Optimal. Leaf size=21 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+x^2}}{a}\right )}{a} \]
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Rubi [A] time = 0.0127425, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 207} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+x^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a^2+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a^2+x}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-a^2+x^2} \, dx,x,\sqrt{a^2+x^2}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+x^2}}{a}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0032219, size = 21, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2+x^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 1.7 \begin{align*} -{\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}+{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86862, size = 90, normalized size = 4.29 \begin{align*} -\frac{\log \left (a - x + \sqrt{a^{2} + x^{2}}\right ) - \log \left (-a - x + \sqrt{a^{2} + x^{2}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01594, size = 7, normalized size = 0.33 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{a}{x} \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06523, size = 50, normalized size = 2.38 \begin{align*} -\frac{\log \left (a + \sqrt{a^{2} + x^{2}}\right )}{2 \, a} + \frac{\log \left (-a + \sqrt{a^{2} + x^{2}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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