Optimal. Leaf size=25 \[ \frac {3}{\left (4 e \left (1-e^x\right )+4 \left (-\frac {15}{2}+e^4\right ) x\right )^2} \]
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Rubi [A] time = 0.30, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 166, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6688, 12, 6686} \begin {gather*} \frac {3}{4 \left (-\left (\left (15-2 e^4\right ) x\right )-2 e^{x+1}+2 e\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45+6 e^4-6 e^{1+x}}{-16 e^3+16 e^{3+3 x}+360 e^2 x-2700 e x^2+\left (6750-16 e^{12}\right ) x^3+e^{2 x} \left (-48 e^3+360 e^2 x-48 e^6 x\right )+e^4 \left (-48 e^2 x+720 e x^2-2700 x^3\right )+e^8 \left (-48 e x^2+360 x^3\right )+e^x \left (48 e^3-720 e^2 x+2700 e x^2+48 e^9 x^2+e^4 \left (96 e^2 x-720 e x^2\right )\right )} \, dx\\ &=\int \frac {3 \left (2 e^{1+x}+15 \left (1-\frac {2 e^4}{15}\right )\right )}{2 \left (2 e-2 e^{1+x}-15 \left (1-\frac {2 e^4}{15}\right ) x\right )^3} \, dx\\ &=\frac {3}{2} \int \frac {2 e^{1+x}+15 \left (1-\frac {2 e^4}{15}\right )}{\left (2 e-2 e^{1+x}-15 \left (1-\frac {2 e^4}{15}\right ) x\right )^3} \, dx\\ &=\frac {3}{4 \left (2 e-2 e^{1+x}-\left (15-2 e^4\right ) x\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 1.04 \begin {gather*} \frac {3}{4 \left (2 e-2 e^{1+x}+\left (-15+2 e^4\right ) x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 65, normalized size = 2.60 \begin {gather*} \frac {3}{4 \, {\left (4 \, x^{2} e^{8} - 60 \, x^{2} e^{4} + 225 \, x^{2} + 8 \, x e^{5} - 60 \, x e - 4 \, {\left (2 \, x e^{4} - 15 \, x + 2 \, e\right )} e^{\left (x + 1\right )} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 66, normalized size = 2.64 \begin {gather*} \frac {3}{4 \, {\left (4 \, x^{2} e^{8} - 60 \, x^{2} e^{4} + 225 \, x^{2} + 8 \, x e^{5} - 60 \, x e - 8 \, x e^{\left (x + 5\right )} + 60 \, x e^{\left (x + 1\right )} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )} - 8 \, e^{\left (x + 2\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 24, normalized size = 0.96
method | result | size |
norman | \(\frac {3}{4 \left (2 \,{\mathrm e} \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}+15 x \right )^{2}}\) | \(24\) |
risch | \(\frac {3}{4 \left (2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{x +1}+2 \,{\mathrm e}-15 x \right )^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 63, normalized size = 2.52 \begin {gather*} \frac {3}{4 \, {\left (x^{2} {\left (4 \, e^{8} - 60 \, e^{4} + 225\right )} + 4 \, x {\left (2 \, e^{5} - 15 \, e\right )} - 4 \, {\left (x {\left (2 \, e^{5} - 15 \, e\right )} + 2 \, e^{2}\right )} e^{x} + 4 \, e^{2} + 4 \, e^{\left (2 \, x + 2\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.79, size = 118, normalized size = 4.72 \begin {gather*} -\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{2}-3\,{\mathrm {e}}^x-\frac {x\,{\mathrm {e}}^{x-1}\,\left (6\,{\mathrm {e}}^4-45\right )}{2}+\frac {x\,{\mathrm {e}}^{-1}\,\left (6\,{\mathrm {e}}^4-45\right )}{2}+\frac {x^2\,{\mathrm {e}}^{-2}\,\left (12\,{\mathrm {e}}^8-180\,{\mathrm {e}}^4+675\right )}{8}}{8\,{\mathrm {e}}^2-16\,{\mathrm {e}}^{x+2}+8\,{\mathrm {e}}^{2\,x+2}+120\,x\,{\mathrm {e}}^{x+1}-16\,x\,{\mathrm {e}}^{x+5}-120\,x\,\mathrm {e}+16\,x\,{\mathrm {e}}^5-120\,x^2\,{\mathrm {e}}^4+8\,x^2\,{\mathrm {e}}^8+450\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 73, normalized size = 2.92 \begin {gather*} \frac {3}{- 240 x^{2} e^{4} + 900 x^{2} + 16 x^{2} e^{8} - 240 e x + 32 x e^{5} + \left (- 32 x e^{5} + 240 e x - 32 e^{2}\right ) e^{x} + 16 e^{2} e^{2 x} + 16 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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