Optimal. Leaf size=79 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
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Rubi [A] time = 0.0301928, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {212, 206, 203} \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{2 (a+b)+x^4} \, dx &=-\frac{\int \frac{1}{\sqrt{2} \sqrt{-a-b}-x^2} \, dx}{2 \sqrt{2} \sqrt{-a-b}}-\frac{\int \frac{1}{\sqrt{2} \sqrt{-a-b}+x^2} \, dx}{2 \sqrt{2} \sqrt{-a-b}}\\ &=-\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0174291, size = 128, normalized size = 1.62 \[ \frac{-\log \left (-2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )+\log \left (2 \sqrt [4]{2} x \sqrt [4]{a+b}+2 \sqrt{a+b}+\sqrt{2} x^2\right )-2 \tan ^{-1}\left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{a+b}}+1\right )}{8 \sqrt [4]{2} (a+b)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., 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 ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}+1 \right ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}}+{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{2\,a+2\,b}}}}-1 \right ) \left ( 2\,a+2\,b \right ) ^{-{\frac{3}{4}}}} \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 = 1.66262, size = 782, normalized size = 9.9 \begin{align*} \left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \arctan \left (-4 \, \left (\frac{1}{8}\right )^{\frac{3}{4}}{\left (a^{2} + 2 \, a b + b^{2}\right )} x \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{3}{4}} + 4 \, \left (\frac{1}{8}\right )^{\frac{3}{4}}{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{x^{2} + 2 \, \sqrt{\frac{1}{2}}{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{3}{4}}\right ) + \frac{1}{4} \, \left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \log \left (2 \, \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (a + b\right )} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} + x\right ) - \frac{1}{4} \, \left (\frac{1}{8}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} \log \left (-2 \, \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (a + b\right )} \left (-\frac{1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac{1}{4}} + x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.304061, size = 42, normalized size = 0.53 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (2048 a^{3} + 6144 a^{2} b + 6144 a b^{2} + 2048 b^{3}\right ) + 1, \left ( t \mapsto t \log{\left (8 t a + 8 t b + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13084, size = 296, normalized size = 3.75 \begin{align*} \frac{{\left (2 \, a + 2 \, b\right )}^{\frac{1}{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{1}{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{1}{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{1}{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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