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 = 0.0255669, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, 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 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{2 a+2 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*}
Mathematica [A] time = 0.0243773, 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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Maple [B] time = 0.002, size = 137, normalized size = 1.7 \begin{align*}{\frac{\sqrt{2}}{8}\ln \left ({ \left ({x}^{2}-\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) \left ({x}^{2}+\sqrt [4]{2\,a+2\,b}x\sqrt{2}+\sqrt{2\,a+2\,b} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ){\frac{1}{\sqrt [4]{2\,a+2\,b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5113, size = 448, normalized size = 5.67 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.202555, size = 29, normalized size = 0.37 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1662, size = 296, normalized size = 3.75 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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