Optimal. Leaf size=39 \[ \frac {1}{4} x^2 \sqrt {x^4-1}-\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x^4-1}}{x^2+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {275, 195, 217, 206} \begin {gather*} \frac {1}{4} x^2 \sqrt {x^4-1}-\frac {1}{4} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 275
Rubi steps
\begin {align*} \int x \sqrt {-1+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \sqrt {-1+x^4}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \sqrt {-1+x^4}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-1+x^4}}\right )\\ &=\frac {1}{4} x^2 \sqrt {-1+x^4}-\frac {1}{4} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-1+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.08 \begin {gather*} \frac {\left (x^4-1\right ) \left (\sin ^{-1}\left (x^2\right )+\sqrt {1-x^4} x^2\right )}{4 \sqrt {-\left (x^4-1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^2 \sqrt {-1+x^4}-\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {-1+x^4}}{1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} - 1} x^{2} + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} - 1} x^{2} + \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{4} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 28, normalized size = 0.72
method | result | size |
default | \(\frac {x^{2} \sqrt {x^{4}-1}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-1}\right )}{4}\) | \(28\) |
risch | \(\frac {x^{2} \sqrt {x^{4}-1}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-1}\right )}{4}\) | \(28\) |
elliptic | \(\frac {x^{2} \sqrt {x^{4}-1}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-1}\right )}{4}\) | \(28\) |
trager | \(\frac {x^{2} \sqrt {x^{4}-1}}{4}+\frac {\ln \left (x^{2}-\sqrt {x^{4}-1}\right )}{4}\) | \(30\) |
meijerg | \(\frac {i \sqrt {\mathrm {signum}\left (x^{4}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{2} \sqrt {-x^{4}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{2}\right )\right )}{8 \sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \sqrt {\pi }}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 58, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {x^{4} - 1}}{4 \, x^{2} {\left (\frac {x^{4} - 1}{x^{4}} - 1\right )}} - \frac {1}{8} \, \log \left (\frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {1}{8} \, \log \left (\frac {\sqrt {x^{4} - 1}}{x^{2}} - 1\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 = -0.03 \begin {gather*} \int x\,\sqrt {x^4-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.28, size = 60, normalized size = 1.54 \begin {gather*} \begin {cases} \frac {x^{6}}{4 \sqrt {x^{4} - 1}} - \frac {x^{2}}{4 \sqrt {x^{4} - 1}} - \frac {\operatorname {acosh}{\left (x^{2} \right )}}{4} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {i x^{2} \sqrt {1 - x^{4}}}{4} + \frac {i \operatorname {asin}{\left (x^{2} \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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