Integrand size = 20, antiderivative size = 9 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {1}{\sqrt {1+\frac {1}{x^2}}} \]
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Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {25, 267} \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {1}{\sqrt {\frac {1}{x^2}+1}} \]
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Rule 25
Rule 267
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (1+\frac {1}{x^2}\right )^{3/2} x^3} \, dx \\ & = \frac {1}{\sqrt {1+\frac {1}{x^2}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(20\) vs. \(2(9)=18\).
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {\sqrt {1+\frac {1}{x^2}} x^2}{1+x^2} \]
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Time = 1.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.33
method | result | size |
gosper | \(\frac {1}{\sqrt {\frac {x^{2}+1}{x^{2}}}}\) | \(12\) |
default | \(\frac {1}{\sqrt {\frac {x^{2}+1}{x^{2}}}}\) | \(12\) |
risch | \(\frac {1}{\sqrt {\frac {x^{2}+1}{x^{2}}}}\) | \(12\) |
trager | \(\frac {x^{2} \sqrt {-\frac {-x^{2}-1}{x^{2}}}}{x^{2}+1}\) | \(26\) |
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (7) = 14\).
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 3.11 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {x^{2} \sqrt {\frac {x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \]
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Time = 1.40 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {1}{\sqrt {1 + \frac {1}{x^{2}}}} \]
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\[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )} x \sqrt {\frac {1}{x^{2}} + 1}} \,d x } \]
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none
Time = 0.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {x}{\sqrt {x^{2} + 1} \mathrm {sgn}\left (x\right )} \]
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Time = 18.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {1+\frac {1}{x^2}} x \left (1+x^2\right )} \, dx=\frac {x^2\,\sqrt {\frac {1}{x^2}+1}}{x^2+1} \]
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