Integrand size = 12, antiderivative size = 79 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
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Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {218, 212, 209} \[ \int \frac {1}{2 a+2 b+x^4} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{-a-b}}\right )}{2\ 2^{3/4} (-a-b)^{3/4}} \]
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Rule 209
Rule 212
Rule 218
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.62 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )+2 \arctan \left (1+\frac {\sqrt [4]{2} x}{\sqrt [4]{a+b}}\right )-\log \left (2 \sqrt {a+b}-2 \sqrt [4]{2} \sqrt [4]{a+b} x+\sqrt {2} x^2\right )+\log \left (2 \sqrt {a+b}+2 \sqrt [4]{2} \sqrt [4]{a+b} x+\sqrt {2} x^2\right )}{8 \sqrt [4]{2} (a+b)^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.34
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 a +2 b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{4}\) | \(27\) |
default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (2 a +2 b \right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {2 a +2 b}}{x^{2}-\left (2 a +2 b \right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {2 a +2 b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (2 a +2 b \right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (2 a +2 b \right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (2 a +2 b \right )^{\frac {3}{4}}}\) | \(113\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.46 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} \log \left (2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (a + b\right )} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (a + b\right )} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} + x\right ) - \frac {1}{4} i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (i \, a + i \, b\right )} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} + x\right ) + \frac {1}{4} i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} \log \left (-2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (-i \, a - i \, b\right )} \left (-\frac {1}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}\right )^{\frac {1}{4}} + x\right ) \]
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Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.53 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (2048 a^{3} + 6144 a^{2} b + 6144 a b^{2} + 2048 b^{3}\right ) + 1, \left ( t \mapsto t \log {\left (8 t a + 8 t b + x \right )} \right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (61) = 122\).
Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.27 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (x^{2} + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (x^{2} - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (2 \, a + 2 \, b\right )}^{\frac {3}{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (61) = 122\).
Time = 0.30 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.77 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\frac {{\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} + \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}\right )}}{2 \, {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}}}\right )}{4 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} + \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} - \frac {{\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} {\left (2 \, a + 2 \, b\right )}^{\frac {1}{4}} x + \sqrt {2 \, a + 2 \, b}\right )}{8 \, {\left (\sqrt {2} a + \sqrt {2} b\right )}} \]
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Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.53 \[ \int \frac {1}{2 a+2 b+x^4} \, dx=\frac {2^{1/4}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{\left (\frac {\sqrt {2}\,a}{{\left (-a-b\right )}^{3/2}}+\frac {\sqrt {2}\,b}{{\left (-a-b\right )}^{3/2}}\right )\,{\left (-a-b\right )}^{3/4}}\right )}{4\,{\left (-a-b\right )}^{3/4}}+\frac {2^{1/4}\,\mathrm {atanh}\left (\frac {2^{1/4}\,x}{\left (\frac {\sqrt {2}\,a}{{\left (-a-b\right )}^{3/2}}+\frac {\sqrt {2}\,b}{{\left (-a-b\right )}^{3/2}}\right )\,{\left (-a-b\right )}^{3/4}}\right )}{4\,{\left (-a-b\right )}^{3/4}} \]
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