Optimal. Leaf size=35 \[ \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right )-\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {329, 331, 298, 203, 206} \begin {gather*} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right )-\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\left (-2+x^2\right )^{3/4}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{\left (-2+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )\\ &=-\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right )-\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 35, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-2+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 83, normalized size = 2.37 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {{\left (x^{2} - 2\right )}^{\frac {3}{4}} x^{\frac {3}{2}} - {\left (x^{2} - 2\right )}^{\frac {5}{4}} \sqrt {x}}{2 \, {\left (x^{3} - 2 \, x\right )}}\right ) + \frac {1}{2} \, \log \left (-x^{2} - {\left (x^{2} - 2\right )}^{\frac {1}{4}} x^{\frac {3}{2}} - \sqrt {x^{2} - 2} x - {\left (x^{2} - 2\right )}^{\frac {3}{4}} \sqrt {x} + 1\right ) \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x}}{{\left (x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 42, normalized size = 1.20
method | result | size |
meijerg | \(\frac {\left (-\mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )\right )^{\frac {3}{4}} 2^{\frac {1}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], \frac {x^{2}}{2}\right ) x^{\frac {3}{2}}}{3 \mathrm {signum}\left (-1+\frac {x^{2}}{2}\right )^{\frac {3}{4}}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 45, normalized size = 1.29 \begin {gather*} \arctan \left (\frac {{\left (x^{2} - 2\right )}^{\frac {1}{4}}}{\sqrt {x}}\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{2} - 2\right )}^{\frac {1}{4}}}{\sqrt {x}} + 1\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{2} - 2\right )}^{\frac {1}{4}}}{\sqrt {x}} - 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 \frac {\sqrt {x}}{{\left (x^2-2\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.87, size = 41, normalized size = 1.17 \begin {gather*} \frac {\sqrt [4]{2} x^{\frac {3}{2}} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{2}}{2}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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