Optimal. Leaf size=133 \[ \frac {x \sec ^{-1}(x)^3}{8 \sqrt {x^2}}-\frac {3 \sqrt {x^2-1} \sec ^{-1}(x)^2}{8 x^2}+\frac {9 x \sec ^{-1}(x)}{64 \sqrt {x^2}}-\frac {3 \sec ^{-1}(x)}{8 x \sqrt {x^2}}+\frac {\sqrt {x^2-1} \left (17 x^2-2\right )}{64 x^4}-\frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac {\left (x^2-1\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt {x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 172, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {5242, 4650, 4648, 4642, 4628, 321, 216, 4678, 195} \[ \frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{32 \sqrt {x^2}}+\frac {15 \sqrt {1-\frac {1}{x^2}}}{64 \sqrt {x^2}}-\frac {9 \sqrt {x^2} \csc ^{-1}(x)}{64 x}+\frac {\sqrt {x^2} \sec ^{-1}(x)^3}{8 x}-\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt {x^2}}-\frac {3 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^2 \sqrt {x^2} \sec ^{-1}(x)}{8 x}-\frac {3 \sqrt {x^2} \sec ^{-1}(x)}{8 x^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 321
Rule 4628
Rule 4642
Rule 4648
Rule 4650
Rule 4678
Rule 5242
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx &=-\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \cos ^{-1}(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} \operatorname {Subst}\left (\int x \left (1-x^2\right ) \cos ^{-1}(x) \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \sqrt {1-x^2} \cos ^{-1}(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} \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\cos ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{8 x}-\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int x \cos ^{-1}(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 ) \operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\frac {1}{x}\right )}{32 x}-\frac {\left (3 \sqrt {x^2}\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{64 x}-\frac {\left (3 \sqrt {x^2}\right ) \operatorname {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*}
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Mathematica [A] time = 0.31, size = 84, normalized size = 0.63 \[ \frac {\sqrt {x^2-1} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (\cos \left (4 \sec ^{-1}(x)\right )-16 \cos \left (2 \sec ^{-1}(x)\right )\right )+8 \sec ^{-1}(x)^2 \left (\sin \left (4 \sec ^{-1}(x)\right )-8 \sin \left (2 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )\right )}{256 \sqrt {1-\frac {1}{x^2}} x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)^2}{x^5} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.12, size = 59, normalized size = 0.44 \[ \frac {8 \, x^{4} \operatorname {arcsec}\relax (x)^{3} + {\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname {arcsec}\relax (x) - {\left (8 \, {\left (5 \, x^{2} - 2\right )} \operatorname {arcsec}\relax (x)^{2} - 17 \, x^{2} + 2\right )} \sqrt {x^{2} - 1}}{64 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\relax (x)^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.70, size = 386, normalized size = 2.90
method | result | size |
default | \(\frac {\sqrt {\frac {x^{2}-1}{x^{2}}}\, x \mathrm {arcsec}\relax (x )^{3}}{8 \sqrt {x^{2}-1}}+\frac {\left (-5 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}+x^{6}+20 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-13 x^{4}-16 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +28 x^{2}-16\right ) \left (4 i \mathrm {arcsec}\relax (x )+8 \mathrm {arcsec}\relax (x )^{2}-1\right )}{1024 \sqrt {x^{2}-1}\, x^{4}}-\frac {\left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (2 \mathrm {arcsec}\relax (x )^{2}-1+2 i \mathrm {arcsec}\relax (x )\right )}{32 \sqrt {x^{2}-1}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 x^{2}+2\right ) \left (2 \mathrm {arcsec}\relax (x )^{2}-1-2 i \mathrm {arcsec}\relax (x )\right )}{16 \sqrt {x^{2}-1}\, x^{2}}-\frac {\left (-5 x^{2}+4+3 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+x^{4}-4 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \left (-4 i \mathrm {arcsec}\relax (x )+8 \mathrm {arcsec}\relax (x )^{2}-1\right )}{1024 \sqrt {x^{2}-1}\, x^{2}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (7 i \mathrm {arcsec}\relax (x )+8 \mathrm {arcsec}\relax (x )^{2}-4\right ) \cos \left (4 \,\mathrm {arcsec}\relax (x )\right )}{128 \sqrt {x^{2}-1}}+\frac {\left (i x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \left (32 i \mathrm {arcsec}\relax (x )+24 \mathrm {arcsec}\relax (x )^{2}-15\right ) \sin \left (4 \,\mathrm {arcsec}\relax (x )\right )}{512 \sqrt {x^{2}-1}}\) | \(386\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\relax (x)^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^2\,{\left (x^2-1\right )}^{3/2}}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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