Integrand size = 21, antiderivative size = 82 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}}-2 \sqrt {a} \arctan \left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2144, 470, 335, 218, 212, 209} \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=-2 \sqrt {a} \arctan \left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )+2 \sqrt {\sqrt {a^2+x^2}+x} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 470
Rule 2144
Rubi steps \begin{align*} \text {integral}& = \text {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 ) \text {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 ) \text {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) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )-(2 a) \text {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} \arctan \left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=2 \sqrt {x+\sqrt {a^2+x^2}}-2 \sqrt {a} \arctan \left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.30
method | result | size |
meijerg | \(2 \sqrt {2}\, \sqrt {x}\, {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{4};\frac {1}{2},\frac {3}{4};-\frac {a^{2}}{x^{2}}\right )\) | \(25\) |
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none
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.63 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\left [-2 \, \sqrt {a} \arctan \left (\frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a}}\right ) + \sqrt {a} \log \left (\frac {a^{2} + \sqrt {a^{2} + x^{2}} a - {\left ({\left (a - x\right )} \sqrt {a} + \sqrt {a^{2} + x^{2}} \sqrt {a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}, 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{a}\right ) + \sqrt {-a} \log \left (-\frac {a^{2} - \sqrt {a^{2} + x^{2}} a + {\left (\sqrt {-a} {\left (a + x\right )} - \sqrt {a^{2} + x^{2}} \sqrt {-a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right ) + 2 \, \sqrt {x + \sqrt {a^{2} + x^{2}}}\right ] \]
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Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\frac {\sqrt {x} \Gamma ^{2}\left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {3}{4} \end {matrix}\middle | {\frac {a^{2} e^{i \pi }}{x^{2}}} \right )}}{8 \pi \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int { \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x} \,d x } \]
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\[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int { \frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \, dx=\int \frac {\sqrt {x+\sqrt {a^2+x^2}}}{x} \,d x \]
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