Optimal. Leaf size=70 \[ -\frac {\sqrt {-1+x^4}}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {1}{2} \sqrt {-1+x^4} \sec ^{-1}(x)+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{x^2}} x}{\sqrt {-1+x^4}}\right ) \]
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Rubi [A]
time = 0.09, 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 = {267, 5354, 12,
1586, 1266, 879, 889, 209} \begin {gather*} \frac {\sqrt {1-x^2} \text {ArcTan}\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)-\frac {\sqrt {x^4-1}}{2 \sqrt {1-\frac {1}{x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 267
Rule 879
Rule 889
Rule 1266
Rule 1586
Rule 5354
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} \text {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} \text {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} \text {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 = 0.07, size = 88, normalized size = 1.26 \begin {gather*} \frac {1}{2} \left (-\frac {\sqrt {1-\frac {1}{x^2}} x \sqrt {-1+x^4}}{-1+x^2}+\sqrt {-1+x^4} \sec ^{-1}(x)-\log \left (x-x^3\right )+\log \left (1-x^2-\sqrt {1-\frac {1}{x^2}} x \sqrt {-1+x^4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.53, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \mathrm {arcsec}\left (x \right )}{\sqrt {x^{4}-1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (54) = 108\).
time = 0.47, size = 110, normalized size = 1.57 \begin {gather*} \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}\left (x\right ) - \sqrt {x^{2} - 1}\right )}}{4 \, {\left (x^{2} - 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \operatorname {asec}{\left (x \right )}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 52, normalized size = 0.74 \begin {gather*} \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}\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 = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {acos}\left (\frac {1}{x}\right )}{\sqrt {x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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