3.37.78 \(\int \frac {e^{20-4 e^x} (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6)+e^{10-2 e^x} (12 x+36 x^2+24 x^3+e^x (-12 x^2-24 x^3-12 x^4))}{1+e^{10-2 e^x} (-48 x-96 x^2-48 x^3)+e^{20-4 e^x} (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6)} \, dx\)

Optimal. Leaf size=27 \[ \frac {x}{-4+\frac {e^{-10+2 e^x}}{6 x (1+x)^2}} \]

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Rubi [F]  time = 8.57, 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^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(20 - 4*E^x)*(-144*x^2 - 576*x^3 - 864*x^4 - 576*x^5 - 144*x^6) + E^(10 - 2*E^x)*(12*x + 36*x^2 + 24*x^
3 + E^x*(-12*x^2 - 24*x^3 - 12*x^4)))/(1 + E^(10 - 2*E^x)*(-48*x - 96*x^2 - 48*x^3) + E^(20 - 4*E^x)*(576*x^2
+ 2304*x^3 + 3456*x^4 + 2304*x^5 + 576*x^6)),x]

[Out]

144*E^20*Defer[Int][x^2/(E^(2*E^x) - 24*E^10*x*(1 + x)^2)^2, x] - 12*E^10*Defer[Int][(E^(2*E^x + x)*x^2)/(E^(2
*E^x) - 24*E^10*x*(1 + x)^2)^2, x] + 864*E^20*Defer[Int][x^3/(E^(2*E^x) - 24*E^10*x*(1 + x)^2)^2, x] - 24*E^10
*Defer[Int][(E^(2*E^x + x)*x^3)/(E^(2*E^x) - 24*E^10*x*(1 + x)^2)^2, x] + 1728*E^20*Defer[Int][x^4/(E^(2*E^x)
- 24*E^10*x*(1 + x)^2)^2, x] - 12*E^10*Defer[Int][(E^(2*E^x + x)*x^4)/(E^(2*E^x) - 24*E^10*x*(1 + x)^2)^2, x]
+ 1440*E^20*Defer[Int][x^5/(E^(2*E^x) - 24*E^10*x*(1 + x)^2)^2, x] + 432*E^20*Defer[Int][x^6/(E^(2*E^x) - 24*E
^10*x*(1 + x)^2)^2, x] - 12*E^10*Defer[Int][x/(-E^(2*E^x) + 24*E^10*x*(1 + x)^2), x] - 36*E^10*Defer[Int][x^2/
(-E^(2*E^x) + 24*E^10*x*(1 + x)^2), x] - 24*E^10*Defer[Int][x^3/(-E^(2*E^x) + 24*E^10*x*(1 + x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 e^{10} x (1+x) \left (-e^{2 e^x+x} x (1+x)-12 e^{10} x (1+x)^3+e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=\left (12 e^{10}\right ) \int \frac {x (1+x) \left (-e^{2 e^x+x} x (1+x)-12 e^{10} x (1+x)^3+e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=\left (12 e^{10}\right ) \int \left (-\frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x-48 e^{10} x^2-24 e^{10} x^3\right )^2}-\frac {x (1+x) \left (-e^{2 e^x}+12 e^{10} x-2 e^{2 e^x} x+36 e^{10} x^2+36 e^{10} x^3+12 e^{10} x^4\right )}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x-48 e^{10} x^2-24 e^{10} x^3\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {x (1+x) \left (-e^{2 e^x}+12 e^{10} x-2 e^{2 e^x} x+36 e^{10} x^2+36 e^{10} x^3+12 e^{10} x^4\right )}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {x (1+x) \left (12 e^{10} x (1+x)^3-e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \left (\frac {e^{2 e^x+x} x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {2 e^{2 e^x+x} x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {e^{2 e^x+x} x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\right )-\left (12 e^{10}\right ) \int \left (-\frac {12 e^{10} x^2 (1+x)^3 (1+3 x)}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {x \left (1+3 x+2 x^2\right )}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x \left (1+3 x+2 x^2\right )}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2 (1+x)^3 (1+3 x)}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x \left (-1-3 x-2 x^2\right )}{e^{2 e^x}-24 e^{10} x (1+x)^2} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2 (1+x)^3 (1+3 x)}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \left (\frac {x}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}+\frac {3 x^2}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}+\frac {2 x^3}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}\right ) \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (144 e^{20}\right ) \int \left (\frac {x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {6 x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {12 x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {10 x^5}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {3 x^6}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {x}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (24 e^{10}\right ) \int \frac {x^3}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (36 e^{10}\right ) \int \frac {x^2}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx+\left (144 e^{20}\right ) \int \frac {x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (432 e^{20}\right ) \int \frac {x^6}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (864 e^{20}\right ) \int \frac {x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (1440 e^{20}\right ) \int \frac {x^5}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (1728 e^{20}\right ) \int \frac {x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (24 e^{10}\right ) \int \frac {x^3}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx-\left (36 e^{10}\right ) \int \frac {x^2}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (432 e^{20}\right ) \int \frac {x^6}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (864 e^{20}\right ) \int \frac {x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (1440 e^{20}\right ) \int \frac {x^5}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (1728 e^{20}\right ) \int \frac {x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.41, size = 36, normalized size = 1.33 \begin {gather*} -\frac {12 e^{10} x^2 (1+x)^2}{-2 e^{2 e^x}+48 e^{10} x (1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(20 - 4*E^x)*(-144*x^2 - 576*x^3 - 864*x^4 - 576*x^5 - 144*x^6) + E^(10 - 2*E^x)*(12*x + 36*x^2 +
24*x^3 + E^x*(-12*x^2 - 24*x^3 - 12*x^4)))/(1 + E^(10 - 2*E^x)*(-48*x - 96*x^2 - 48*x^3) + E^(20 - 4*E^x)*(57
6*x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 + 576*x^6)),x]

