Optimal. Leaf size=40 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right ) \]
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Rubi [A]
time = 0.07, 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 = {2320, 738, 212}
\begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 2320
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} \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} \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 = 0.12, size = 42, normalized size = 1.05 \begin {gather*} -\frac {4}{3} \tanh ^{-1}\left (1-\frac {1}{2} e^{3 x/4}+\frac {1}{2} \sqrt {-2+e^{3 x/4}+e^{3 x/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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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 = 2.79, size = 39, normalized size = 0.98 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 46, normalized size = 1.15 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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