Integrand size = 37, antiderivative size = 81 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=-\frac {\log \left (e-2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}+\frac {\log \left (e+2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 1178, 642} \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\log \left (2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}-\frac {\log \left (-2 \sqrt {-d} \sqrt {f} x+e+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \]
[In]
[Out]
Rule 6
Rule 642
Rule 1178
Rubi steps \begin{align*} \text {integral}& = \int \frac {e-2 f x^2}{e^2+4 (d+e) f x^2+4 f^2 x^4} \, dx \\ & = -\frac {\int \frac {\frac {\sqrt {-d}}{\sqrt {f}}+2 x}{-\frac {e}{2 f}-\frac {\sqrt {-d} x}{\sqrt {f}}-x^2} \, dx}{4 \sqrt {-d} \sqrt {f}}-\frac {\int \frac {\frac {\sqrt {-d}}{\sqrt {f}}-2 x}{-\frac {e}{2 f}+\frac {\sqrt {-d} x}{\sqrt {f}}-x^2} \, dx}{4 \sqrt {-d} \sqrt {f}} \\ & = -\frac {\log \left (e-2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}}+\frac {\log \left (e+2 \sqrt {-d} \sqrt {f} x+2 f x^2\right )}{4 \sqrt {-d} \sqrt {f}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(81)=162\).
Time = 0.09 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.36 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {-\frac {\left (-d-2 e+\sqrt {d} \sqrt {d+2 e}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {d+e-\sqrt {d} \sqrt {d+2 e}}}\right )}{\sqrt {d+e-\sqrt {d} \sqrt {d+2 e}}}-\frac {\left (d+2 e+\sqrt {d} \sqrt {d+2 e}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {f} x}{\sqrt {d+e+\sqrt {d} \sqrt {d+2 e}}}\right )}{\sqrt {d+e+\sqrt {d} \sqrt {d+2 e}}}}{2 \sqrt {2} \sqrt {d} \sqrt {d+2 e} \sqrt {f}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\ln \left (-2 \sqrt {-d f}\, f \,x^{2}-2 d f x -\sqrt {-d f}\, e \right )}{4 \sqrt {-d f}}+\frac {\ln \left (-2 \sqrt {-d f}\, f \,x^{2}+2 d f x -\sqrt {-d f}\, e \right )}{4 \sqrt {-d f}}\) | \(74\) |
default | \(f^{2} \left (-\frac {\left (d f +2 e f -\sqrt {d \,f^{2} \left (d +2 e \right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {f x \sqrt {2}}{\sqrt {-d f -e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )}{4 f^{2} \sqrt {d \,f^{2} \left (d +2 e \right )}\, \sqrt {-d f -e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}+\frac {\left (-d f -2 e f -\sqrt {d \,f^{2} \left (d +2 e \right )}\right ) \sqrt {2}\, \arctan \left (\frac {f x \sqrt {2}}{\sqrt {d f +e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )}{4 f^{2} \sqrt {d \,f^{2} \left (d +2 e \right )}\, \sqrt {d f +e f +\sqrt {d \,f^{2} \left (d +2 e \right )}}}\right )\) | \(193\) |
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, f^{2} x^{4} - 4 \, {\left (d - e\right )} f x^{2} + e^{2} + 4 \, {\left (2 \, f x^{3} + e x\right )} \sqrt {-d f}}{4 \, f^{2} x^{4} + 4 \, {\left (d + e\right )} f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{d}\right ) - \sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{3} + {\left (2 \, d + e\right )} x\right )} \sqrt {d f}}{d e}\right )}{2 \, d f}\right ] \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\sqrt {- \frac {1}{d f}} \log {\left (- d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} - \frac {\sqrt {- \frac {1}{d f}} \log {\left (d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{2} \right )}}{4} \]
[In]
[Out]
\[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\int { -\frac {2 \, f x^{2} - e}{4 \, f^{2} x^{4} + 4 \, d f x^{2} + 4 \, e f x^{2} + e^{2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (61) = 122\).
Time = 0.40 (sec) , antiderivative size = 266, normalized size of antiderivative = 3.28 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\sqrt {2} {\left (d e f {\left | f \right |} - \sqrt {d^{2} + 2 \, d e} {\left (d + e\right )} f^{2} + \sqrt {d^{2} + 2 \, d e} d f^{2}\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {4 \, d f + 4 \, e f + \sqrt {-16 \, e^{2} f^{2} + 16 \, {\left (d f + e f\right )}^{2}}}{f^{2}}}}\right )}{4 \, {\left (d^{2} + d e - \sqrt {d^{2} + 2 \, d e} d\right )} \sqrt {{\left (d + e + \sqrt {d^{2} + 2 \, d e}\right )} f} f^{2}} + \frac {\sqrt {2} {\left (d e f {\left | f \right |} + \sqrt {d^{2} + 2 \, d e} {\left (d + e\right )} f^{2} - \sqrt {d^{2} + 2 \, d e} d f^{2}\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {4 \, d f + 4 \, e f - \sqrt {-16 \, e^{2} f^{2} + 16 \, {\left (d f + e f\right )}^{2}}}{f^{2}}}}\right )}{4 \, {\left (d^{2} + d e + \sqrt {d^{2} + 2 \, d e} d\right )} \sqrt {{\left (d + e - \sqrt {d^{2} + 2 \, d e}\right )} f} f^{2}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.62 \[ \int \frac {e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx=\frac {\mathrm {atan}\left (\frac {2\,f^{3/2}\,x^3+2\,d\,\sqrt {f}\,x+e\,\sqrt {f}\,x}{\sqrt {d}\,e}\right )-\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {d}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]
[In]
[Out]