Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {-9+16 x^2}}{9 x} \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16 x^2-9}}{9 x} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-9+16 x^2}}{9 x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {-9+16 x^2}}{9 x} \]
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Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {16 x^{2}-9}}{9 x}\) | \(15\) |
trager | \(\frac {\sqrt {16 x^{2}-9}}{9 x}\) | \(15\) |
risch | \(\frac {\sqrt {16 x^{2}-9}}{9 x}\) | \(15\) |
pseudoelliptic | \(\frac {\sqrt {16 x^{2}-9}}{9 x}\) | \(15\) |
gosper | \(\frac {\left (4 x -3\right ) \left (4 x +3\right )}{9 x \sqrt {16 x^{2}-9}}\) | \(25\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, \sqrt {1-\frac {16 x^{2}}{9}}}{3 \sqrt {\operatorname {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, x}\) | \(37\) |
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none
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {4 \, x + \sqrt {16 \, x^{2} - 9}}{9 \, x} \]
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Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\begin {cases} \frac {4 i \sqrt {-1 + \frac {9}{16 x^{2}}}}{9} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {16}{9} \\\frac {4 \sqrt {1 - \frac {9}{16 x^{2}}}}{9} & \text {otherwise} \end {cases} \]
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none
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16 \, x^{2} - 9}}{9 \, x} \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {8}{{\left (4 \, x - \sqrt {16 \, x^{2} - 9}\right )}^{2} + 9} \]
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Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16\,x^2-9}}{9\,x} \]
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