[Out]

(-12*E^10*x^2*(1 + x)^2)/(-2*E^(2*E^x) + 48*E^10*x*(1 + x)^2)

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fricas [A]  time = 0.78, size = 44, normalized size = 1.63 \begin {gather*} -\frac {6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (-2 \, e^{x} + 10\right )}}{24 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{\left (-2 \, e^{x} + 10\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-12*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+3
6*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6+2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*
x)*exp(5-exp(x))^2+1),x, algorithm="fricas")

[Out]

-6*(x^4 + 2*x^3 + x^2)*e^(-2*e^x + 10)/(24*(x^3 + 2*x^2 + x)*e^(-2*e^x + 10) - 1)

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giac [B]  time = 0.22, size = 51, normalized size = 1.89 \begin {gather*} -\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-12*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+3
6*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6+2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*
x)*exp(5-exp(x))^2+1),x, algorithm="giac")

[Out]

-6*(x^4*e^10 + 2*x^3*e^10 + x^2*e^10)/(24*x^3*e^10 + 48*x^2*e^10 + 24*x*e^10 - e^(2*e^x))

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maple [A]  time = 0.08, size = 46, normalized size = 1.70

method	result	size
risch	\(-\frac {x}{4}-\frac {x}{4 \left (24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{3}+48 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{2}+24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x -1\right )}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-12*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+36*x^2+
12*x)*exp(5-exp(x))^2)/((576*x^6+2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*x)*exp
(5-exp(x))^2+1),x,method=_RETURNVERBOSE)

[Out]

-1/4*x-1/4*x/(24*exp(-2*exp(x)+10)*x^3+48*exp(-2*exp(x)+10)*x^2+24*exp(-2*exp(x)+10)*x-1)

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maxima [B]  time = 0.46, size = 51, normalized size = 1.89 \begin {gather*} -\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x^6-576*x^5-864*x^4-576*x^3-144*x^2)*exp(5-exp(x))^4+((-12*x^4-24*x^3-12*x^2)*exp(x)+24*x^3+3
6*x^2+12*x)*exp(5-exp(x))^2)/((576*x^6+2304*x^5+3456*x^4+2304*x^3+576*x^2)*exp(5-exp(x))^4+(-48*x^3-96*x^2-48*
x)*exp(5-exp(x))^2+1),x, algorithm="maxima")

[Out]

-6*(x^4*e^10 + 2*x^3*e^10 + x^2*e^10)/(24*x^3*e^10 + 48*x^2*e^10 + 24*x*e^10 - e^(2*e^x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{10-2\,{\mathrm {e}}^x}\,\left (12\,x-{\mathrm {e}}^x\,\left (12\,x^4+24\,x^3+12\,x^2\right )+36\,x^2+24\,x^3\right )-{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (144\,x^6+576\,x^5+864\,x^4+576\,x^3+144\,x^2\right )}{{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (576\,x^6+2304\,x^5+3456\,x^4+2304\,x^3+576\,x^2\right )-{\mathrm {e}}^{10-2\,{\mathrm {e}}^x}\,\left (48\,x^3+96\,x^2+48\,x\right )+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10 - 2*exp(x))*(12*x - exp(x)*(12*x^2 + 24*x^3 + 12*x^4) + 36*x^2 + 24*x^3) - exp(20 - 4*exp(x))*(144
*x^2 + 576*x^3 + 864*x^4 + 576*x^5 + 144*x^6))/(exp(20 - 4*exp(x))*(576*x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 +
576*x^6) - exp(10 - 2*exp(x))*(48*x + 96*x^2 + 48*x^3) + 1),x)

[Out]

int((exp(10 - 2*exp(x))*(12*x - exp(x)*(12*x^2 + 24*x^3 + 12*x^4) + 36*x^2 + 24*x^3) - exp(20 - 4*exp(x))*(144
*x^2 + 576*x^3 + 864*x^4 + 576*x^5 + 144*x^6))/(exp(20 - 4*exp(x))*(576*x^2 + 2304*x^3 + 3456*x^4 + 2304*x^5 +
576*x^6) - exp(10 - 2*exp(x))*(48*x + 96*x^2 + 48*x^3) + 1), x)

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sympy [A]  time = 0.36, size = 29, normalized size = 1.07 \begin {gather*} - \frac {x}{4} - \frac {x}{\left (96 x^{3} + 192 x^{2} + 96 x\right ) e^{10 - 2 e^{x}} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*x**6-576*x**5-864*x**4-576*x**3-144*x**2)*exp(5-exp(x))**4+((-12*x**4-24*x**3-12*x**2)*exp(x)
+24*x**3+36*x**2+12*x)*exp(5-exp(x))**2)/((576*x**6+2304*x**5+3456*x**4+2304*x**3+576*x**2)*exp(5-exp(x))**4+(
-48*x**3-96*x**2-48*x)*exp(5-exp(x))**2+1),x)

[Out]

-x/4 - x/((96*x**3 + 192*x**2 + 96*x)*exp(10 - 2*exp(x)) - 4)

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