∫ 140−184x−64x2+ex(28−20x−16x2)+eex (8575−11270x+2863x2+1172x3−304x4−64x5+e3x(−343x+588x2−336x3+64x4)+e2x(343−4018x+5530x2−2248x3−32x4+128x5)+ex(3430−13769x+13454x2−2831x3−1300x4+304x5+64x6))_____________________________________________________________________________________________________________________________________________________________________ −8575x2+11270x3−2863x4−1172x5+304x6+64x7+e2x(−343x2+588x3−336x4+64x5)+ex(−3430x2+5194x3−2184x4−32x5+128x6) dx

3.11.8 \(\int \frac {140-184 x-64 x^2+e^x (28-20 x-16 x^2)+e^{e^x} (8575-11270 x+2863 x^2+1172 x^3-304 x^4-64 x^5+e^{3 x} (-343 x+588 x^2-336 x^3+64 x^4)+e^{2 x} (343-4018 x+5530 x^2-2248 x^3-32 x^4+128 x^5)+e^x (3430-13769 x+13454 x^2-2831 x^3-1300 x^4+304 x^5+64 x^6))}{-8575 x^2+11270 x^3-2863 x^4-1172 x^5+304 x^6+64 x^7+e^{2 x} (-343 x^2+588 x^3-336 x^4+64 x^5)+e^x (-3430 x^2+5194 x^3-2184 x^4-32 x^5+128 x^6)} \, dx\)

Optimal. Leaf size=27 \[ \frac {e^{e^x}+\frac {4}{(7-4 x)^2 \left (5+e^x+x\right )}}{x} \]

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Rubi [F] time = 4.10, 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 {140-184 x-64 x^2+e^x \left (28-20 x-16 x^2\right )+e^{e^x} \left (8575-11270 x+2863 x^2+1172 x^3-304 x^4-64 x^5+e^{3 x} \left (-343 x+588 x^2-336 x^3+64 x^4\right )+e^{2 x} \left (343-4018 x+5530 x^2-2248 x^3-32 x^4+128 x^5\right )+e^x \left (3430-13769 x+13454 x^2-2831 x^3-1300 x^4+304 x^5+64 x^6\right )\right )}{-8575 x^2+11270 x^3-2863 x^4-1172 x^5+304 x^6+64 x^7+e^{2 x} \left (-343 x^2+588 x^3-336 x^4+64 x^5\right )+e^x \left (-3430 x^2+5194 x^3-2184 x^4-32 x^5+128 x^6\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(140 - 184*x - 64*x^2 + E^x*(28 - 20*x - 16*x^2) + E^E^x*(8575 - 11270*x + 2863*x^2 + 1172*x^3 - 304*x^4 -

64*x^5 + E^(3*x)*(-343*x + 588*x^2 - 336*x^3 + 64*x^4) + E^(2*x)*(343 - 4018*x + 5530*x^2 - 2248*x^3 - 32*x^4

+ 128*x^5) + E^x*(3430 - 13769*x + 13454*x^2 - 2831*x^3 - 1300*x^4 + 304*x^5 + 64*x^6)))/(-8575*x^2 + 11270*x

^3 - 2863*x^4 - 1172*x^5 + 304*x^6 + 64*x^7 + E^(2*x)*(-343*x^2 + 588*x^3 - 336*x^4 + 64*x^5) + E^x*(-3430*x^2

+ 5194*x^3 - 2184*x^4 - 32*x^5 + 128*x^6)),x]

[Out]

-Defer[Int][E^E^x/x^2, x] + Defer[Int][E^(E^x + x)/x, x] + (16*Defer[Int][1/(x*(5 + E^x + x)^2), x])/49 - (4*D

efer[Int][1/(x^2*(5 + E^x + x)), x])/49 - (4*Defer[Int][1/(x*(5 + E^x + x)), x])/49 - (128*Defer[Int][1/((5 +

E^x + x)*(-7 + 4*x)^3), x])/7 + (92*Defer[Int][1/((5 + E^x + x)^2*(-7 + 4*x)^2), x])/7 - (48*Defer[Int][1/((5

+ E^x + x)*(-7 + 4*x)^2), x])/49 - (64*Defer[Int][1/((5 + E^x + x)^2*(-7 + 4*x)), x])/49 + (16*Defer[Int][1/((

