Integrand size = 39, antiderivative size = 40 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2320, 738, 212} \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]
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Rule 212
Rule 738
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {4}{3} \text {Subst}\left (\int \frac {1}{(-2+x) \sqrt {-2+x+x^2}} \, dx,x,e^{3 x/4}\right ) \\ & = -\left (\frac {8}{3} \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {-2+5 e^{3 x/4}}{\sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right )\right ) \\ & = \frac {2}{3} \text {arctanh}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {4}{3} \text {arctanh}\left (1-\frac {1}{2} e^{3 x/4}+\frac {1}{2} \sqrt {-2+e^{3 x/4}+e^{3 x/2}}\right ) \]
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\[\int \frac {{\mathrm e}^{\frac {3 x}{4}}}{\left (-2+{\mathrm e}^{\frac {3 x}{4}}\right ) \sqrt {-2+{\mathrm e}^{\frac {3 x}{4}}+{\mathrm e}^{\frac {3 x}{2}}}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4\right ) + \frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )}\right ) \]
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\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int \frac {e^{\frac {3 x}{4}}}{\left (e^{\frac {3 x}{4}} - 2\right ) \sqrt {e^{\frac {3 x}{4}} + e^{\frac {3 x}{2}} - 2}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {2}{3} \, \log \left (\frac {4 \, \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + \frac {8}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + 5\right ) \]
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\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int { \frac {e^{\left (\frac {3}{4} \, x\right )}}{\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} {\left (e^{\left (\frac {3}{4} \, x\right )} - 2\right )}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int \frac {{\mathrm {e}}^{\frac {3\,x}{4}}}{\left ({\mathrm {e}}^{\frac {3\,x}{4}}-2\right )\,\sqrt {{\mathrm {e}}^{\frac {3\,x}{2}}+{\mathrm {e}}^{\frac {3\,x}{4}}-2}} \,d x \]
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