Optimal. Leaf size=41 \[ -\frac {1}{9 \left (x^2\right )^{3/2}}+\frac {1}{3 \sqrt {x^2}}+\frac {\left (-1+x^2\right )^{3/2} \sec ^{-1}(x)}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {270, 5346, 12,
14} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 5346
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^2} \sec ^{-1}(x)}{x^4} \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 1.17 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.48, size = 329, normalized size = 8.02
method | result | size |
default | \(-\frac {\sqrt {x^{2}-1}\, \left (\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-5 i x^{4}-12 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+20 i x^{2}+16 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -16 i\right )}{144 \left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) x^{3}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{5}-8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}+4 x^{4}+8 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -12 x^{2}+8\right ) \mathrm {arcsec}\left (x \right )}{48 x^{3} \sqrt {x^{2}-1}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (-13 i+3 \,\mathrm {arcsec}\left (x \right )\right )}{72 \sqrt {x^{2}-1}\, x}-\frac {\left (5 i+3 \,\mathrm {arcsec}\left (x \right )\right ) \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) \sqrt {x^{2}-1}}{72 x}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (7 i+9 \,\mathrm {arcsec}\left (x \right )\right ) \cos \left (3 \,\mathrm {arcsec}\left (x \right )\right )}{144 \sqrt {x^{2}-1}}-\frac {\left (i x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -i\right ) \left (3 i+\mathrm {arcsec}\left (x \right )\right ) \sin \left (3 \,\mathrm {arcsec}\left (x \right )\right )}{48 \sqrt {x^{2}-1}}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.76, size = 27, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 23, normalized size = 0.56 \begin {gather*} \frac {3 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} \operatorname {arcsec}\left (x\right ) + 3 \, x^{2} - 1}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (32) = 64\).
time = 0.67, size = 75, normalized size = 1.83 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acos}\left (\frac {1}{x}\right )\,\sqrt {x^2-1}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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