Integrand size = 23, antiderivative size = 120 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}} \]
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Time = 0.01 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {406} \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {a^{3/4} \left (\frac {\sqrt {a+3 x^2}}{\sqrt {a}}+1\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}} \]
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Rule 406
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\frac {-\arctan \left (\frac {-3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}{2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}{3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}\right )}{4 \sqrt {3} a^{3/4}} \]
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\[\int \frac {1}{\left (3 x^{2}+a \right )^{\frac {1}{4}} \left (3 x^{2}+2 a \right )}d x\]
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Result contains complex when optimal does not.
Time = 4.03 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} + 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {-18 i \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) \]
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\[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\int \frac {1}{\sqrt [4]{a + 3 x^{2}} \cdot \left (2 a + 3 x^{2}\right )}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right )} \, dx=\int \frac {1}{\left (3\,x^2+2\,a\right )\,{\left (3\,x^2+a\right )}^{1/4}} \,d x \]
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