Optimal. Leaf size=24 \[ \frac{\log (x)}{a}+x e^{\text{sech}^{-1}(a x)}-\frac{\text{sech}^{-1}(a x)}{a} \]
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Rubi [A] time = 0.136978, antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6329, 1962, 208} \[ \frac{\log (x)}{a}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )}{a}+x e^{\text{sech}^{-1}(a x)} \]
Warning: Unable to verify antiderivative.
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Rule 6329
Rule 1962
Rule 208
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}(a x)} \, dx &=e^{\text{sech}^{-1}(a x)} x+\frac{\log (x)}{a}+\frac{\int \frac{\sqrt{\frac{1-a x}{1+a x}}}{x (1-a x)} \, dx}{a}\\ &=e^{\text{sech}^{-1}(a x)} x+\frac{\log (x)}{a}-4 \operatorname{Subst}\left (\int \frac{1}{2 a-2 a x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=e^{\text{sech}^{-1}(a x)} x-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )}{a}+\frac{\log (x)}{a}\\ \end{align*}
Mathematica [B] time = 0.0402629, size = 79, normalized size = 3.29 \[ \frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)+2 \log (a x)-\log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 79, normalized size = 3.3 \begin{align*}{\frac{\ln \left ( x \right ) }{a}}+{x\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( \sqrt{-{a}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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 \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 [B] time = 1.83216, size = 258, normalized size = 10.75 \begin{align*} \frac{2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) + 2 \, \log \left (x\right )}{2 \, a} \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 \frac{1}{x}\, dx + \int a \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 \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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