Integrand size = 24, antiderivative size = 79 \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {641, 44, 53, 65, 212} \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e}+\frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (e x+2)} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(6-3 e x)^{3/2} (2+e x)^2} \, dx \\ & = -\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}+\frac {3}{8} \int \frac {1}{(6-3 e x)^{3/2} (2+e x)} \, dx \\ & = \frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}+\frac {1}{32} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = \frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{48 e} \\ & = \frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {4+6 e x-3 \sqrt {2+e x} \sqrt {4-e^2 x^2} \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{96 e \sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \]
[In]
[Out]
Time = 2.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {\sqrt {-3 x^{2} e^{2}+12}\, \left (\sqrt {3}\, \sqrt {-3 e x +6}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}-6 e x -4\right )}{288 \left (e x +2\right )^{\frac {3}{2}} \left (e x -2\right ) e}\) | \(93\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (59) = 118\).
Time = 0.40 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {3 \, \sqrt {3} {\left (e^{3} x^{3} + 2 \, e^{2} x^{2} - 4 \, e x - 8\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (3 \, e x + 2\right )} \sqrt {e x + 2}}{576 \, {\left (e^{4} x^{3} + 2 \, e^{3} x^{2} - 4 \, e^{2} x - 8 \, e\right )}} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \int \frac {1}{- e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} \sqrt {e x + 2}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {-e x + 2} + 2\right )}{192 \, e} + \frac {\sqrt {3} \log \left (-\sqrt {-e x + 2} + 2\right )}{192 \, e} - \frac {\sqrt {3} {\left (3 \, e x + 2\right )}}{144 \, {\left ({\left (-e x + 2\right )}^{\frac {3}{2}} - 4 \, \sqrt {-e x + 2}\right )} e} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{3/2}\,\sqrt {e\,x+2}} \,d x \]
[In]
[Out]