Optimal. Leaf size=19 \[ 16 x \left (4+e^{e^{-x} (4+e)}+x\right )^4 \]
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Rubi [F] time = 7.43, 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 e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int 16 e^{-x} \left (4+e^{e^{-x} (4+e)}+x\right )^3 \left (e^{e^{-x} \left (4+e+e^x x\right )}-16 \left (1+\frac {e}{4}\right ) e^{e^{-x} (4+e)} x+e^x (4+5 x)\right ) \, dx\\ &=16 \int e^{-x} \left (4+e^{e^{-x} (4+e)}+x\right )^3 \left (e^{e^{-x} \left (4+e+e^x x\right )}-16 \left (1+\frac {e}{4}\right ) e^{e^{-x} (4+e)} x+e^x (4+5 x)\right ) \, dx\\ &=16 \int \left (4 (-4-e) e^{e^{-x} (4+e)-x} x \left (4+e^{e^{1-x}+4 e^{-x}}+x\right )^3+\left (4+e^{e^{1-x}+4 e^{-x}}+x\right )^3 \left (4+e^{e^{1-x}+4 e^{-x}}+5 x\right )\right ) \, dx\\ &=16 \int \left (4+e^{e^{1-x}+4 e^{-x}}+x\right )^3 \left (4+e^{e^{1-x}+4 e^{-x}}+5 x\right ) \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x \left (4+e^{e^{1-x}+4 e^{-x}}+x\right )^3 \, dx\\ &=16 \int \left (4+e^{e^{-x} (4+e)}+x\right )^3 \left (4+e^{e^{-x} (4+e)}+5 x\right ) \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x \left (4+e^{e^{-x} (4+e)}+x\right )^3 \, dx\\ &=16 \int \left (e^{4 e^{-x} (4+e)}+8 e^{3 e^{-x} (4+e)} (2+x)+16 e^{e^{-x} (4+e)} (1+x) (4+x)^2+(4+x)^3 (4+5 x)+6 e^{2 e^{-x} (4+e)} \left (16+16 x+3 x^2\right )\right ) \, dx-(64 (4+e)) \int \left (e^{4 e^{-x} (4+e)-x} x+3 e^{3 e^{-x} (4+e)-x} x (4+x)+3 e^{2 e^{-x} (4+e)-x} x (4+x)^2+e^{e^{-x} (4+e)-x} x (4+x)^3\right ) \, dx\\ &=16 \int e^{4 e^{-x} (4+e)} \, dx+16 \int (4+x)^3 (4+5 x) \, dx+96 \int e^{2 e^{-x} (4+e)} \left (16+16 x+3 x^2\right ) \, dx+128 \int e^{3 e^{-x} (4+e)} (2+x) \, dx+256 \int e^{e^{-x} (4+e)} (1+x) (4+x)^2 \, dx-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x (4+x)^3 \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x (4+x) \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x (4+x)^2 \, dx\\ &=16 x (4+x)^4-16 \operatorname {Subst}\left (\int \frac {e^{4 (4+e) x}}{x} \, dx,x,e^{-x}\right )+96 \int \left (16 e^{2 e^{-x} (4+e)}+16 e^{2 e^{-x} (4+e)} x+3 e^{2 e^{-x} (4+e)} x^2\right ) \, dx+128 \int \left (2 e^{3 e^{-x} (4+e)}+e^{3 e^{-x} (4+e)} x\right ) \, dx+256 \int \left (-3 e^{e^{-x} (4+e)} (4+x)^2+e^{e^{-x} (4+e)} (4+x)^3\right ) \, dx-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int \left (-4 e^{e^{-x} (4+e)-x} (4+x)^3+e^{e^{-x} (4+e)-x} (4+x)^4\right ) \, dx-(192 (4+e)) \int \left (4 e^{3 e^{-x} (4+e)-x} x+e^{3 e^{-x} (4+e)-x} x^2\right ) \, dx-(192 (4+e)) \int \left (16 e^{2 e^{-x} (4+e)-x} x+8 e^{2 e^{-x} (4+e)-x} x^2+e^{2 e^{-x} (4+e)-x} x^3\right ) \, dx\\ &=16 x (4+x)^4-16 \text {Ei}\left (4 e^{-x} (4+e)\right )+128 \int e^{3 e^{-x} (4+e)} x \, dx+256 \int e^{3 e^{-x} (4+e)} \, dx+256 \int e^{e^{-x} (4+e)} (4+x)^3 \, dx+288 \int e^{2 e^{-x} (4+e)} x^2 \, dx-768 \int e^{e^{-x} (4+e)} (4+x)^2 \, dx+1536 \int e^{2 e^{-x} (4+e)} \, dx+1536 \int e^{2 e^{-x} (4+e)} x \, dx-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} (4+x)^4 \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x^2 \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^3 \, dx+(256 (4+e)) \int e^{e^{-x} (4+e)-x} (4+x)^3 \, dx-(768 (4+e)) \int e^{3 e^{-x} (4+e)-x} x \, dx-(1536 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^2 \, dx-(3072 (4+e)) \int e^{2 e^{-x} (4+e)-x} x \, dx\\ &=16 x (4+x)^4-16 \text {Ei}\left (4 e^{-x} (4+e)\right )+128 \int e^{3 e^{-x} (4+e)} x \, dx+256 \int \left (64 e^{e^{-x} (4+e)}+48 e^{e^{-x} (4+e)} x+12 e^{e^{-x} (4+e)} x^2+e^{e^{-x} (4+e)} x^3\right ) \, dx-256 \operatorname {Subst}\left (\int \frac {e^{3 (4+e) x}}{x} \, dx,x,e^{-x}\right )+288 \int e^{2 e^{-x} (4+e)} x^2 \, dx-768 \int \left (16 e^{e^{-x} (4+e)}+8 e^{e^{-x} (4+e)} x+e^{e^{-x} (4+e)} x^2\right ) \, dx+1536 \int e^{2 e^{-x} (4+e)} x \, dx-1536 \operatorname {Subst}\left (\int \frac {e^{2 (4+e) x}}{x} \, dx,x,e^{-x}\right )-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int \left (256 e^{e^{-x} (4+e)-x}+256 e^{e^{-x} (4+e)-x} x+96 e^{e^{-x} (4+e)-x} x^2+16 e^{e^{-x} (4+e)-x} x^3+e^{e^{-x} (4+e)-x} x^4\right ) \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x^2 \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^3 \, dx+(256 (4+e)) \int \left (64 e^{e^{-x} (4+e)-x}+48 e^{e^{-x} (4+e)-x} x+12 e^{e^{-x} (4+e)-x} x^2+e^{e^{-x} (4+e)-x} x^3\right ) \, dx-(768 (4+e)) \int e^{3 e^{-x} (4+e)-x} x \, dx-(1536 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^2 \, dx-(3072 (4+e)) \int e^{2 e^{-x} (4+e)-x} x \, dx\\ &=16 x (4+x)^4-1536 \text {Ei}\left (2 e^{-x} (4+e)\right )-256 \text {Ei}\left (3 e^{-x} (4+e)\right )-16 \text {Ei}\left (4 e^{-x} (4+e)\right )+128 \int e^{3 e^{-x} (4+e)} x \, dx+256 \int e^{e^{-x} (4+e)} x^3 \, dx+288 \int e^{2 e^{-x} (4+e)} x^2 \, dx-768 \int e^{e^{-x} (4+e)} x^2 \, dx+1536 \int e^{2 e^{-x} (4+e)} x \, dx+3072 \int e^{e^{-x} (4+e)} x^2 \, dx-6144 \int e^{e^{-x} (4+e)} x \, dx-12288 \int e^{e^{-x} (4+e)} \, dx+12288 \int e^{e^{-x} (4+e)} x \, dx+16384 \int e^{e^{-x} (4+e)} \, dx-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x^4 \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x^2 \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^3 \, dx+(256 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(768 (4+e)) \int e^{3 e^{-x} (4+e)-x} x \, dx-(1024 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(1536 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^2 \, dx-(3072 (4+e)) \int e^{2 e^{-x} (4+e)-x} x \, dx+(3072 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx-(6144 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx+(12288 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx-(16384 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx\\ &=16 x (4+x)^4-1536 \text {Ei}\left (2 e^{-x} (4+e)\right )-256 \text {Ei}\left (3 e^{-x} (4+e)\right )-16 \text {Ei}\left (4 e^{-x} (4+e)\right )+128 \int e^{3 e^{-x} (4+e)} x \, dx+256 \int e^{e^{-x} (4+e)} x^3 \, dx+288 \int e^{2 e^{-x} (4+e)} x^2 \, dx-768 \int e^{e^{-x} (4+e)} x^2 \, dx+1536 \int e^{2 e^{-x} (4+e)} x \, dx+3072 \int e^{e^{-x} (4+e)} x^2 \, dx-6144 \int e^{e^{-x} (4+e)} x \, dx+12288 \int