Optimal. Leaf size=23 \[ \frac {9 (3+x)^2}{4 \left (5+e^x-x\right )^2 \log ^4(5)} \]
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Rubi [F]
time = 0.82, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {216+72 x+e^x \left (-54-45 x-9 x^2\right )}{2 e^{3 x} \log ^4(5)+e^{2 x} (30-6 x) \log ^4(5)+e^x \left (150-60 x+6 x^2\right ) \log ^4(5)+\left (250-150 x+30 x^2-2 x^3\right ) \log ^4(5)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 (3+x) \left (8-e^x (2+x)\right )}{2 \left (5+e^x-x\right )^3 \log ^4(5)} \, dx\\ &=\frac {9 \int \frac {(3+x) \left (8-e^x (2+x)\right )}{\left (5+e^x-x\right )^3} \, dx}{2 \log ^4(5)}\\ &=\frac {9 \int \left (-\frac {(-6+x) (3+x)^2}{\left (5+e^x-x\right )^3}-\frac {6+5 x+x^2}{\left (5+e^x-x\right )^2}\right ) \, dx}{2 \log ^4(5)}\\ &=-\frac {9 \int \frac {(-6+x) (3+x)^2}{\left (5+e^x-x\right )^3} \, dx}{2 \log ^4(5)}-\frac {9 \int \frac {6+5 x+x^2}{\left (5+e^x-x\right )^2} \, dx}{2 \log ^4(5)}\\ &=-\frac {9 \int \left (\frac {6}{\left (5+e^x-x\right )^2}+\frac {5 x}{\left (5+e^x-x\right )^2}+\frac {x^2}{\left (5+e^x-x\right )^2}\right ) \, dx}{2 \log ^4(5)}-\frac {9 \int \left (-\frac {54}{\left (5+e^x-x\right )^3}-\frac {27 x}{\left (5+e^x-x\right )^3}+\frac {x^3}{\left (5+e^x-x\right )^3}\right ) \, dx}{2 \log ^4(5)}\\ &=-\frac {9 \int \frac {x^2}{\left (5+e^x-x\right )^2} \, dx}{2 \log ^4(5)}-\frac {9 \int \frac {x^3}{\left (5+e^x-x\right )^3} \, dx}{2 \log ^4(5)}-\frac {45 \int \frac {x}{\left (5+e^x-x\right )^2} \, dx}{2 \log ^4(5)}-\frac {27 \int \frac {1}{\left (5+e^x-x\right )^2} \, dx}{\log ^4(5)}+\frac {243 \int \frac {x}{\left (5+e^x-x\right )^3} \, dx}{2 \log ^4(5)}+\frac {243 \int \frac {1}{\left (5+e^x-x\right )^3} \, dx}{\log ^4(5)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.35, size = 23, normalized size = 1.00 \begin {gather*} \frac {9 (3+x)^2}{4 \left (5+e^x-x\right )^2 \log ^4(5)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 24, normalized size = 1.04
method | result | size |
risch | \(\frac {\frac {9}{4} x^{2}+\frac {27}{2} x +\frac {81}{4}}{\ln \left (5\right )^{4} \left (-{\mathrm e}^{x}+x -5\right )^{2}}\) | \(24\) |
norman | \(\frac {\frac {36 x}{\ln \left (5\right )}-\frac {9 \,{\mathrm e}^{2 x}}{4 \ln \left (5\right )}-\frac {45 \,{\mathrm e}^{x}}{2 \ln \left (5\right )}+\frac {9 \,{\mathrm e}^{x} x}{2 \ln \left (5\right )}-\frac {36}{\ln \left (5\right )}}{\ln \left (5\right )^{3} \left (-{\mathrm e}^{x}+x -5\right )^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (20) = 40\).
time = 0.53, size = 60, normalized size = 2.61 \begin {gather*} \frac {9 \, {\left (x^{2} + 6 \, x + 9\right )}}{4 \, {\left (x^{2} \log \left (5\right )^{4} - 10 \, x \log \left (5\right )^{4} + e^{\left (2 \, x\right )} \log \left (5\right )^{4} + 25 \, \log \left (5\right )^{4} - 2 \, {\left (x \log \left (5\right )^{4} - 5 \, \log \left (5\right )^{4}\right )} e^{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (20) = 40\).
time = 0.36, size = 48, normalized size = 2.09 \begin {gather*} -\frac {9 \, {\left (x^{2} + 6 \, x + 9\right )}}{4 \, {\left (2 \, {\left (x - 5\right )} e^{x} \log \left (5\right )^{4} - {\left (x^{2} - 10 \, x + 25\right )} \log \left (5\right )^{4} - e^{\left (2 \, x\right )} \log \left (5\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (20) = 40\).
time = 0.09, size = 65, normalized size = 2.83 \begin {gather*} \frac {9 x^{2} + 54 x + 81}{4 x^{2} \log {\left (5 \right )}^{4} - 40 x \log {\left (5 \right )}^{4} + \left (- 8 x \log {\left (5 \right )}^{4} + 40 \log {\left (5 \right )}^{4}\right ) e^{x} + 4 e^{2 x} \log {\left (5 \right )}^{4} + 100 \log {\left (5 \right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (20) = 40\).
time = 0.41, size = 60, normalized size = 2.61 \begin {gather*} \frac {9 \, {\left (x^{2} + 6 \, x + 9\right )}}{4 \, {\left (x^{2} \log \left (5\right )^{4} - 2 \, x e^{x} \log \left (5\right )^{4} - 10 \, x \log \left (5\right )^{4} + e^{\left (2 \, x\right )} \log \left (5\right )^{4} + 10 \, e^{x} \log \left (5\right )^{4} + 25 \, \log \left (5\right )^{4}\right )}} \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 = -0.04 \begin {gather*} \int \frac {72\,x-{\mathrm {e}}^x\,\left (9\,x^2+45\,x+54\right )+216}{2\,{\mathrm {e}}^{3\,x}\,{\ln \left (5\right )}^4-{\ln \left (5\right )}^4\,\left (2\,x^3-30\,x^2+150\,x-250\right )+{\mathrm {e}}^x\,{\ln \left (5\right )}^4\,\left (6\,x^2-60\,x+150\right )-{\mathrm {e}}^{2\,x}\,{\ln \left (5\right )}^4\,\left (6\,x-30\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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