Optimal. Leaf size=27 \[ \frac {4 x^2}{7+\frac {e^{3-x^2}}{3}-\frac {4}{x^2}} \]
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Rubi [F]
time = 2.84, 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^{2 x^2} \left (-576 x^3+504 x^5\right )+e^{3+x^2} \left (24 x^5+24 x^7\right )}{e^6 x^4+e^{3+x^2} \left (-24 x^2+42 x^4\right )+e^{2 x^2} \left (144-504 x^2+441 x^4\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 {24 e^{x^2} x^3 \left (3 e^{x^2} \left (-8+7 x^2\right )+e^3 \left (x^2+x^4\right )\right )}{\left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )^2} \, dx\\ &=24 \int \frac {e^{x^2} x^3 \left (3 e^{x^2} \left (-8+7 x^2\right )+e^3 \left (x^2+x^4\right )\right )}{\left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )^2} \, dx\\ &=12 \text {Subst}\left (\int \frac {e^x x \left (e^3 x (1+x)+3 e^x (-8+7 x)\right )}{\left (e^3 x+3 e^x (-4+7 x)\right )^2} \, dx,x,x^2\right )\\ &=12 \text {Subst}\left (\int \left (\frac {e^x x (-8+7 x)}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )}+\frac {e^{3+x} x^2 \left (4-4 x+7 x^2\right )}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2}\right ) \, dx,x,x^2\right )\\ &=12 \text {Subst}\left (\int \frac {e^x x (-8+7 x)}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )} \, dx,x,x^2\right )+12 \text {Subst}\left (\int \frac {e^{3+x} x^2 \left (4-4 x+7 x^2\right )}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )\\ &=12 \text {Subst}\left (\int \left (\frac {16 e^{3+x}}{49 \left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {4 e^{3+x} x}{7 \left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {e^{3+x} x^3}{\left (-12 e^x+e^3 x+21 e^x x\right )^2}+\frac {64 e^{3+x}}{49 (-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2}\right ) \, dx,x,x^2\right )+12 \text {Subst}\left (\int \left (-\frac {4 e^x}{7 \left (-12 e^x+e^3 x+21 e^x x\right )}+\frac {e^x x}{-12 e^x+e^3 x+21 e^x x}-\frac {16 e^x}{7 (-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {192}{49} \text {Subst}\left (\int \frac {e^{3+x}}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )+\frac {48}{7} \text {Subst}\left (\int \frac {e^{3+x} x}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )-\frac {48}{7} \text {Subst}\left (\int \frac {e^x}{-12 e^x+e^3 x+21 e^x x} \, dx,x,x^2\right )+12 \text {Subst}\left (\int \frac {e^{3+x} x^3}{\left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )+12 \text {Subst}\left (\int \frac {e^x x}{-12 e^x+e^3 x+21 e^x x} \, dx,x,x^2\right )+\frac {768}{49} \text {Subst}\left (\int \frac {e^{3+x}}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )^2} \, dx,x,x^2\right )-\frac {192}{7} \text {Subst}\left (\int \frac {e^x}{(-4+7 x) \left (-12 e^x+e^3 x+21 e^x x\right )} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(27)=54\).
time = 4.16, size = 56, normalized size = 2.07 \begin {gather*} \frac {4 \left (-4 e^3 x^2+3 e^{x^2} \left (16-28 x^2+49 x^4\right )\right )}{49 \left (e^3 x^2+3 e^{x^2} \left (-4+7 x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 34, normalized size = 1.26
method | result | size |
norman | \(\frac {12 x^{4} {\mathrm e}^{x^{2}}}{x^{2} {\mathrm e}^{3}+21 x^{2} {\mathrm e}^{x^{2}}-12 \,{\mathrm e}^{x^{2}}}\) | \(34\) |
risch | \(\frac {4 x^{2}}{7}+\frac {64}{343 \left (x^{2}-\frac {4}{7}\right )}-\frac {4 \,{\mathrm e}^{3} x^{6}}{\left (7 x^{2}-4\right ) \left (x^{2} {\mathrm e}^{3}+21 x^{2} {\mathrm e}^{x^{2}}-12 \,{\mathrm e}^{x^{2}}\right )}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (24) = 48\).
time = 0.44, size = 50, normalized size = 1.85 \begin {gather*} -\frac {4 \, {\left (4 \, x^{2} e^{3} - 3 \, {\left (49 \, x^{4} - 28 \, x^{2} + 16\right )} e^{\left (x^{2}\right )}\right )}}{49 \, {\left (x^{2} e^{3} + 3 \, {\left (7 \, x^{2} - 4\right )} e^{\left (x^{2}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (24) = 48\).
time = 0.36, size = 54, normalized size = 2.00 \begin {gather*} -\frac {4 \, {\left (4 \, x^{2} e^{6} - 3 \, {\left (49 \, x^{4} - 28 \, x^{2} + 16\right )} e^{\left (x^{2} + 3\right )}\right )}}{49 \, {\left (x^{2} e^{6} + 3 \, {\left (7 \, x^{2} - 4\right )} e^{\left (x^{2} + 3\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (20) = 40\).
time = 0.15, size = 56, normalized size = 2.07 \begin {gather*} - \frac {4 x^{6} e^{3}}{7 x^{4} e^{3} - 4 x^{2} e^{3} + \left (147 x^{4} - 168 x^{2} + 48\right ) e^{x^{2}}} + \frac {4 x^{2}}{7} + \frac {64}{343 x^{2} - 196} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (24) = 48\).
time = 0.40, size = 58, normalized size = 2.15 \begin {gather*} \frac {4 \, {\left (147 \, x^{4} e^{\left (x^{2}\right )} - 4 \, x^{2} e^{3} - 84 \, x^{2} e^{\left (x^{2}\right )} + 48 \, e^{\left (x^{2}\right )}\right )}}{49 \, {\left (x^{2} e^{3} + 21 \, x^{2} e^{\left (x^{2}\right )} - 12 \, e^{\left (x^{2}\right )}\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}}^{2\,x^2}\,\left (576\,x^3-504\,x^5\right )-{\mathrm {e}}^{x^2+3}\,\left (24\,x^7+24\,x^5\right )}{x^4\,{\mathrm {e}}^6+{\mathrm {e}}^{2\,x^2}\,\left (441\,x^4-504\,x^2+144\right )-{\mathrm {e}}^{x^2+3}\,\left (24\,x^2-42\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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