+x)^2 \left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-x+\frac{2 \operatorname{Subst}\left (\int -\frac{4 x}{(-1+x)^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-x-\frac{8 \operatorname{Subst}\left (\int \frac{x}{(-1+x)^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-x-\frac{8 \operatorname{Subst}\left (\int \left (\frac{1}{2 (-1+x)^2}-\frac{1}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-x-\frac{4}{a \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ &=-x-\frac{4}{a \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}+\frac{4 \tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0948087, size = 75, normalized size = 1.32 \[ -\frac{a^2 x^2+2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)+2 a x \tan ^{-1}\left (\frac{a x}{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}\right )+2}{a^2 x} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.209, size = 98, normalized size = 1.7 \begin{align*} -x-2\,{\frac{1}{{a}^{2}x}}-2\,{\frac{{\it csgn} \left ( a \right ) }{a\sqrt{-{a}^{2}{x}^{2}+1}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( \arctan \left ({\frac{{\it csgn} \left ( a \right ) ax}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) xa+{\it csgn} \left ( a \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \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*} -x + \frac{2 \,{\left (-\frac{a^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{x}\right )}}{a^{2}} + \frac{-\frac{1}{x}}{a^{2}} - \frac{1}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.3013, size = 186, normalized size = 3.26 \begin{align*} -\frac{a^{2} x^{2} + 2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2 \, a x \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right ) + 2}{a^{2} x} \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 - a^{2}\, dx + \int \frac{2}{x^{2}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{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{\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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