Optimal. Leaf size=24 \[ e^5-\frac {x}{3 \left (1+60 e^x (-2+x)^2 x\right )} \]
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Rubi [F]
time = 1.85, 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 {-1+e^x \left (-120 x^3+60 x^4\right )}{3+e^x \left (1440 x-1440 x^2+360 x^3\right )+e^{2 x} \left (172800 x^2-345600 x^3+259200 x^4-86400 x^5+10800 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+60 e^x (-2+x) x^3}{3 \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-1+60 e^x (-2+x) x^3}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {-2+x+x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {-2+x+x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )} \, dx\\ &=\frac {1}{3} \int \frac {x^2}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )} \, dx-\frac {1}{3} \int \left (\frac {3}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {4}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {x}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\right )+\frac {1}{3} \int \left (\frac {2}{1+240 e^x x-240 e^x x^2+60 e^x x^3}+\frac {4}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )}+\frac {x}{1+240 e^x x-240 e^x x^2+60 e^x x^3}\right ) \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx-\int \frac {1}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x}{1+240 e^x x-240 e^x x^2+60 e^x x^3} \, dx+\frac {2}{3} \int \frac {1}{1+240 e^x x-240 e^x x^2+60 e^x x^3} \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx+\frac {4}{3} \int \frac {1}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )} \, dx-\int \frac {1}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x}{1+60 e^x (-2+x)^2 x} \, dx+\frac {2}{3} \int \frac {1}{1+60 e^x (-2+x)^2 x} \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx+\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )} \, dx-\int \frac {1}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.20, size = 20, normalized size = 0.83 \begin {gather*} -\frac {x}{3 \left (1+60 e^x (-2+x)^2 x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 27, normalized size = 1.12
method | result | size |
norman | \(-\frac {x}{3 \left (60 \,{\mathrm e}^{x} x^{3}-240 \,{\mathrm e}^{x} x^{2}+240 \,{\mathrm e}^{x} x +1\right )}\) | \(27\) |
risch | \(-\frac {x}{3 \left (60 \,{\mathrm e}^{x} x^{3}-240 \,{\mathrm e}^{x} x^{2}+240 \,{\mathrm e}^{x} x +1\right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 23, normalized size = 0.96 \begin {gather*} -\frac {x}{3 \, {\left (60 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{x} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 23, normalized size = 0.96 \begin {gather*} -\frac {x}{3 \, {\left (60 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{x} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 20, normalized size = 0.83 \begin {gather*} - \frac {x}{\left (180 x^{3} - 720 x^{2} + 720 x\right ) e^{x} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 1.08 \begin {gather*} -\frac {x}{3 \, {\left (60 \, x^{3} e^{x} - 240 \, x^{2} e^{x} + 240 \, x e^{x} + 1\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 {{\mathrm {e}}^x\,\left (120\,x^3-60\,x^4\right )+1}{{\mathrm {e}}^{2\,x}\,\left (10800\,x^6-86400\,x^5+259200\,x^4-345600\,x^3+172800\,x^2\right )+{\mathrm {e}}^x\,\left (360\,x^3-1440\,x^2+1440\,x\right )+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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