Optimal. Leaf size=40 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2282, 724, 206} \[ \frac {2}{3} \tanh ^{-1}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 2282
Rubi steps
\begin {align*} \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} \operatorname {Subst}\left (\int \frac {1}{(-2+x) \sqrt {-2+x+x^2}} \, dx,x,e^{3 x/4}\right )\\ &=-\left (\frac {8}{3} \operatorname {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} \tanh ^{-1}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \[ -\frac {2}{3} \tanh ^{-1}\left (\frac {5 e^{3 x/4}-2}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.03, size = 46, normalized size = 1.15 \[ -\frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4\right ) + \frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 48, normalized size = 1.20 \[ -\frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4 \right |}\right ) + \frac {2}{3} \, \log \left ({\left | \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\frac {3 x}{4}}}{\left (-2+{\mathrm e}^{\frac {3 x}{4}}\right ) \sqrt {-2+{\mathrm e}^{\frac {3 x}{4}}+{\mathrm e}^{\frac {3 x}{2}}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 39, normalized size = 0.98 \[ -\frac {2}{3} \, \log \left (\frac {4 \, \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + \frac {8}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{\frac {3\,x}{4}}}{\left ({\mathrm {e}}^{\frac {3\,x}{4}}-2\right )\,\sqrt {{\mathrm {e}}^{\frac {3\,x}{2}}+{\mathrm {e}}^{\frac {3\,x}{4}}-2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\frac {3 x}{4}}}{\left (e^{\frac {3 x}{4}} - 2\right ) \sqrt {e^{\frac {3 x}{4}} + e^{\frac {3 x}{2}} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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