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 = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 203
Rule 206
Rule 212
Rule 329
Rule 459
Rule 2119
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.77, size = 216, normalized size = 2.63 \[ \left [-2 \, \sqrt {a} \arctan \left (\frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a}}\right ) + \sqrt {a} \log \left (\frac {a^{2} + \sqrt {a^{2} + x^{2}} a - {\left ({\left (a - x\right )} \sqrt {a} + \sqrt {a^{2} + x^{2}} \sqrt {a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}, 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{a}\right ) + \sqrt {-a} \log \left (-\frac {a^{2} - \sqrt {a^{2} + x^{2}} a + {\left (\sqrt {-a} {\left (a + x\right )} - \sqrt {a^{2} + x^{2}} \sqrt {-a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 25, normalized size = 0.30 \[ 2 \sqrt {2}\, \sqrt {x}\, \hypergeom \left (\left [-\frac {1}{4}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], -\frac {a^{2}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.32, size = 51, normalized size = 0.62 \[ \frac {\sqrt {x} \Gamma ^{2}\left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {3}{4} \end {matrix}\middle | {\frac {a^{2} e^{i \pi }}{x^{2}}} \right )}}{8 \pi \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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