Optimal. Leaf size=28 \[ -e^{x+\frac {3}{4+x^2}} x^2+\frac {4+x}{1+x} \]
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Rubi [F] time = 1.26, 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 {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 \left (4+x^2\right )^2-e^{x+\frac {3}{4+x^2}} x (1+x)^2 \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+4 x+x^2+x^3\right )^2} \, dx\\ &=\int \left (-\frac {3}{(1+x)^2}-\frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2}\right ) \, dx\\ &=\frac {3}{1+x}-\int \frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2} \, dx\\ &=\frac {3}{1+x}-\int \left (2 e^{x+\frac {3}{4+x^2}} x+e^{x+\frac {3}{4+x^2}} x^2+\frac {24 e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2}-\frac {6 e^{x+\frac {3}{4+x^2}} x}{4+x^2}\right ) \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \frac {e^{x+\frac {3}{4+x^2}} x}{4+x^2} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \left (-\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i-x)}+\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i+x)}\right ) \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx-3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i-x} \, dx+3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i+x} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} -e^{x+\frac {3}{4+x^2}} x^2+\frac {3}{1+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 34, normalized size = 1.21 \begin {gather*} -\frac {{\left (x^{3} + x^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x + 3}{x^{2} + 4}\right )} - 3}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 69, normalized size = 2.46 \begin {gather*} -\frac {x^{3} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} + x^{2} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} - 3}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 31, normalized size = 1.11
method | result | size |
risch | \(\frac {3}{x +1}-x^{2} {\mathrm e}^{\frac {x^{3}+4 x +3}{x^{2}+4}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 91, normalized size = 3.25 \begin {gather*} -x^{2} e^{\left (x + \frac {3}{x^{2} + 4}\right )} + \frac {6 \, {\left (11 \, x^{2} - 5 \, x + 24\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} + \frac {3 \, {\left (7 \, x^{2} - 10 \, x - 12\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} - \frac {12 \, {\left (x^{2} - 5 \, x - 16\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.09, size = 25, normalized size = 0.89 \begin {gather*} \frac {3}{x+1}-x^2\,{\mathrm {e}}^{\frac {3}{x^2+4}}\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.77, size = 19, normalized size = 0.68 \begin {gather*} - x^{2} e^{x} e^{\frac {3}{x^{2} + 4}} + \frac {3}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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