5 + E^x + x)*(-7 + 4*x)), x])/49

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{e^x+3 x} x (-7+4 x)^3+e^{e^x} (5+x)^2 (-7+4 x)^3-e^{e^x+2 x} (-7+4 x)^3 \left (-1+10 x+2 x^2\right )+4 e^x \left (-7+5 x+4 x^2\right )+4 \left (-35+46 x+16 x^2\right )-e^{e^x+x} (-7+4 x)^3 \left (-10+23 x+10 x^2+x^3\right )}{(7-4 x)^3 x^2 \left (5+e^x+x\right )^2} \, dx\\ &=\int \left (-\frac {e^{e^x}}{x^2}+\frac {e^{e^x+x}}{x}+\frac {4 (4+x)}{x \left (5+e^x+x\right )^2 (-7+4 x)^2}-\frac {4 \left (-7+5 x+4 x^2\right )}{x^2 \left (5+e^x+x\right ) (-7+4 x)^3}\right ) \, dx\\ &=4 \int \frac {4+x}{x \left (5+e^x+x\right )^2 (-7+4 x)^2} \, dx-4 \int \frac {-7+5 x+4 x^2}{x^2 \left (5+e^x+x\right ) (-7+4 x)^3} \, dx-\int \frac {e^{e^x}}{x^2} \, dx+\int \frac {e^{e^x+x}}{x} \, dx\\ &=4 \int \left (\frac {4}{49 x \left (5+e^x+x\right )^2}+\frac {23}{7 \left (5+e^x+x\right )^2 (-7+4 x)^2}-\frac {16}{49 \left (5+e^x+x\right )^2 (-7+4 x)}\right ) \, dx-4 \int \left (\frac {1}{49 x^2 \left (5+e^x+x\right )}+\frac {1}{49 x \left (5+e^x+x\right )}+\frac {32}{7 \left (5+e^x+x\right ) (-7+4 x)^3}+\frac {12}{49 \left (5+e^x+x\right ) (-7+4 x)^2}-\frac {4}{49 \left (5+e^x+x\right ) (-7+4 x)}\right ) \, dx-\int \frac {e^{e^x}}{x^2} \, dx+\int \frac {e^{e^x+x}}{x} \, dx\\ &=-\left (\frac {4}{49} \int \frac {1}{x^2 \left (5+e^x+x\right )} \, dx\right )-\frac {4}{49} \int \frac {1}{x \left (5+e^x+x\right )} \, dx+\frac {16}{49} \int \frac {1}{x \left (5+e^x+x\right )^2} \, dx+\frac {16}{49} \int \frac {1}{\left (5+e^x+x\right ) (-7+4 x)} \, dx-\frac {48}{49} \int \frac {1}{\left (5+e^x+x\right ) (-7+4 x)^2} \, dx-\frac {64}{49} \int \frac {1}{\left (5+e^x+x\right )^2 (-7+4 x)} \, dx+\frac {92}{7} \int \frac {1}{\left (5+e^x+x\right )^2 (-7+4 x)^2} \, dx-\frac {128}{7} \int \frac {1}{\left (5+e^x+x\right ) (-7+4 x)^3} \, dx-\int \frac {e^{e^x}}{x^2} \, dx+\int \frac {e^{e^x+x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.11, size = 27, normalized size = 1.00 \begin {gather*} \frac {e^{e^x}+\frac {4}{(7-4 x)^2 \left (5+e^x+x\right )}}{x} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(140 - 184*x - 64*x^2 + E^x*(28 - 20*x - 16*x^2) + E^E^x*(8575 - 11270*x + 2863*x^2 + 1172*x^3 - 304

*x^4 - 64*x^5 + E^(3*x)*(-343*x + 588*x^2 - 336*x^3 + 64*x^4) + E^(2*x)*(343 - 4018*x + 5530*x^2 - 2248*x^3 -

32*x^4 + 128*x^5) + E^x*(3430 - 13769*x + 13454*x^2 - 2831*x^3 - 1300*x^4 + 304*x^5 + 64*x^6)))/(-8575*x^2 + 1

1270*x^3 - 2863*x^4 - 1172*x^5 + 304*x^6 + 64*x^7 + E^(2*x)*(-343*x^2 + 588*x^3 - 336*x^4 + 64*x^5) + E^x*(-34

30*x^2 + 5194*x^3 - 2184*x^4 - 32*x^5 + 128*x^6)),x]

[Out]

(E^E^x + 4/((7 - 4*x)^2*(5 + E^x + x)))/x

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fricas [B] time = 0.92, size = 73, normalized size = 2.70 \begin {gather*} \frac {{\left (16 \, x^{3} + 24 \, x^{2} + {\left (16 \, x^{2} - 56 \, x + 49\right )} e^{x} - 231 \, x + 245\right )} e^{\left (e^{x}\right )} + 4}{16 \, x^{4} + 24 \, x^{3} - 231 \, x^{2} + {\left (16 \, x^{3} - 56 \, x^{2} + 49 \, x\right )} e^{x} + 245 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^4-336*x^3+588*x^2-343*x)*exp(x)^3+(128*x^5-32*x^4-2248*x^3+5530*x^2-4018*x+343)*exp(x)^2+(64

