Optimal. Leaf size=79 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {298, 203, 206} \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rubi steps
\begin {align*} \int \frac {x^2}{2 (a+b)+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\sqrt {2} \sqrt {-a-b}-x^2} \, dx\right )+\frac {1}{2} \int \frac {1}{\sqrt {2} \sqrt {-a-b}+x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-a-b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 128, normalized size = 1.62 \[ \frac {\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt {a+b}+\sqrt {2} x^2\right )-\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt {a+b}+\sqrt {2} x^2\right )-2 \tan ^{-1}\left (1-\frac {\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{4\ 2^{3/4} \sqrt [4]{a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 136, normalized size = 1.72 \[ -\left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a + b}\right )^{\frac {1}{4}} \arctan \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} x \left (-\frac {1}{a + b}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {x^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} \sqrt {-\frac {1}{a + b}}} \left (-\frac {1}{a + b}\right )^{\frac {1}{4}}\right ) + \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a + b}\right )^{\frac {1}{4}} \log \left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a + b\right )} \left (-\frac {1}{a + b}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{4} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (-\frac {1}{a + b}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a + b\right )} \left (-\frac {1}{a + b}\right )^{\frac {3}{4}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 219, normalized size = 2.77 \[ \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} + \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} - \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} + \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 137, normalized size = 1.73 \[ \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (2 a +2 b \right )^{\frac {1}{4}}}-1\right )}{4 \left (2 a +2 b \right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (2 a +2 b \right )^{\frac {1}{4}}}+1\right )}{4 \left (2 a +2 b \right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (2 a +2 b \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {2 a +2 b}}{x^{2}+\left (2 a +2 b \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {2 a +2 b}}\right )}{8 \left (2 a +2 b \right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.11, size = 179, normalized size = 2.27 \[ \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (x^{2} + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (x^{2} - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 53, normalized size = 0.67 \[ \frac {8^{1/4}\,\mathrm {atan}\left (\frac {8^{1/4}\,x}{2\,{\left (-a-b\right )}^{1/4}}\right )-8^{1/4}\,\mathrm {atanh}\left (\frac {8^{1/4}\,x}{2\,{\left (-a-b\right )}^{1/4}}\right )}{4\,{\left (-a-b\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 29, normalized size = 0.37 \[ \operatorname {RootSum} {\left (t^{4} \left (512 a + 512 b\right ) + 1, \left (t \mapsto t \log {\left (128 t^{3} a + 128 t^{3} b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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