Optimal. Leaf size=27 \[ \frac {(4-x) x}{13-e^{\left (1+x^2\right )^2}-x^2} \]
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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 {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\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 {52-26 x+4 x^2+2 e^{\left (1+x^2\right )^2} \left (-2+x+8 x^2-2 x^3+8 x^4-2 x^5\right )}{\left (13-e^{\left (1+x^2\right )^2}-x^2\right )^2} \, dx\\ &=\int \left (\frac {2 x^2 \left (108-27 x+96 x^2-24 x^3-8 x^4+2 x^5\right )}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {2 \left (2-x-8 x^2+2 x^3-8 x^4+2 x^5\right )}{-13+e^{\left (1+x^2\right )^2}+x^2}\right ) \, dx\\ &=2 \int \frac {x^2 \left (108-27 x+96 x^2-24 x^3-8 x^4+2 x^5\right )}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-2 \int \frac {2-x-8 x^2+2 x^3-8 x^4+2 x^5}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx\\ &=2 \int \left (\frac {108 x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {27 x^3}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}+\frac {96 x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {24 x^5}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {8 x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}+\frac {2 x^7}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}\right ) \, dx-2 \int \left (\frac {2}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {x}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {8 x^2}{-13+e^{\left (1+x^2\right )^2}+x^2}+\frac {2 x^3}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {8 x^4}{-13+e^{\left (1+x^2\right )^2}+x^2}+\frac {2 x^5}{-13+e^{\left (1+x^2\right )^2}+x^2}\right ) \, dx\\ &=2 \int \frac {x}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+4 \int \frac {x^7}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-4 \int \frac {1}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-4 \int \frac {x^3}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-4 \int \frac {x^5}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-16 \int \frac {x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+16 \int \frac {x^2}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+16 \int \frac {x^4}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-48 \int \frac {x^5}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-54 \int \frac {x^3}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+192 \int \frac {x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+216 \int \frac {x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx\\ &=2 \text {Subst}\left (\int \frac {x^3}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {x}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {x^2}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )-4 \int \frac {1}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-16 \int \frac {x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+16 \int \frac {x^2}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+16 \int \frac {x^4}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-24 \text {Subst}\left (\int \frac {x^2}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )-27 \text {Subst}\left (\int \frac {x}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )+192 \int \frac {x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+216 \int \frac {x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+\text {Subst}\left (\int \frac {1}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 2.15, size = 21, normalized size = 0.78 \begin {gather*} \frac {(-4+x) x}{-13+e^{\left (1+x^2\right )^2}+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 21, normalized size = 0.78
method | result | size |
risch | \(\frac {\left (x -4\right ) x}{x^{2}+{\mathrm e}^{\left (x^{2}+1\right )^{2}}-13}\) | \(21\) |
norman | \(\frac {x^{2}-4 x}{x^{2}+{\mathrm e}^{x^{4}+2 x^{2}+1}-13}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{2} - 4 x}{x^{2} + e^{x^{4} + 2 x^{2} + 1} - 13} \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 = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 29, normalized size = 1.07 \begin {gather*} -\frac {4\,x-x^2}{{\mathrm {e}}^{x^4+2\,x^2+1}+x^2-13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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