Optimal. Leaf size=29 \[ \frac {x^2}{\left (-e^{x^2}+\frac {x^3}{e^{42+x}+x^2}\right )^2} \]
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Rubi [F] time = 7.30, 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 {e^{84+2 x} \left (4 x^4-2 x^5\right )+e^{42+x} \left (4 x^6-2 x^7\right )+e^{x^2} \left (2 x^7-4 x^9+e^{126+3 x} \left (2 x-4 x^3\right )+e^{84+2 x} \left (6 x^3-12 x^5\right )+e^{42+x} \left (6 x^5-12 x^7\right )\right )}{-x^9+e^{3 x^2} \left (e^{126+3 x}+3 e^{84+2 x} x^2+3 e^{42+x} x^4+x^6\right )+e^{2 x^2} \left (-3 e^{84+2 x} x^3-6 e^{42+x} x^5-3 x^7\right )+e^{x^2} \left (3 e^{42+x} x^6+3 x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (e^{42+x}+x^2\right ) \left (-e^{42+x} (-2+x) x^3-e^{84+2 x+x^2} \left (-1+2 x^2\right )-2 e^{42+x+x^2} x^2 \left (-1+2 x^2\right )-e^{x^2} x^4 \left (-1+2 x^2\right )\right )}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\\ &=2 \int \frac {x \left (e^{42+x}+x^2\right ) \left (-e^{42+x} (-2+x) x^3-e^{84+2 x+x^2} \left (-1+2 x^2\right )-2 e^{42+x+x^2} x^2 \left (-1+2 x^2\right )-e^{x^2} x^4 \left (-1+2 x^2\right )\right )}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\\ &=2 \int \left (-\frac {x \left (e^{42+x}+x^2\right )^2 \left (-1+2 x^2\right )}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2}+\frac {x^4 \left (-3 e^{84+2 x}+e^{84+2 x} x-4 e^{42+x} x^2+2 e^{84+2 x} x^2+e^{42+x} x^3-x^4+4 e^{42+x} x^4+2 x^6\right )}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (e^{42+x}+x^2\right )^2 \left (-1+2 x^2\right )}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2} \, dx\right )+2 \int \frac {x^4 \left (-3 e^{84+2 x}+e^{84+2 x} x-4 e^{42+x} x^2+2 e^{84+2 x} x^2+e^{42+x} x^3-x^4+4 e^{42+x} x^4+2 x^6\right )}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx\\ &=2 \int \left (\frac {3 e^{84+2 x} x^4}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}-\frac {e^{84+2 x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}+\frac {4 e^{42+x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}-\frac {2 e^{84+2 x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}-\frac {e^{42+x} x^7}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}-\frac {4 e^{42+x} x^8}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3}-\frac {x^8}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3}+\frac {2 x^{10}}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3}\right ) \, dx-2 \int \left (-\frac {x \left (e^{42+x}+x^2\right )^2}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2}+\frac {2 x^3 \left (e^{42+x}+x^2\right )^2}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{84+2 x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\right )-2 \int \frac {e^{42+x} x^7}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-2 \int \frac {x^8}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx+2 \int \frac {x \left (e^{42+x}+x^2\right )^2}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2} \, dx-4 \int \frac {e^{84+2 x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+4 \int \frac {x^{10}}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx-4 \int \frac {x^3 \left (e^{42+x}+x^2\right )^2}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2} \, dx+6 \int \frac {e^{84+2 x} x^4}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+8 \int \frac {e^{42+x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-8 \int \frac {e^{42+x} x^8}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\\ &=-\left (2 \int \frac {e^{84+2 x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\right )-2 \int \frac {e^{42+x} x^7}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-2 \int \frac {x^8}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx+2 \int \left (\frac {e^{84+2 x} x}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2}+\frac {2 e^{42+x} x^3}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2}+\frac {x^5}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2}\right ) \, dx-4 \int \frac {e^{84+2 x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+4 \int \frac {x^{10}}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx-4 \int \left (\frac {e^{84+2 x} x^3}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2}+\frac {2 e^{42+x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2}+\frac {x^7}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2}\right ) \, dx+6 \int \frac {e^{84+2 x} x^4}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+8 \int \frac {e^{42+x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-8 \int \frac {e^{42+x} x^8}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\\ &=-\left (2 \int \frac {e^{84+2 x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx\right )-2 \int \frac {e^{42+x} x^7}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+2 \int \frac {e^{84+2 x} x}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2} \, dx-2 \int \frac {x^8}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx+2 \int \frac {x^5}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2} \, dx-4 \int \frac {e^{84+2 x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+4 \int \frac {e^{42+x} x^3}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2} \, dx-4 \int \frac {e^{84+2 x} x^3}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2} \, dx+4 \int \frac {x^{10}}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^3} \, dx-4 \int \frac {x^7}{\left (-e^{42+x+x^2}-e^{x^2} x^2+x^3\right )^2} \, dx+6 \int \frac {e^{84+2 x} x^4}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx+8 \int \frac {e^{42+x} x^6}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-8 \int \frac {e^{42+x} x^8}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^3} \, dx-8 \int \frac {e^{42+x} x^5}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 40, normalized size = 1.38 \begin {gather*} \frac {x^2 \left (e^{42+x}+x^2\right )^2}{\left (e^{42+x+x^2}+e^{x^2} x^2-x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 74, normalized size = 2.55 \begin {gather*} \frac {x^{6} + 2 \, x^{4} e^{\left (x + 42\right )} + x^{2} e^{\left (2 \, x + 84\right )}}{x^{6} + {\left (x^{4} + 2 \, x^{2} e^{\left (x + 42\right )} + e^{\left (2 \, x + 84\right )}\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{5} + x^{3} e^{\left (x + 42\right )}\right )} e^{\left (x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.38, size = 114, normalized size = 3.93 \begin {gather*} \frac {x^{6} e^{\left (2 \, x^{2}\right )} + 2 \, x^{4} e^{\left (2 \, x^{2} + x + 42\right )} + x^{2} e^{\left (2 \, x^{2} + 2 \, x + 84\right )}}{x^{6} e^{\left (2 \, x^{2}\right )} - 2 \, x^{5} e^{\left (3 \, x^{2}\right )} + x^{4} e^{\left (4 \, x^{2}\right )} - 2 \, x^{3} e^{\left (3 \, x^{2} + x + 42\right )} + 2 \, x^{2} e^{\left (4 \, x^{2} + x + 42\right )} + e^{\left (4 \, x^{2} + 2 \, x + 84\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 69, normalized size = 2.38
method | result | size |
risch | \(x^{2} {\mathrm e}^{-2 x^{2}}-\frac {\left (x^{3}-2 x^{2} {\mathrm e}^{x^{2}}-2 \,{\mathrm e}^{x^{2}+x +42}\right ) x^{5} {\mathrm e}^{-2 x^{2}}}{\left (x^{3}-x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}+x +42}\right )^{2}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.10, size = 74, normalized size = 2.55 \begin {gather*} \frac {x^{6} + 2 \, x^{4} e^{\left (x + 42\right )} + x^{2} e^{\left (2 \, x + 84\right )}}{x^{6} + {\left (x^{4} + 2 \, x^{2} e^{\left (x + 42\right )} + e^{\left (2 \, x + 84\right )}\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{5} + x^{3} e^{\left (x + 42\right )}\right )} e^{\left (x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.45, size = 175, normalized size = 6.03 \begin {gather*} \frac {{\mathrm {e}}^{x+42}\,\left (6\,x^{10}+x^9-5\,x^8\right )-7\,x^6\,{\mathrm {e}}^{2\,x+84}+2\,x^7\,{\mathrm {e}}^{2\,x+84}+6\,x^8\,{\mathrm {e}}^{2\,x+84}-3\,x^4\,{\mathrm {e}}^{3\,x+126}+x^5\,{\mathrm {e}}^{3\,x+126}+2\,x^6\,{\mathrm {e}}^{3\,x+126}-x^{10}+2\,x^{12}}{\left ({\mathrm {e}}^{2\,x^2}\,{\left ({\mathrm {e}}^{x+42}+x^2\right )}^2+x^6-2\,x^3\,{\mathrm {e}}^{x^2}\,\left ({\mathrm {e}}^{x+42}+x^2\right )\right )\,\left (x^3\,{\mathrm {e}}^{x+42}-3\,x^2\,{\mathrm {e}}^{x+42}+2\,x^4\,{\mathrm {e}}^{x+42}-x^4+2\,x^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.50, size = 112, normalized size = 3.86 \begin {gather*} x^{2} e^{- 2 x^{2}} + \frac {- x^{8} + 2 x^{7} e^{x^{2}} + 2 x^{5} e^{x^{2}} e^{x + 42}}{x^{6} e^{2 x^{2}} - 2 x^{5} e^{3 x^{2}} + x^{4} e^{4 x^{2}} + \left (- 2 x^{3} e^{3 x^{2}} + 2 x^{2} e^{4 x^{2}}\right ) e^{x + 42} + e^{4 x^{2}} e^{2 x + 84}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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