Optimal. Leaf size=82 \[ 2 \sqrt{\sqrt{a^2+x^2}+x}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0743947, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2119, 459, 329, 212, 206, 203} \[ 2 \sqrt{\sqrt{a^2+x^2}+x}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 2119
Rule 459
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{x+\sqrt{a^2+x^2}}}{x} \, dx &=\operatorname{Subst}\left (\int \frac{a^2+x^2}{\sqrt{x} \left (-a^2+x^2\right )} \, dx,x,x+\sqrt{a^2+x^2}\right )\\ &=2 \sqrt{x+\sqrt{a^2+x^2}}+\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-a^2+x^2\right )} \, dx,x,x+\sqrt{a^2+x^2}\right )\\ &=2 \sqrt{x+\sqrt{a^2+x^2}}+\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a^2+x^4} \, dx,x,\sqrt{x+\sqrt{a^2+x^2}}\right )\\ &=2 \sqrt{x+\sqrt{a^2+x^2}}-(2 a) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\sqrt{x+\sqrt{a^2+x^2}}\right )-(2 a) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\sqrt{x+\sqrt{a^2+x^2}}\right )\\ &=2 \sqrt{x+\sqrt{a^2+x^2}}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{x+\sqrt{a^2+x^2}}}{\sqrt{a}}\right )-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{x+\sqrt{a^2+x^2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.17944, size = 127, normalized size = 1.55 \[ -\frac{2 \sqrt{a^2+x^2} \left (\sqrt{a^2+x^2}+x\right ) \left (-\sqrt{\sqrt{a^2+x^2}+x}+\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+x^2}+x}}{\sqrt{a}}\right )\right )}{x \left (\sqrt{a^2+x^2}+x\right )+a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.037, size = 25, normalized size = 0.3 \begin{align*} 2\,\sqrt{2}\sqrt{x}{\mbox{$_3$F$_2$}(-1/4,-1/4,1/4;\,1/2,3/4;\,-{\frac{{a}^{2}}{{x}^{2}}})} \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{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68155, size = 552, normalized size = 6.73 \begin{align*} \left [-2 \, \sqrt{a} \arctan \left (\frac{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{\sqrt{a}}\right ) + \sqrt{a} \log \left (\frac{a^{2} + \sqrt{a^{2} + x^{2}} a -{\left ({\left (a - x\right )} \sqrt{a} + \sqrt{a^{2} + x^{2}} \sqrt{a}\right )} \sqrt{x + \sqrt{a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}}, 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{x + \sqrt{a^{2} + x^{2}}}}{a}\right ) + \sqrt{-a} \log \left (-\frac{a^{2} - \sqrt{a^{2} + x^{2}} a +{\left (\sqrt{-a}{\left (a + x\right )} - \sqrt{a^{2} + x^{2}} \sqrt{-a}\right )} \sqrt{x + \sqrt{a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.49162, size = 51, normalized size = 0.62 \begin{align*} \frac{\sqrt{x} \Gamma ^{2}\left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right ){{}_{3}F_{2}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4}, \frac{1}{4} \\ \frac{1}{2}, \frac{3}{4} \end{matrix}\middle |{\frac{a^{2} e^{i \pi }}{x^{2}}} \right )}}{8 \pi \Gamma \left (\frac{3}{4}\right )} \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{\sqrt{x + \sqrt{a^{2} + x^{2}}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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