Integrand size = 15, antiderivative size = 41 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=-\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {1}{3 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {270, 5346, 12, 14} \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\frac {1}{3 \sqrt {x^2}}-\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \]
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Rule 12
Rule 14
Rule 270
Rule 5346
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \frac {-1+x^2}{3 x^4} \, dx}{\sqrt {x^2}} \\ & = \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \frac {-1+x^2}{x^4} \, dx}{3 \sqrt {x^2}} \\ & = \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3}-\frac {x \int \left (-\frac {1}{x^4}+\frac {1}{x^2}\right ) \, dx}{3 \sqrt {x^2}} \\ & = -\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {1}{3 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\frac {\sqrt {1-\frac {1}{x^2}} x \left (-1+3 x^2\right )+3 \left (-1+x^2\right )^2 \sec ^{-1}(x)}{9 x^3 \sqrt {-1+x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.48 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.07
method | result | size |
default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, \left (3 \,\operatorname {arcsec}\left (x \right ) x^{4}+3 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-6 \,\operatorname {arcsec}\left (x \right ) x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x +3 \,\operatorname {arcsec}\left (x \right )\right )}{9 \left (x^{2}-1\right ) x^{2}}\) | \(85\) |
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none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\frac {3 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right ) + 3 \, x^{2} - 1}{9 \, x^{3}} \]
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Timed out. \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )}{3 \, x^{3}} + \frac {3 \, x^{2} - 1}{9 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (32) = 64\).
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=-\frac {2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right )}{3 \, \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{4} + 1\right )} \arccos \left (\frac {1}{x}\right )}{3 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}^{3}} + \frac {3 \, x^{2} - 1}{9 \, x^{3} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx=\int \frac {\mathrm {acos}\left (\frac {1}{x}\right )\,\sqrt {x^2-1}}{x^4} \,d x \]
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