Integrand size = 20, antiderivative size = 136 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {a \arctan \left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \arctan \left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]
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Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1890, 217, 1179, 642, 1176, 631, 210, 1262, 649, 209, 266} \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=-\frac {a \arctan \left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \arctan \left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}+\frac {1}{12} d \log \left (3 x^4+2\right ) \]
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Rule 209
Rule 210
Rule 217
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1262
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{2+3 x^4}+\frac {x \left (b+d x^2\right )}{2+3 x^4}\right ) \, dx \\ & = a \int \frac {1}{2+3 x^4} \, dx+\int \frac {x \left (b+d x^2\right )}{2+3 x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {b+d x}{2+3 x^2} \, dx,x,x^2\right )+\frac {a \int \frac {\sqrt {2}-\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {2}}+\frac {a \int \frac {\sqrt {2}+\sqrt {3} x^2}{2+3 x^4} \, dx}{2 \sqrt {2}} \\ & = \frac {a \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt {6}}+\frac {a \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt {6}}-\frac {a \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}-\frac {a \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right )+\frac {1}{2} d \text {Subst}\left (\int \frac {x}{2+3 x^2} \, dx,x,x^2\right ) \\ & = \frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {a \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {1}{12} d \log \left (2+3 x^4\right )+\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {a \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}} \\ & = \frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\frac {1}{48} \left (-2 \sqrt {6} \left (\sqrt [4]{6} a+2 b\right ) \arctan \left (1-\sqrt [4]{6} x\right )+2 \sqrt {6} \left (\sqrt [4]{6} a-2 b\right ) \arctan \left (1+\sqrt [4]{6} x\right )-6^{3/4} a \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+6^{3/4} a \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+4 d \log \left (2+3 x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.25
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +\textit {\_R} b +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(34\) |
default | \(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}+\frac {b \arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{12}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\) | \(121\) |
meijerg | \(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{2}\right )}{12}+\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\) | \(193\) |
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Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 17085, normalized size of antiderivative = 125.62 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\text {Too large to display} \]
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Time = 0.84 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (165888 t^{4} - 55296 t^{3} d + t^{2} \cdot \left (3456 b^{2} + 6912 d^{2}\right ) + t \left (- 864 a^{2} b - 576 b^{2} d - 384 d^{3}\right ) + 27 a^{4} + 72 a^{2} b d + 18 b^{4} + 24 b^{2} d^{2} + 8 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {27648 t^{3} b^{2} + 1728 t^{2} a^{2} b - 6912 t^{2} b^{2} d + 216 t a^{4} - 288 t a^{2} b d + 288 t b^{4} + 576 t b^{2} d^{2} - 18 a^{4} d - 90 a^{2} b^{3} + 12 a^{2} b d^{2} - 24 b^{4} d - 16 b^{2} d^{3}}{27 a^{5} - 72 a b^{4}} \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d + 3 \, a\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{24} \, \sqrt {3} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} a - 2 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \, \sqrt {3} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (6^{\frac {3}{4}} a - 2 \, \sqrt {6} b\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \, \sqrt {6} b\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a + 4 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 4 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
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Time = 9.48 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.26 \[ \int \frac {a+b x+d x^3}{2+3 x^4} \, dx=\sum _{k=1}^4\ln \left (x\,\left (9\,a^2\,d+9\,b^3+6\,b\,d^2\right )+9\,a\,b^2-6\,a\,d^2-\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (3456\,b^2+6912\,d^2\right )}{165888}-\frac {z\,\left (864\,a^2\,b+576\,b^2\,d+384\,d^3\right )}{165888}+\frac {a^2\,b\,d}{2304}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (3456\,b^2+6912\,d^2\right )}{165888}-\frac {z\,\left (864\,a^2\,b+576\,b^2\,d+384\,d^3\right )}{165888}+\frac {a^2\,b\,d}{2304}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right )\,\left (864\,a-864\,b\,x\right )-144\,a\,d+x\,\left (108\,a^2+144\,b\,d\right )\right )\right )\,\mathrm {root}\left (z^4-\frac {d\,z^3}{3}+\frac {z^2\,\left (3456\,b^2+6912\,d^2\right )}{165888}-\frac {z\,\left (864\,a^2\,b+576\,b^2\,d+384\,d^3\right )}{165888}+\frac {a^2\,b\,d}{2304}+\frac {b^2\,d^2}{6912}+\frac {d^4}{20736}+\frac {b^4}{9216}+\frac {a^4}{6144},z,k\right ) \]
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