Optimal. Leaf size=18 \[ \frac {e^x+x}{9+x}-\log (4+x) \]
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Rubi [A]
time = 0.36, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps
used = 10, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6820, 6874,
46, 78, 90, 2228} \begin {gather*} \frac {e^x}{x+9}-\frac {9}{x+9}-\log (x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 78
Rule 90
Rule 2228
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45-9 x-x^2+e^x \left (32+12 x+x^2\right )}{(4+x) (9+x)^2} \, dx\\ &=\int \left (-\frac {45}{(4+x) (9+x)^2}-\frac {9 x}{(4+x) (9+x)^2}-\frac {x^2}{(4+x) (9+x)^2}+\frac {e^x (8+x)}{(9+x)^2}\right ) \, dx\\ &=-\left (9 \int \frac {x}{(4+x) (9+x)^2} \, dx\right )-45 \int \frac {1}{(4+x) (9+x)^2} \, dx-\int \frac {x^2}{(4+x) (9+x)^2} \, dx+\int \frac {e^x (8+x)}{(9+x)^2} \, dx\\ &=\frac {e^x}{9+x}-9 \int \left (-\frac {4}{25 (4+x)}+\frac {9}{5 (9+x)^2}+\frac {4}{25 (9+x)}\right ) \, dx-45 \int \left (\frac {1}{25 (4+x)}-\frac {1}{5 (9+x)^2}-\frac {1}{25 (9+x)}\right ) \, dx-\int \left (\frac {16}{25 (4+x)}-\frac {81}{5 (9+x)^2}+\frac {9}{25 (9+x)}\right ) \, dx\\ &=-\frac {9}{9+x}+\frac {e^x}{9+x}-\log (4+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 23, normalized size = 1.28 \begin {gather*} -\frac {9}{9+x}+\frac {e^x}{9+x}-\log (4+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 23, normalized size = 1.28
method | result | size |
norman | \(\frac {-9+{\mathrm e}^{x}}{x +9}-\ln \left (4+x \right )\) | \(18\) |
default | \(\frac {{\mathrm e}^{x}}{x +9}-\frac {9}{x +9}-\ln \left (4+x \right )\) | \(23\) |
risch | \(\frac {{\mathrm e}^{x}}{x +9}-\frac {9}{x +9}-\ln \left (4+x \right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 22, normalized size = 1.22 \begin {gather*} \frac {e^{x}}{x + 9} - \frac {9}{x + 9} - \log \left (x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 21, normalized size = 1.17 \begin {gather*} -\frac {{\left (x + 9\right )} \log \left (x + 4\right ) - e^{x} + 9}{x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 0.83 \begin {gather*} - \log {\left (x + 4 \right )} + \frac {e^{x}}{x + 9} - \frac {9}{x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 25, normalized size = 1.39 \begin {gather*} -\frac {x \log \left (x + 4\right ) - e^{x} + 9 \, \log \left (x + 4\right ) + 9}{x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.20, size = 17, normalized size = 0.94 \begin {gather*} \frac {x+{\mathrm {e}}^x}{x+9}-\ln \left (x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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