Optimal. Leaf size=31 \[ 1-e^{\left (8+e^4+x\right )^2}+\frac {1}{4} \left (4+\frac {x}{-x+x^2}\right ) \]
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Rubi [A]
time = 0.26, antiderivative size = 24, normalized size of antiderivative = 0.77, number of steps
used = 6, number of rules used = 5, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {27, 12, 6820,
2259, 2240} \begin {gather*} -e^{\left (x+e^4+8\right )^2}-\frac {1}{4 (1-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 2240
Rule 2259
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+e^{64+e^8+16 x+x^2+e^4 (16+2 x)} \left (-64+120 x-48 x^2-8 x^3+e^4 \left (-8+16 x-8 x^2\right )\right )}{4 (-1+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {-1+e^{64+e^8+16 x+x^2+e^4 (16+2 x)} \left (-64+120 x-48 x^2-8 x^3+e^4 \left (-8+16 x-8 x^2\right )\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {1}{(-1+x)^2}-8 e^{\left (8+e^4\right )^2+2 \left (8+e^4\right ) x+x^2} \left (8+e^4+x\right )\right ) \, dx\\ &=-\frac {1}{4 (1-x)}-2 \int e^{\left (8+e^4\right )^2+2 \left (8+e^4\right ) x+x^2} \left (8+e^4+x\right ) \, dx\\ &=-\frac {1}{4 (1-x)}-2 \int e^{\left (8+e^4+x\right )^2} \left (8+e^4+x\right ) \, dx\\ &=-e^{\left (8+e^4+x\right )^2}-\frac {1}{4 (1-x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 22, normalized size = 0.71 \begin {gather*} -e^{\left (8+e^4+x\right )^2}+\frac {1}{4 (-1+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.44, size = 167, normalized size = 5.39
method | result | size |
risch | \(\frac {1}{4 x -4}-{\mathrm e}^{2 x \,{\mathrm e}^{4}+x^{2}+16 \,{\mathrm e}^{4}+{\mathrm e}^{8}+16 x +64}\) | \(31\) |
norman | \(\frac {-{\mathrm e}^{{\mathrm e}^{8}+\left (2 x +16\right ) {\mathrm e}^{4}+x^{2}+16 x +64} x +\frac {1}{4}+{\mathrm e}^{{\mathrm e}^{8}+\left (2 x +16\right ) {\mathrm e}^{4}+x^{2}+16 x +64}}{x -1}\) | \(54\) |
default | \(-{\mathrm e}^{x^{2}+\left (2 \,{\mathrm e}^{4}+16\right ) x +{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+64}-\frac {i \left (2 \,{\mathrm e}^{4}+16\right ) \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+64-\frac {\left (2 \,{\mathrm e}^{4}+16\right )^{2}}{4}} \erf \left (i x +\frac {i \left (2 \,{\mathrm e}^{4}+16\right )}{2}\right )}{2}+i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+64-\frac {\left (2 \,{\mathrm e}^{4}+16\right )^{2}}{4}} \erf \left (i x +\frac {i \left (2 \,{\mathrm e}^{4}+16\right )}{2}\right )+8 i \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+64-\frac {\left (2 \,{\mathrm e}^{4}+16\right )^{2}}{4}} \erf \left (i x +\frac {i \left (2 \,{\mathrm e}^{4}+16\right )}{2}\right )+\frac {1}{4 x -4}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{4 \, {\left (x - 1\right )}} - e^{\left (x^{2} + 2 \, x e^{4} + 16 \, x + e^{8} + 16 \, e^{4} + 64\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 32, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (x - 1\right )} e^{\left (x^{2} + 2 \, {\left (x + 8\right )} e^{4} + 16 \, x + e^{8} + 64\right )} - 1}{4 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 27, normalized size = 0.87 \begin {gather*} - e^{x^{2} + 16 x + \left (2 x + 16\right ) e^{4} + 64 + e^{8}} + \frac {1}{4 x - 4} \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. 54 vs.
\(2 (24) = 48\).
time = 0.40, size = 54, normalized size = 1.74 \begin {gather*} -\frac {4 \, x e^{\left (x^{2} + 2 \, x e^{4} + 16 \, x + e^{8} + 16 \, e^{4} + 64\right )} - 4 \, e^{\left (x^{2} + 2 \, x e^{4} + 16 \, x + e^{8} + 16 \, e^{4} + 64\right )} - 1}{4 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 34, normalized size = 1.10 \begin {gather*} \frac {1}{4\,\left (x-1\right )}-{\mathrm {e}}^{16\,{\mathrm {e}}^4}\,{\mathrm {e}}^{16\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{64}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^4}\,{\mathrm {e}}^{{\mathrm {e}}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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