Optimal. Leaf size=27 \[ x+\frac {25 e^x (1-x) \log ^2\left (1+\frac {2}{x}\right )}{2 x^2} \]
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Rubi [F]
time = 3.17, 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 {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{4 x^3+2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{x^3 (4+2 x)} \, dx\\ &=\int \left (1+\frac {25 e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-4 \log \left (\frac {2+x}{x}\right )+2 x \log \left (\frac {2+x}{x}\right )-x^3 \log \left (\frac {2+x}{x}\right )\right )}{2 x^3 (2+x)}\right ) \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-4 \log \left (\frac {2+x}{x}\right )+2 x \log \left (\frac {2+x}{x}\right )-x^3 \log \left (\frac {2+x}{x}\right )\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-\left (4-2 x+x^3\right ) \log \left (\frac {2+x}{x}\right )\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25}{2} \int \left (\frac {4 e^x (-1+x) \log \left (1+\frac {2}{x}\right )}{x^3 (2+x)}+\frac {e^x \left (-2+2 x-x^2\right ) \log ^2\left (1+\frac {2}{x}\right )}{x^3}\right ) \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \left (-2+2 x-x^2\right ) \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+50 \int \frac {e^x (-1+x) \log \left (1+\frac {2}{x}\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \left (-\frac {2 e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3}+\frac {2 e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2}-\frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x}\right ) \, dx-50 \int \frac {2 e^{2+x} (-1+2 x)-e^2 x^2 \text {Ei}(x)-3 x^2 \text {Ei}(2+x)}{4 e^2 x^3 (2+x)} \, dx\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \frac {2 e^{2+x} (-1+2 x)-e^2 x^2 \text {Ei}(x)-3 x^2 \text {Ei}(2+x)}{x^3 (2+x)} \, dx}{2 e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \left (\frac {2 e^{2+x} (-1+2 x)}{x^3 (2+x)}+\frac {-e^2 \text {Ei}(x)-3 \text {Ei}(2+x)}{x (2+x)}\right ) \, dx}{2 e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \frac {-e^2 \text {Ei}(x)-3 \text {Ei}(2+x)}{x (2+x)} \, dx}{2 e^2}-\frac {25 \int \frac {e^{2+x} (-1+2 x)}{x^3 (2+x)} \, dx}{e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \left (-\frac {e^2 \text {Ei}(x)}{x (2+x)}-\frac {3 \text {Ei}(2+x)}{x (2+x)}\right ) \, dx}{2 e^2}-\frac {25 \int \left (-\frac {e^{2+x}}{2 x^3}+\frac {5 e^{2+x}}{4 x^2}-\frac {5 e^{2+x}}{8 x}+\frac {5 e^{2+x}}{8 (2+x)}\right ) \, dx}{e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \frac {\text {Ei}(x)}{x (2+x)} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x^3} \, dx}{2 e^2}+\frac {125 \int \frac {e^{2+x}}{x} \, dx}{8 e^2}-\frac {125 \int \frac {e^{2+x}}{2+x} \, dx}{8 e^2}-\frac {125 \int \frac {e^{2+x}}{x^2} \, dx}{4 e^2}+\frac {75 \int \frac {\text {Ei}(2+x)}{x (2+x)} \, dx}{2 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {125 e^x}{4 x}+x+\frac {125 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \left (\frac {\text {Ei}(x)}{2 x}-\frac {\text {Ei}(x)}{2 (2+x)}\right ) \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x^2} \, dx}{4 e^2}-\frac {125 \int \frac {e^{2+x}}{x} \, dx}{4 e^2}+\frac {75 \int \left (\frac {\text {Ei}(2+x)}{2 x}-\frac {\text {Ei}(2+x)}{2 (2+x)}\right ) \, dx}{2 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {125 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{4} \int \frac {\text {Ei}(x)}{x} \, dx-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x} \, dx}{4 e^2}+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {75 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{4} (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {25}{4} \int \frac {E_1(-x)}{x} \, dx-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {75 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25}{4} x \, _3F_3(1,1,1;2,2,2;x)+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{8} \log ^2(-x)+\frac {25}{4} \gamma \log (x)+\frac {25}{4} (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 6.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{4 x^3+2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.41, size = 1010, normalized size = 37.41
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1010\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 43, normalized size = 1.59 \begin {gather*} x - \frac {25 \, {\left ({\left (x - 1\right )} e^{x} \log \left (x + 2\right )^{2} - 2 \, {\left (x - 1\right )} e^{x} \log \left (x + 2\right ) \log \left (x\right ) + {\left (x - 1\right )} e^{x} \log \left (x\right )^{2}\right )}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 28, normalized size = 1.04 \begin {gather*} -\frac {25 \, {\left (x - 1\right )} e^{x} \log \left (\frac {x + 2}{x}\right )^{2} - 2 \, x^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 31, normalized size = 1.15 \begin {gather*} x + \frac {\left (- 25 x \log {\left (\frac {x + 2}{x} \right )}^{2} + 25 \log {\left (\frac {x + 2}{x} \right )}^{2}\right ) e^{x}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 40, normalized size = 1.48 \begin {gather*} -\frac {25 \, x e^{x} \log \left (\frac {x + 2}{x}\right )^{2} - 2 \, x^{3} - 25 \, e^{x} \log \left (\frac {x + 2}{x}\right )^{2}}{2 \, x^{2}} \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 {4\,x^3+2\,x^4-{\mathrm {e}}^x\,{\ln \left (\frac {x+2}{x}\right )}^2\,\left (25\,x^3-50\,x+100\right )+{\mathrm {e}}^x\,\ln \left (\frac {x+2}{x}\right )\,\left (100\,x-100\right )}{2\,x^4+4\,x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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