e^{e^{-x} (4+e)} x \, dx+12288 \operatorname {Subst}\left (\int \frac {e^{(4+e) x}}{x} \, dx,x,e^{-x}\right )-16384 \operatorname {Subst}\left (\int \frac {e^{(4+e) x}}{x} \, dx,x,e^{-x}\right )-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x^4 \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x^2 \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^3 \, dx+(256 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(768 (4+e)) \int e^{3 e^{-x} (4+e)-x} x \, dx-(1024 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(1536 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^2 \, dx-(3072 (4+e)) \int e^{2 e^{-x} (4+e)-x} x \, dx+(3072 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx-(6144 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx+(12288 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx-(16384 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx\\ &=16 x (4+x)^4-4096 \text {Ei}\left (e^{-x} (4+e)\right )-1536 \text {Ei}\left (2 e^{-x} (4+e)\right )-256 \text {Ei}\left (3 e^{-x} (4+e)\right )-16 \text {Ei}\left (4 e^{-x} (4+e)\right )+128 \int e^{3 e^{-x} (4+e)} x \, dx+256 \int e^{e^{-x} (4+e)} x^3 \, dx+288 \int e^{2 e^{-x} (4+e)} x^2 \, dx-768 \int e^{e^{-x} (4+e)} x^2 \, dx+1536 \int e^{2 e^{-x} (4+e)} x \, dx+3072 \int e^{e^{-x} (4+e)} x^2 \, dx-6144 \int e^{e^{-x} (4+e)} x \, dx+12288 \int e^{e^{-x} (4+e)} x \, dx-(64 (4+e)) \int e^{4 e^{-x} (4+e)-x} x \, dx-(64 (4+e)) \int e^{e^{-x} (4+e)-x} x^4 \, dx-(192 (4+e)) \int e^{3 e^{-x} (4+e)-x} x^2 \, dx-(192 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^3 \, dx+(256 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(768 (4+e)) \int e^{3 e^{-x} (4+e)-x} x \, dx-(1024 (4+e)) \int e^{e^{-x} (4+e)-x} x^3 \, dx-(1536 (4+e)) \int e^{2 e^{-x} (4+e)-x} x^2 \, dx-(3072 (4+e)) \int e^{2 e^{-x} (4+e)-x} x \, dx+(3072 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx-(6144 (4+e)) \int e^{e^{-x} (4+e)-x} x^2 \, dx+(12288 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx-(16384 (4+e)) \int e^{e^{-x} (4+e)-x} x \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 3.27, size = 19, normalized size = 1.00 \begin {gather*} 16 x \left (4+e^{e^{-x} (4+e)}+x\right )^4 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 112, normalized size = 5.89 \begin {gather*} 16 \, x^{5} + 256 \, x^{4} + 1536 \, x^{3} + 4096 \, x^{2} + 16 \, x e^{\left (4 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 64 \, {\left (x^{2} + 4 \, x\right )} e^{\left (3 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 96 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{\left (2 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 64 \, {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )} e^{\left ({\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4096 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -16 \, {\left ({\left (4 \, x e + 16 \, x - e^{x}\right )} e^{\left (4 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (12 \, x^{2} + 3 \, {\left (x^{2} + 4 \, x\right )} e - 2 \, {\left (x + 2\right )} e^{x} + 48 \, x\right )} e^{\left (3 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 6 \, {\left (8 \, x^{3} + 64 \, x^{2} + 2 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e - {\left (3 \, x^{2} + 16 \, x + 16\right )} e^{x} + 128 \, x\right )} e^{\left (2 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (4 \, x^{4} + 48 \, x^{3} + 192 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )} e - 4 \, {\left (x^{3} + 9 \, x^{2} + 24 \, x + 16\right )} e^{x} + 256 \, x\right )} e^{\left ({\left (e + 4\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{4} + 64 \, x^{3} + 288 \, x^{2} + 512 \, x + 256\right )} e^{x}\right )} e^{\left (-x\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 104, normalized size = 5.47
method | result | size |
risch | \(16 x^{5}+16 \,{\mathrm e}^{4 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +256 x^{4}+1536 x^{3}+4096 x^{2}+4096 x +64 \left (4+x \right ) x \,{\mathrm e}^{3 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}+96 \left (x^{2}+8 x +16\right ) x \,{\mathrm e}^{2 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}+64 \left (x^{3}+12 x^{2}+48 x +64\right ) x \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 16 \, x^{5} + 256 \, x^{4} + 1536 \, x^{3} + 4096 \, x^{2} + 4096 \, x - 4096 \, {\rm Ei}\left ({\left (e + 4\right )} e^{\left (-x\right )}\right ) + 16 \, \int -{\left (4 \, x {\left (e + 4\right )} - e^{x}\right )} e^{\left (-x + 16 \, e^{\left (-x\right )} + 4 \, e^{\left (-x + 1\right )}\right )}\,{d x} + 16 \, \int -4 \, {\left (3 \, x^{2} {\left (e + 4\right )} + 12 \, x {\left (e + 4\right )} - 2 \, {\left (x + 2\right )} e^{x}\right )} e^{\left (-x + 12 \, e^{\left (-x\right )} + 3 \, e^{\left (-x + 1\right )}\right )}\,{d x} + 16 \, \int -6 \, {\left (2 \, x^{3} {\left (e + 4\right )} + 16 \, x^{2} {\left (e + 4\right )} + 32 \, x {\left (e + 4\right )} - {\left (3 \, x^{2} + 16 \, x + 16\right )} e^{x}\right )} e^{\left (-x + 8 \, e^{\left (-x\right )} + 2 \, e^{\left (-x + 1\right )}\right )}\,{d x} + 16 \, \int -4 \, {\left (x^{4} {\left (e + 4\right )} + 12 \, x^{3} {\left (e + 4\right )} + 48 \, x^{2} {\left (e + 4\right )} + 64 \, x {\left (e + 4\right )} - 4 \, {\left (x^{3} + 9 \, x^{2} + 24 \, x\right )} e^{x}\right )} e^{\left (-x + 4 \, e^{\left (-x\right )} + e^{\left (-x + 1\right )}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 23, normalized size = 1.21 \begin {gather*} 16\,x\,{\left (x+{\mathrm {e}}^{4\,{\mathrm {e}}^{-x}+{\mathrm {e}}^{-x}\,\mathrm {e}}+4\right )}^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.57, size = 112, normalized size = 5.89 \begin {gather*} 16 x^{5} + 256 x^{4} + 1536 x^{3} + 4096 x^{2} + 16 x e^{4 \left (e + 4\right ) e^{- x}} + 4096 x + \left (64 x^{2} + 256 x\right ) e^{3 \left (e + 4\right ) e^{- x}} + \left (96 x^{3} + 768 x^{2} + 1536 x\right ) e^{2 \left (e + 4\right ) e^{- x}} + \left (64 x^{4} + 768 x^{3} + 3072 x^{2} + 4096 x\right ) e^{\left (e + 4\right ) e^{- x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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