3.96.50 \(\int \frac {-1+e^{-10+2 e^x-2 x} (-16+8 x-x^2)+e^{-5+e^x-x} (2+e^x (-8+2 x))}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} (16-8 x+x^2)} \, dx\)

Optimal. Leaf size=28 \[ 3+\frac {4}{2+2 e^{-5+e^x-x} (4-x)}-x \]

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Rubi [F]  time = 8.23, 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 {-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )}{1+e^{-5+e^x-x} (8-2 x)+e^{-10+2 e^x-2 x} \left (16-8 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + E^(-10 + 2*E^x - 2*x)*(-16 + 8*x - x^2) + E^(-5 + E^x - x)*(2 + E^x*(-8 + 2*x)))/(1 + E^(-5 + E^x -
x)*(8 - 2*x) + E^(-10 + 2*E^x - 2*x)*(16 - 8*x + x^2)),x]

[Out]

-x + 2*Defer[Int][E^(5 - E^x + x)/(-4 + x)^2, x] - 2*Defer[Int][E^(5 - E^x + x)/(-4 + x), x] + 10*Defer[Int][E
^(10 + 2*x)/((-4 + x)*(4*E^E^x + E^(5 + x) - E^E^x*x)^2), x] - 2*Defer[Int][E^(10 + 2*x)/(-4*E^E^x - E^(5 + x)
+ E^E^x*x)^2, x] - 8*Defer[Int][E^(5 + E^x + 2*x)/(-4*E^E^x - E^(5 + x) + E^E^x*x)^2, x] - 8*Defer[Int][E^(10
+ 2*x)/((-4 + x)*(-4*E^E^x - E^(5 + x) + E^E^x*x)^2), x] + 2*Defer[Int][(E^(5 + E^x + 2*x)*x)/(-4*E^E^x - E^(
5 + x) + E^E^x*x)^2, x] + 2*Defer[Int][E^(10 - E^x + 2*x)/((-4 + x)^2*(-4*E^E^x - E^(5 + x) + E^E^x*x)), x] -
2*Defer[Int][E^(10 - E^x + 2*x)/((-4 + x)*(-4*E^E^x - E^(5 + x) + E^E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{10+2 x} \left (-1+e^{-10+2 e^x-2 x} \left (-16+8 x-x^2\right )+e^{-5+e^x-x} \left (2+e^x (-8+2 x)\right )\right )}{\left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx\\ &=\int \left (-1-\frac {2 e^{5-e^x+x} (-5+x)}{(-4+x)^2}-\frac {2 e^{10-e^x+2 x} (-5+x)}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )}-\frac {2 e^{5+2 x} \left (-5 e^5-16 e^{e^x}+e^5 x+8 e^{e^x} x-e^{e^x} x^2\right )}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2}\right ) \, dx\\ &=-x-2 \int \frac {e^{5-e^x+x} (-5+x)}{(-4+x)^2} \, dx-2 \int \frac {e^{10-e^x+2 x} (-5+x)}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \frac {e^{5+2 x} \left (-5 e^5-16 e^{e^x}+e^5 x+8 e^{e^x} x-e^{e^x} x^2\right )}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx\\ &=-x-2 \int \left (-\frac {e^{5-e^x+x}}{(-4+x)^2}+\frac {e^{5-e^x+x}}{-4+x}\right ) \, dx-2 \int \frac {e^{5+2 x} \left (-e^5 (-5+x)+e^{e^x} (-4+x)^2\right )}{\left (e^{5+x}-e^{e^x} (-4+x)\right )^2 (4-x)} \, dx-2 \int \left (-\frac {e^{10-e^x+2 x}}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )}+\frac {e^{10-e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )}\right ) \, dx\\ &=-x+2 \int \frac {e^{5-e^x+x}}{(-4+x)^2} \, dx-2 \int \frac {e^{5-e^x+x}}{-4+x} \, dx+2 \int \frac {e^{10-e^x+2 x}}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \frac {e^{10-e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \left (-\frac {5 e^{10+2 x}}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2}+\frac {e^{10+2 x} x}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2}-\frac {16 e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}+\frac {8 e^{5+e^x+2 x} x}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}-\frac {e^{5+e^x+2 x} x^2}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}\right ) \, dx\\ &=-x+2 \int \frac {e^{5-e^x+x}}{(-4+x)^2} \, dx-2 \int \frac {e^{5-e^x+x}}{-4+x} \, dx-2 \int \frac {e^{10+2 x} x}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx+2 \int \frac {e^{5+e^x+2 x} x^2}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+2 \int \frac {e^{10-e^x+2 x}}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \frac {e^{10-e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx+10 \int \frac {e^{10+2 x}}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx-16 \int \frac {e^{5+e^x+2 x} x}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+32 \int \frac {e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx\\ &=-x+2 \int \frac {e^{5-e^x+x}}{(-4+x)^2} \, dx-2 \int \frac {e^{5-e^x+x}}{-4+x} \, dx+2 \int \frac {e^{10-e^x+2 x}}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \frac {e^{10-e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \left (\frac {e^{10+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}+\frac {4 e^{10+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}\right ) \, dx+2 \int \left (\frac {4 e^{5+e^x+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}+\frac {16 e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}+\frac {e^{5+e^x+2 x} x}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}\right ) \, dx+10 \int \frac {e^{10+2 x}}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx-16 \int \left (\frac {e^{5+e^x+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}+\frac {4 e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2}\right ) \, dx+32 \int \frac {e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx\\ &=-x+2 \int \frac {e^{5-e^x+x}}{(-4+x)^2} \, dx-2 \int \frac {e^{5-e^x+x}}{-4+x} \, dx-2 \int \frac {e^{10+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+2 \int \frac {e^{5+e^x+2 x} x}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+2 \int \frac {e^{10-e^x+2 x}}{(-4+x)^2 \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx-2 \int \frac {e^{10-e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )} \, dx+8 \int \frac {e^{5+e^x+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx-8 \int \frac {e^{10+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+10 \int \frac {e^{10+2 x}}{(-4+x) \left (4 e^{e^x}+e^{5+x}-e^{e^x} x\right )^2} \, dx-16 \int \frac {e^{5+e^x+2 x}}{\left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx+2 \left (32 \int \frac {e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx\right )-64 \int \frac {e^{5+e^x+2 x}}{(-4+x) \left (-4 e^{e^x}-e^{5+x}+e^{e^x} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.64, size = 30, normalized size = 1.07 \begin {gather*} -\frac {2 e^{5+x}}{-e^{5+x}+e^{e^x} (-4+x)}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^(-10 + 2*E^x - 2*x)*(-16 + 8*x - x^2) + E^(-5 + E^x - x)*(2 + E^x*(-8 + 2*x)))/(1 + E^(-5 +
E^x - x)*(8 - 2*x) + E^(-10 + 2*E^x - 2*x)*(16 - 8*x + x^2)),x]

