Integrand size = 28, antiderivative size = 39 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \arctan \left (\frac {a+2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b}-\frac {\log (a-b x)}{b} \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1882, 31, 631, 210} \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \arctan \left (\frac {a+2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b}-\frac {\log (a-b x)}{b} \]
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Rule 31
Rule 210
Rule 631
Rule 1882
Rubi steps \begin{align*} \text {integral}& = \frac {a \int \frac {1}{\frac {a^2}{b^2}+\frac {a x}{b}+x^2} \, dx}{b^2}-\frac {\int \frac {1}{-\frac {a}{b}+x} \, dx}{b} \\ & = -\frac {\log (a-b x)}{b}-\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 b x}{a}\right )}{b} \\ & = \frac {2 \tan ^{-1}\left (\frac {a+2 b x}{\sqrt {3} a}\right )}{\sqrt {3} b}-\frac {\log (a-b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.82 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {a+2 b x}{\sqrt {3} a}\right )-2 \log (a-b x)+\log \left (a^2+a b x+b^2 x^2\right )-\log \left (a^3-b^3 x^3\right )}{3 b} \]
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Time = 1.51 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {2 \arctan \left (\frac {\left (2 b x +a \right ) \sqrt {3}}{3 a}\right ) \sqrt {3}}{3 b}-\frac {\ln \left (b x -a \right )}{b}\) | \(38\) |
default | \(-\frac {\ln \left (-b x +a \right )}{b}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 b^{2} x +a b \right ) \sqrt {3}}{3 a b}\right )}{3 b}\) | \(44\) |
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none
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b x + a\right )}}{3 \, a}\right ) - 3 \, \log \left (b x - a\right )}{3 \, b} \]
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Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=- \frac {\frac {\sqrt {3} i \log {\left (x + \frac {a - \sqrt {3} i a}{2 b} \right )}}{3} - \frac {\sqrt {3} i \log {\left (x + \frac {a + \sqrt {3} i a}{2 b} \right )}}{3} + \log {\left (- \frac {a}{b} + x \right )}}{b} \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b^{2} x + a b\right )}}{3 \, a b}\right )}{3 \, b} - \frac {\log \left (b x - a\right )}{b} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, b x + a\right )}}{3 \, a}\right )}{3 \, b} - \frac {\log \left ({\left | b x - a \right |}\right )}{b} \]
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Time = 9.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.21 \[ \int \frac {2 a^2+b^2 x^2}{a^3-b^3 x^3} \, dx=\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {4\,\sqrt {3}\,a^3\,b^4}{4\,a^3\,b^4-4\,a^2\,b^5\,x}+\frac {4\,\sqrt {3}\,a^2\,b^5\,x}{4\,a^3\,b^4-4\,a^2\,b^5\,x}\right )}{3\,b}-\frac {\ln \left (a-b\,x\right )}{b} \]
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