Integrand size = 37, antiderivative size = 38 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=\frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2118, 214} \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=\frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rule 214
Rule 2118
Rubi steps \begin{align*} \text {integral}& = \left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{e^2-16 d e^2 f x^2} \, dx,x,\frac {x}{2 e+4 f x^3}\right ) \\ & = \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.29 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=-\frac {\text {RootSum}\left [e^2-4 d f \text {$\#$1}^2+4 e f \text {$\#$1}^3+4 f^2 \text {$\#$1}^6\&,\frac {-e \log (x-\text {$\#$1})+4 f \log (x-\text {$\#$1}) \text {$\#$1}^3}{-2 d \text {$\#$1}+3 e \text {$\#$1}^2+6 f \text {$\#$1}^5}\&\right ]}{4 f} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 f^{2} \textit {\_Z}^{6}+4 e f \,\textit {\_Z}^{3}-4 d f \,\textit {\_Z}^{2}+e^{2}\right )}{\sum }\frac {\left (4 \textit {\_R}^{3} f -e \right ) \ln \left (x -\textit {\_R} \right )}{-6 f \,\textit {\_R}^{5}-3 e \,\textit {\_R}^{2}+2 d \textit {\_R}}}{4 f}\) | \(72\) |
risch | \(\frac {\ln \left (\left (-32 f \left (d f \right )^{\frac {3}{2}} d +54 e^{2} f^{2} d \right ) x^{3}+\left (54 e^{2} \left (d f \right )^{\frac {3}{2}}-32 d^{3} f^{2}\right ) x -16 e \left (d f \right )^{\frac {3}{2}} d +27 e^{3} f d \right )}{4 \sqrt {d f}}-\frac {\ln \left (\left (-32 f \left (d f \right )^{\frac {3}{2}} d -54 e^{2} f^{2} d \right ) x^{3}+\left (54 e^{2} \left (d f \right )^{\frac {3}{2}}+32 d^{3} f^{2}\right ) x -16 e \left (d f \right )^{\frac {3}{2}} d -27 e^{3} f d \right )}{4 \sqrt {d f}}\) | \(140\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 4.08 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=\left [\frac {\sqrt {d f} \log \left (\frac {4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2} + 4 \, {\left (2 \, f x^{4} + e x\right )} \sqrt {d f}}{4 \, f^{2} x^{6} + 4 \, e f x^{3} - 4 \, d f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (\frac {\sqrt {-d f} x^{2}}{d}\right ) - \sqrt {-d f} \arctan \left (\frac {{\left (2 \, f x^{5} + e x^{2} - 2 \, d x\right )} \sqrt {-d f}}{d e}\right )}{2 \, d f}\right ] \]
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Time = 0.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=- \frac {\sqrt {\frac {1}{d f}} \log {\left (- d x \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} + \frac {\sqrt {\frac {1}{d f}} \log {\left (d x \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} \]
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\[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=\int { -\frac {4 \, f x^{3} - e}{4 \, f^{2} x^{6} + 4 \, e f x^{3} - 4 \, d f x^{2} + e^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).
Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=\frac {\sqrt {d f} \log \left ({\left | 2 \, f x^{3} + 2 \, \sqrt {d f} x + e \right |}\right )}{4 \, d f} - \frac {\sqrt {d f} \log \left ({\left | 2 \, f x^{3} - 2 \, \sqrt {d f} x + e \right |}\right )}{4 \, d f} \]
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Time = 17.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {e-4 f x^3}{e^2-4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx=-\frac {\mathrm {atanh}\left (\frac {-32\,f\,d^2\,x+27\,e^3+54\,f\,e^2\,x^3}{16\,d^{3/2}\,e\,\sqrt {f}+32\,d^{3/2}\,f^{3/2}\,x^3-54\,\sqrt {d}\,e^2\,\sqrt {f}\,x}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]
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