*x^6+304*x^5-1300*x^4-2831*x^3+13454*x^2-13769*x+3430)*exp(x)-64*x^5-304*x^4+1172*x^3+2863*x^2-11270*x+8575)*e

xp(exp(x))+(-16*x^2-20*x+28)*exp(x)-64*x^2-184*x+140)/((64*x^5-336*x^4+588*x^3-343*x^2)*exp(x)^2+(128*x^6-32*x

^5-2184*x^4+5194*x^3-3430*x^2)*exp(x)+64*x^7+304*x^6-1172*x^5-2863*x^4+11270*x^3-8575*x^2),x, algorithm="frica

s")

[Out]

((16*x^3 + 24*x^2 + (16*x^2 - 56*x + 49)*e^x - 231*x + 245)*e^(e^x) + 4)/(16*x^4 + 24*x^3 - 231*x^2 + (16*x^3

- 56*x^2 + 49*x)*e^x + 245*x)

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giac [B] time = 0.47, size = 126, normalized size = 4.67 \begin {gather*} \frac {16 \, x^{3} e^{\left (x + e^{x}\right )} + 16 \, x^{2} e^{\left (2 \, x + e^{x}\right )} + 24 \, x^{2} e^{\left (x + e^{x}\right )} - 56 \, x e^{\left (2 \, x + e^{x}\right )} - 231 \, x e^{\left (x + e^{x}\right )} + 49 \, e^{\left (2 \, x + e^{x}\right )} + 245 \, e^{\left (x + e^{x}\right )} + 4 \, e^{x}}{16 \, x^{4} e^{x} + 16 \, x^{3} e^{\left (2 \, x\right )} + 24 \, x^{3} e^{x} - 56 \, x^{2} e^{\left (2 \, x\right )} - 231 \, x^{2} e^{x} + 49 \, x e^{\left (2 \, x\right )} + 245 \, x e^{x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^4-336*x^3+588*x^2-343*x)*exp(x)^3+(128*x^5-32*x^4-2248*x^3+5530*x^2-4018*x+343)*exp(x)^2+(64

*x^6+304*x^5-1300*x^4-2831*x^3+13454*x^2-13769*x+3430)*exp(x)-64*x^5-304*x^4+1172*x^3+2863*x^2-11270*x+8575)*e

xp(exp(x))+(-16*x^2-20*x+28)*exp(x)-64*x^2-184*x+140)/((64*x^5-336*x^4+588*x^3-343*x^2)*exp(x)^2+(128*x^6-32*x

^5-2184*x^4+5194*x^3-3430*x^2)*exp(x)+64*x^7+304*x^6-1172*x^5-2863*x^4+11270*x^3-8575*x^2),x, algorithm="giac"

)

[Out]

(16*x^3*e^(x + e^x) + 16*x^2*e^(2*x + e^x) + 24*x^2*e^(x + e^x) - 56*x*e^(2*x + e^x) - 231*x*e^(x + e^x) + 49*

e^(2*x + e^x) + 245*e^(x + e^x) + 4*e^x)/(16*x^4*e^x + 16*x^3*e^(2*x) + 24*x^3*e^x - 56*x^2*e^(2*x) - 231*x^2*

e^x + 49*x*e^(2*x) + 245*x*e^x)

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maple [A] time = 0.08, size = 33, normalized size = 1.22


method	result	size

risch	\(\frac {4}{x \left (16 x^{2}-56 x +49\right ) \left ({\mathrm e}^{x}+5+x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{x}}}{x}\)	\(33\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((64*x^4-336*x^3+588*x^2-343*x)*exp(x)^3+(128*x^5-32*x^4-2248*x^3+5530*x^2-4018*x+343)*exp(x)^2+(64*x^6+3

04*x^5-1300*x^4-2831*x^3+13454*x^2-13769*x+3430)*exp(x)-64*x^5-304*x^4+1172*x^3+2863*x^2-11270*x+8575)*exp(exp

(x))+(-16*x^2-20*x+28)*exp(x)-64*x^2-184*x+140)/((64*x^5-336*x^4+588*x^3-343*x^2)*exp(x)^2+(128*x^6-32*x^5-218

4*x^4+5194*x^3-3430*x^2)*exp(x)+64*x^7+304*x^6-1172*x^5-2863*x^4+11270*x^3-8575*x^2),x,method=_RETURNVERBOSE)

