Optimal. Leaf size=94 \[ \frac{(a x+1)^2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}{4 a^2}+\frac{(a x+1) \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{2 a^2}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a^2} \]
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Rubi [A] time = 0.308749, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6337, 819, 639, 203} \[ \frac{(a x+1)^2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}{4 a^2}+\frac{(a x+1) \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}{2 a^2}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 819
Rule 639
Rule 203
Rubi steps
\begin{align*} \int e^{-\text{sech}^{-1}(a x)} x \, dx &=\int \frac{x}{\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}} \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{(-1+x)^2 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 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}{4 a^2}-\frac{\operatorname{Subst}\left (\int \frac{-2+2 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 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}{4 a^2}+\frac{(1+a x) \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ &=\frac{(1+a x)^2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}{4 a^2}+\frac{(1+a x) \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}{2 a^2}+\frac{\tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{a^2}\\ \end{align*}
Mathematica [C] time = 0.0679862, size = 75, normalized size = 0.8 \[ -\frac{-2 a x+a x \sqrt{\frac{1-a x}{a x+1}} (a x+1)+i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{-1}}\, dx \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 \frac{x}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02363, size = 173, normalized size = 1.84 \begin{align*} -\frac{a^{2} x^{2} \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}} \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*} a \int \frac{x^{2}}{a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + 1}\, dx \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 \frac{x}{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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