Integrand size = 17, antiderivative size = 133 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (-2+17 x^2\right )}{64 x^4}-\frac {3 \sec ^{-1}(x)}{8 x \sqrt {x^2}}+\frac {9 x \sec ^{-1}(x)}{64 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt {x^2}}-\frac {3 \sqrt {-1+x^2} \sec ^{-1}(x)^2}{8 x^2}-\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac {x \sec ^{-1}(x)^3}{8 \sqrt {x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {5350, 4744, 4742, 4738, 4724, 327, 222, 4768, 201} \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{32 \sqrt {x^2}}+\frac {15 \sqrt {1-\frac {1}{x^2}}}{64 \sqrt {x^2}}-\frac {9 \sqrt {x^2} \csc ^{-1}(x)}{64 x}+\frac {\sqrt {x^2} \sec ^{-1}(x)^3}{8 x}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {x^2} \sec ^{-1}(x)}{8 x^3} \]
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Rule 201
Rule 222
Rule 327
Rule 4724
Rule 4738
Rule 4742
Rule 4744
Rule 4768
Rule 5350
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {x^2} \text {Subst}\left (\int \left (1-x^2\right )^{3/2} \arccos (x)^2 \, dx,x,\frac {1}{x}\right )}{x} \\ & = -\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}-\frac {\sqrt {x^2} \text {Subst}\left (\int x \left (1-x^2\right ) \arccos (x) \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \sqrt {1-x^2} \arccos (x)^2 \, dx,x,\frac {1}{x}\right )}{4 x} \\ & = \frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}+\frac {\sqrt {x^2} \text {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )}{8 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\arccos (x)^2}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{8 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int x \arccos (x) \, dx,x,\frac {1}{x}\right )}{4 x} \\ & = \frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{32 \sqrt {x^2}}-\frac {3 \sqrt {x^2} \sec ^{-1}(x)}{8 x^3}+\frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}+\frac {\sqrt {x^2} \sec ^{-1}(x)^3}{8 x}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\frac {1}{x}\right )}{32 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{8 x} \\ & = \frac {15 \sqrt {1-\frac {1}{x^2}}}{64 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{32 \sqrt {x^2}}-\frac {3 \sqrt {x^2} \sec ^{-1}(x)}{8 x^3}+\frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}+\frac {\sqrt {x^2} \sec ^{-1}(x)^3}{8 x}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{64 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{16 x} \\ & = \frac {15 \sqrt {1-\frac {1}{x^2}}}{64 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{32 \sqrt {x^2}}-\frac {9 \sqrt {x^2} \csc ^{-1}(x)}{64 x}-\frac {3 \sqrt {x^2} \sec ^{-1}(x)}{8 x^3}+\frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}+\frac {\sqrt {x^2} \sec ^{-1}(x)^3}{8 x} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {\sqrt {-1+x^2} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (-16 \cos \left (2 \sec ^{-1}(x)\right )+\cos \left (4 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )+8 \sec ^{-1}(x)^2 \left (-8 \sin \left (2 \sec ^{-1}(x)\right )+\sin \left (4 \sec ^{-1}(x)\right )\right )\right )}{256 \sqrt {1-\frac {1}{x^2}} x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\operatorname {csgn}\left (x \sqrt {1-\frac {1}{x^{2}}}\right ) \left (8 \operatorname {arcsec}\left (x \right )^{3} x^{4}-40 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+17 \,\operatorname {arcsec}\left (x \right ) x^{4}+16 \operatorname {arcsec}\left (x \right )^{2} \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +17 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-40 \,\operatorname {arcsec}\left (x \right ) x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +8 \,\operatorname {arcsec}\left (x \right )\right )}{64 x^{4}}\) | \(114\) |
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\frac {8 \, x^{4} \operatorname {arcsec}\left (x\right )^{3} + {\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname {arcsec}\left (x\right ) - {\left (8 \, {\left (5 \, x^{2} - 2\right )} \operatorname {arcsec}\left (x\right )^{2} - 17 \, x^{2} + 2\right )} \sqrt {x^{2} - 1}}{64 \, x^{4}} \]
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Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right )^{2}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^2\,{\left (x^2-1\right )}^{3/2}}{x^5} \,d x \]
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