Optimal. Leaf size=18 \[ e^{2-\frac {x \log \left (\frac {9}{4}\right )}{3+4 x}} \]
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Rubi [A] time = 0.11, antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 27, 6706} \begin {gather*} e^2 3^{-\frac {2 x}{4 x+3}} 4^{\frac {x}{4 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (3 \log \left (\frac {9}{4}\right )\right ) \int \frac {e^{\frac {6+8 x-x \log \left (\frac {9}{4}\right )}{3+4 x}}}{9+24 x+16 x^2} \, dx\right )\\ &=-\left (\left (3 \log \left (\frac {9}{4}\right )\right ) \int \frac {e^{\frac {6+8 x-x \log \left (\frac {9}{4}\right )}{3+4 x}}}{(3+4 x)^2} \, dx\right )\\ &=3^{-\frac {2 x}{3+4 x}} 4^{\frac {x}{3+4 x}} e^2\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.10, size = 42, normalized size = 2.33 \begin {gather*} \frac {2^{-\frac {3+2 x}{3+4 x}} 9^{-\frac {x}{3+4 x}} e^2 \log \left (\frac {9}{4}\right )}{\log \left (\frac {3}{2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 1.00 \begin {gather*} e^{\left (\frac {x \log \left (\frac {4}{9}\right ) + 8 \, x + 6}{4 \, x + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 32, normalized size = 1.78 \begin {gather*} e^{\left (\frac {x \log \left (\frac {4}{9}\right )}{4 \, x + 3} + \frac {8 \, x}{4 \, x + 3} + \frac {6}{4 \, x + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 19, normalized size = 1.06
method | result | size |
gosper | \({\mathrm e}^{\frac {x \ln \left (\frac {4}{9}\right )+8 x +6}{3+4 x}}\) | \(19\) |
derivativedivides | \(\frac {\ln \left (\frac {4}{9}\right )^{2} {\mathrm e}^{\frac {\ln \left (\frac {4}{9}\right )}{4}+2-\frac {3 \ln \left (\frac {4}{9}\right )}{4 \left (3+4 x \right )}}}{4 \ln \relax (2)^{2}-8 \ln \relax (2) \ln \relax (3)+4 \ln \relax (3)^{2}}\) | \(42\) |
default | \(\frac {\ln \left (\frac {4}{9}\right )^{2} {\mathrm e}^{\frac {\ln \left (\frac {4}{9}\right )}{4}+2-\frac {3 \ln \left (\frac {4}{9}\right )}{4 \left (3+4 x \right )}}}{4 \ln \relax (2)^{2}-8 \ln \relax (2) \ln \relax (3)+4 \ln \relax (3)^{2}}\) | \(42\) |
risch | \(4^{\frac {x}{3+4 x}} \left (\frac {1}{9}\right )^{\frac {x}{3+4 x}} {\mathrm e}^{2}\) | \(45\) |
norman | \(\frac {4 x \,{\mathrm e}^{\frac {x \ln \left (\frac {4}{9}\right )+8 x +6}{3+4 x}}+3 \,{\mathrm e}^{\frac {x \ln \left (\frac {4}{9}\right )+8 x +6}{3+4 x}}}{3+4 x}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 44, normalized size = 2.44 \begin {gather*} -\frac {\sqrt {3} \sqrt {2} e^{\left (\frac {3 \, \log \relax (3)}{2 \, {\left (4 \, x + 3\right )}} - \frac {3 \, \log \relax (2)}{2 \, {\left (4 \, x + 3\right )}} + 2\right )} \log \left (\frac {4}{9}\right )}{6 \, {\left (\log \relax (3) - \log \relax (2)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 33, normalized size = 1.83 \begin {gather*} {\left (\frac {4}{9}\right )}^{\frac {x}{4\,x+3}}\,{\mathrm {e}}^{\frac {6}{4\,x+3}}\,{\mathrm {e}}^{\frac {8\,x}{4\,x+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 17, normalized size = 0.94 \begin {gather*} e^{\frac {x \log {\left (\frac {4}{9} \right )} + 8 x + 6}{4 x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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