Optimal. Leaf size=53 \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 e^{\text{sech}^{-1}(a x)}+\frac{x}{2 a} \]
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Rubi [A] time = 0.0181567, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 8, 41, 216} \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 e^{\text{sech}^{-1}(a x)}+\frac{x}{2 a} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 8
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} x \, dx &=\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\int 1 \, dx}{2 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 a}\\ &=\frac{x}{2 a}+\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{x}{2 a}+\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \sin ^{-1}(a x)}{2 a^2}\\ \end{align*}
Mathematica [C] time = 0.0759349, size = 75, normalized size = 1.42 \[ \frac{2 a x+a x \sqrt{\frac{1-a x}{a x+1}} (a x+1)+i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.184, size = 92, normalized size = 1.7 \begin{align*}{\frac{x{\it csgn} \left ( a \right ) }{2\,a}\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}}}}+{\frac{x}{a}} \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{x}{a} + \frac{\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}}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80814, size = 171, normalized size = 3.23 \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*} \frac{\int 1\, dx + \int a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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