[Out]

4/x/(16*x^2-56*x+49)/(exp(x)+5+x)+exp(exp(x))/x

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maxima [B] time = 0.61, size = 73, normalized size = 2.70 \begin {gather*} \frac {{\left (16 \, x^{3} + 24 \, x^{2} + {\left (16 \, x^{2} - 56 \, x + 49\right )} e^{x} - 231 \, x + 245\right )} e^{\left (e^{x}\right )} + 4}{16 \, x^{4} + 24 \, x^{3} - 231 \, x^{2} + {\left (16 \, x^{3} - 56 \, x^{2} + 49 \, x\right )} e^{x} + 245 \, x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x^4-336*x^3+588*x^2-343*x)*exp(x)^3+(128*x^5-32*x^4-2248*x^3+5530*x^2-4018*x+343)*exp(x)^2+(64

*x^6+304*x^5-1300*x^4-2831*x^3+13454*x^2-13769*x+3430)*exp(x)-64*x^5-304*x^4+1172*x^3+2863*x^2-11270*x+8575)*e

xp(exp(x))+(-16*x^2-20*x+28)*exp(x)-64*x^2-184*x+140)/((64*x^5-336*x^4+588*x^3-343*x^2)*exp(x)^2+(128*x^6-32*x

^5-2184*x^4+5194*x^3-3430*x^2)*exp(x)+64*x^7+304*x^6-1172*x^5-2863*x^4+11270*x^3-8575*x^2),x, algorithm="maxim

a")

[Out]

((16*x^3 + 24*x^2 + (16*x^2 - 56*x + 49)*e^x - 231*x + 245)*e^(e^x) + 4)/(16*x^4 + 24*x^3 - 231*x^2 + (16*x^3

- 56*x^2 + 49*x)*e^x + 245*x)

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mupad [B] time = 0.95, size = 46, normalized size = 1.70 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{x}+\frac {4\,\left (4\,x^3+9\,x^2-28\,x\right )}{x^2\,{\left (4\,x-7\right )}^3\,\left (x+4\right )\,\left (x+{\mathrm {e}}^x+5\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((184*x + exp(exp(x))*(11270*x + exp(3*x)*(343*x - 588*x^2 + 336*x^3 - 64*x^4) - exp(x)*(13454*x^2 - 13769*

x - 2831*x^3 - 1300*x^4 + 304*x^5 + 64*x^6 + 3430) + exp(2*x)*(4018*x - 5530*x^2 + 2248*x^3 + 32*x^4 - 128*x^5

- 343) - 2863*x^2 - 1172*x^3 + 304*x^4 + 64*x^5 - 8575) + exp(x)*(20*x + 16*x^2 - 28) + 64*x^2 - 140)/(exp(x)

*(3430*x^2 - 5194*x^3 + 2184*x^4 + 32*x^5 - 128*x^6) + exp(2*x)*(343*x^2 - 588*x^3 + 336*x^4 - 64*x^5) + 8575*

x^2 - 11270*x^3 + 2863*x^4 + 1172*x^5 - 304*x^6 - 64*x^7),x)

[Out]

exp(exp(x))/x + (4*(9*x^2 - 28*x + 4*x^3))/(x^2*(4*x - 7)^3*(x + 4)*(x + exp(x) + 5))

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sympy [B] time = 0.38, size = 42, normalized size = 1.56 \begin {gather*} \frac {4}{16 x^{4} + 24 x^{3} - 231 x^{2} + 245 x + \left (16 x^{3} - 56 x^{2} + 49 x\right ) e^{x}} + \frac {e^{e^{x}}}{x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((64*x**4-336*x**3+588*x**2-343*x)*exp(x)**3+(128*x**5-32*x**4-2248*x**3+5530*x**2-4018*x+343)*exp(

x)**2+(64*x**6+304*x**5-1300*x**4-2831*x**3+13454*x**2-13769*x+3430)*exp(x)-64*x**5-304*x**4+1172*x**3+2863*x*

*2-11270*x+8575)*exp(exp(x))+(-16*x**2-20*x+28)*exp(x)-64*x**2-184*x+140)/((64*x**5-336*x**4+588*x**3-343*x**2

)*exp(x)**2+(128*x**6-32*x**5-2184*x**4+5194*x**3-3430*x**2)*exp(x)+64*x**7+304*x**6-1172*x**5-2863*x**4+11270

*x**3-8575*x**2),x)

[Out]

4/(16*x**4 + 24*x**3 - 231*x**2 + 245*x + (16*x**3 - 56*x**2 + 49*x)*exp(x)) + exp(exp(x))/x

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