Optimal. Leaf size=110 \[ \frac {2 \left (1-21 x^2\right )}{27 \left (x^2\right )^{3/2}}+\frac {\left (x^2-1\right ) \sec ^{-1}(x)^2}{3 \left (x^2\right )^{3/2}}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}-\frac {4 \sqrt {x^2-1} \sec ^{-1}(x)}{3 x}+\frac {\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^3}{3 x^3}-\frac {2 \left (x^2-1\right )^{3/2} \sec ^{-1}(x)}{9 x^3} \]
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Rubi [A] time = 0.20, antiderivative size = 146, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5242, 4678, 4650, 4620, 8} \[ \frac {2 \sqrt {x^2}}{27 x^4}-\frac {14}{9 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}+\frac {\left (1-\frac {1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}-\frac {2 \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)}{9 x}-\frac {4 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \sec ^{-1}(x)}{3 x} \]
Antiderivative was successfully verified.
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Rule 8
Rule 4620
Rule 4650
Rule 4678
Rule 5242
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx &=-\frac {\sqrt {x^2} \operatorname {Subst}\left (\int x \sqrt {1-x^2} \cos ^{-1}(x)^3 \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}+\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \left (1-x^2\right ) \cos ^{-1}(x)^2 \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {\left (1-\frac {1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}+\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int x \sqrt {1-x^2} \cos ^{-1}(x) \, dx,x,\frac {1}{x}\right )}{3 x}+\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \cos ^{-1}(x)^2 \, dx,x,\frac {1}{x}\right )}{3 x}\\ &=-\frac {2 \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)}{9 x}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}-\frac {\left (2 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\frac {1}{x}\right )}{9 x}+\frac {\left (4 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x \cos ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{3 x}\\ &=-\frac {2}{9 \sqrt {x^2}}+\frac {2 \sqrt {x^2}}{27 x^4}-\frac {4 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \sec ^{-1}(x)}{3 x}-\frac {2 \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)}{9 x}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}-\frac {\left (4 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{x}\right )}{3 x}\\ &=-\frac {14}{9 \sqrt {x^2}}+\frac {2 \sqrt {x^2}}{27 x^4}-\frac {4 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \sec ^{-1}(x)}{3 x}-\frac {2 \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)}{9 x}+\frac {2 \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right ) \sec ^{-1}(x)^2}{3 \sqrt {x^2}}+\frac {\left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \sec ^{-1}(x)^3}{3 x}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 92, normalized size = 0.84 \[ \frac {2 \sqrt {1-\frac {1}{x^2}} x \left (1-21 x^2\right )+9 \left (x^2-1\right )^2 \sec ^{-1}(x)^3+9 \sqrt {1-\frac {1}{x^2}} x \left (3 x^2-1\right ) \sec ^{-1}(x)^2-6 \left (7 x^4-8 x^2+1\right ) \sec ^{-1}(x)}{27 x^3 \sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)^3}{x^4} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.06, size = 57, normalized size = 0.52 \[ \frac {9 \, {\left (3 \, x^{2} - 1\right )} \operatorname {arcsec}\relax (x)^{2} - 42 \, x^{2} + 3 \, {\left (3 \, {\left (x^{2} - 1\right )} \operatorname {arcsec}\relax (x)^{3} - 2 \, {\left (7 \, x^{2} - 1\right )} \operatorname {arcsec}\relax (x)\right )} \sqrt {x^{2} - 1} + 2}{27 \, x^{3}} \]
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 \[ \int \frac {\sqrt {x^{2} - 1} \operatorname {arcsec}\relax (x)^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 536, normalized size = 4.87
method | result | size |
default | \(\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+4 x^{4}+8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -12 x^{2}+8\right ) \mathrm {arcsec}\relax (x )^{3}}{48 \sqrt {x^{2}-1}\, x^{3}}+\frac {\sqrt {x^{2}-1}\, \left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-5 i x^{4}-12 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+20 i x^{2}+16 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -16 i\right )}{216 \left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) x^{3}}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+4 x^{4}+8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -12 x^{2}+8\right ) \mathrm {arcsec}\relax (x )}{72 \sqrt {x^{2}-1}\, x^{3}}-\frac {\left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-4 i x^{4}-8 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+12 i x^{2}+8 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -8 i\right ) \mathrm {arcsec}\relax (x )^{2}}{48 \sqrt {x^{2}-1}\, x^{3}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (9 \mathrm {arcsec}\relax (x )^{3}-117 i \mathrm {arcsec}\relax (x )^{2}-78 \,\mathrm {arcsec}\relax (x )+242 i\right )}{216 \sqrt {x^{2}-1}\, x}-\frac {\left (45 i \mathrm {arcsec}\relax (x )^{2}+9 \mathrm {arcsec}\relax (x )^{3}-82 i-78 \,\mathrm {arcsec}\relax (x )\right ) \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) \sqrt {x^{2}-1}}{216 x}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (63 i \mathrm {arcsec}\relax (x )^{2}+27 \mathrm {arcsec}\relax (x )^{3}-158 i-162 \,\mathrm {arcsec}\relax (x )\right ) \cos \left (3 \,\mathrm {arcsec}\relax (x )\right )}{432 \sqrt {x^{2}-1}}-\frac {\left (i x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \left (27 i \mathrm {arcsec}\relax (x )^{2}+3 \mathrm {arcsec}\relax (x )^{3}-54 i-50 \,\mathrm {arcsec}\relax (x )\right ) \sin \left (3 \,\mathrm {arcsec}\relax (x )\right )}{144 \sqrt {x^{2}-1}}\) | \(536\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.19, size = 93, normalized size = 0.85 \[ \frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\relax (x)^{3}}{3 \, x^{3}} + \frac {{\left (3 \, x^{2} - 1\right )} \operatorname {arcsec}\relax (x)^{2}}{3 \, x^{3}} - \frac {2 \, {\left ({\left (21 \, x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} + 3 \, {\left (7 \, x^{4} - 8 \, x^{2} + 1\right )} \arctan \left (\sqrt {x + 1} \sqrt {x - 1}\right )\right )}}{27 \, \sqrt {x + 1} \sqrt {x - 1} x^{3}} \]
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 \[ \int \frac {{\mathrm {acos}\left (\frac {1}{x}\right )}^3\,\sqrt {x^2-1}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right )} \operatorname {asec}^{3}{\relax (x )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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