Optimal. Leaf size=70 \[ \frac {1}{2} \sqrt {x^4-1} \sec ^{-1}(x)-\frac {\sqrt {x^4-1}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{x^2}} x}{\sqrt {x^4-1}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 94, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {261, 5246, 12, 1572, 1252, 865, 875, 203} \[ -\frac {\sqrt {x^4-1}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {\sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {x^4-1}}{\sqrt {1-x^2}}\right )}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \sqrt {x^4-1} \sec ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 261
Rule 865
Rule 875
Rule 1252
Rule 1572
Rule 5246
Rubi steps
\begin {align*} \int \frac {x^3 \sec ^{-1}(x)}{\sqrt {-1+x^4}} \, dx &=\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)-\int \frac {\sqrt {-1+x^4}}{2 \sqrt {1-\frac {1}{x^2}} x^2} \, dx\\ &=\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)-\frac {1}{2} \int \frac {\sqrt {-1+x^4}}{\sqrt {1-\frac {1}{x^2}} x^2} \, dx\\ &=\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)-\frac {\sqrt {1-x^2} \int \frac {\sqrt {-1+x^4}}{x \sqrt {1-x^2}} \, dx}{2 \sqrt {1-\frac {1}{x^2}} x}\\ &=\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)-\frac {\sqrt {1-x^2} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {1-x} x} \, dx,x,x^2\right )}{4 \sqrt {1-\frac {1}{x^2}} x}\\ &=-\frac {\sqrt {-1+x^4}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)+\frac {\sqrt {1-x^2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x \sqrt {-1+x^2}} \, dx,x,x^2\right )}{4 \sqrt {1-\frac {1}{x^2}} x}\\ &=-\frac {\sqrt {-1+x^4}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)+\frac {\sqrt {1-x^2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {-1+x^4}}{\sqrt {1-x^2}}\right )}{2 \sqrt {1-\frac {1}{x^2}} x}\\ &=-\frac {\sqrt {-1+x^4}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)+\frac {\sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {-1+x^4}}{\sqrt {1-x^2}}\right )}{2 \sqrt {1-\frac {1}{x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 88, normalized size = 1.26 \[ \frac {1}{2} \left (\sqrt {x^4-1} \sec ^{-1}(x)-\log \left (x-x^3\right )-\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^4-1} x}{x^2-1}+\log \left (-x^2-\sqrt {1-\frac {1}{x^2}} \sqrt {x^4-1} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 110, normalized size = 1.57 \[ \frac {{\left (x^{2} - 1\right )} \log \left (\frac {x^{2} + \sqrt {x^{4} - 1} \sqrt {x^{2} - 1} - 1}{x^{2} - 1}\right ) - {\left (x^{2} - 1\right )} \log \left (-\frac {x^{2} - \sqrt {x^{4} - 1} \sqrt {x^{2} - 1} - 1}{x^{2} - 1}\right ) + 2 \, \sqrt {x^{4} - 1} {\left ({\left (x^{2} - 1\right )} \operatorname {arcsec}\relax (x) - \sqrt {x^{2} - 1}\right )}}{4 \, {\left (x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 52, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {x^{4} - 1} \arccos \left (\frac {1}{x}\right ) - \frac {2 \, \sqrt {x^{2} + 1} - \log \left (\sqrt {x^{2} + 1} + 1\right ) + \log \left (\sqrt {x^{2} + 1} - 1\right )}{4 \, \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \mathrm {arcsec}\relax (x )}{\sqrt {x^{4}-1}}\, dx \]
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 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1} \arctan \left (\sqrt {x + 1} \sqrt {x - 1}\right ) - \int \frac {2 \, {\left (x^{3} e^{\left (\frac {3}{2} \, \log \left (x + 1\right ) + \frac {3}{2} \, \log \left (x - 1\right )\right )} + x^{3} e^{\left (\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right )\right )}\right )} \sqrt {x^{2} + 1} \log \relax (x) + {\left (x^{3} + x\right )} e^{\left (\frac {1}{2} \, \log \left (x^{2} + 1\right ) + \frac {3}{2} \, \log \left (x + 1\right ) + \frac {3}{2} \, \log \left (x - 1\right )\right )}}{{\left (x^{2} + 1\right )} {\left (e^{\left (2 \, \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right )\right )} + e^{\left (\log \left (x + 1\right ) + \log \left (x - 1\right )\right )}\right )}}\,{d x} \]
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 \[ \int \frac {x^3\,\mathrm {acos}\left (\frac {1}{x}\right )}{\sqrt {x^4-1}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {asec}{\relax (x )}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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