[Out]

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

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fricas [A]  time = 0.58, size = 39, normalized size = 1.39 \begin {gather*} -\frac {{\left (x^{2} - 4 \, x\right )} e^{\left (-x + e^{x} - 5\right )} - x + 2}{{\left (x - 4\right )} e^{\left (-x + e^{x} - 5\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)
^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algorithm="fricas")

[Out]

-((x^2 - 4*x)*e^(-x + e^x - 5) - x + 2)/((x - 4)*e^(-x + e^x - 5) - 1)

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giac [B]  time = 0.70, size = 1136, normalized size = 40.57 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)
^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algorithm="giac")

[Out]

-(x^6*e^(1/2*x + 4*e^x) - x^6*e^(-1/2*x + 4*e^x) - 4*x^5*e^(3/2*x + 3*e^x + 5) - 20*x^5*e^(1/2*x + 4*e^x) + 4*
x^5*e^(1/2*x + 3*e^x + 5) + 21*x^5*e^(-1/2*x + 4*e^x) + 6*x^4*e^(5/2*x + 2*e^x + 10) + 66*x^4*e^(3/2*x + 3*e^x
+ 5) - 6*x^4*e^(3/2*x + 2*e^x + 10) + 160*x^4*e^(1/2*x + 4*e^x) - 70*x^4*e^(1/2*x + 3*e^x + 5) - 176*x^4*e^(-
1/2*x + 4*e^x) - 4*x^3*e^(7/2*x + e^x + 15) - 78*x^3*e^(5/2*x + 2*e^x + 10) + 4*x^3*e^(5/2*x + e^x + 15) - 416
*x^3*e^(3/2*x + 3*e^x + 5) + 84*x^3*e^(3/2*x + 2*e^x + 10) - 640*x^3*e^(1/2*x + 4*e^x) + 466*x^3*e^(1/2*x + 3*
e^x + 5) + 736*x^3*e^(-1/2*x + 4*e^x) + x^2*e^(9/2*x + 20) + 38*x^2*e^(7/2*x + e^x + 15) - x^2*e^(7/2*x + 20)
+ 360*x^2*e^(5/2*x + 2*e^x + 10) - 42*x^2*e^(5/2*x + e^x + 15) + 1216*x^2*e^(3/2*x + 3*e^x + 5) - 414*x^2*e^(3
/2*x + 2*e^x + 10) + 1280*x^2*e^(1/2*x + 4*e^x) - 1432*x^2*e^(1/2*x + 3*e^x + 5) - 1536*x^2*e^(-1/2*x + 4*e^x)
- 6*x*e^(9/2*x + 20) - 112*x*e^(7/2*x + e^x + 15) + 7*x*e^(7/2*x + 20) - 672*x*e^(5/2*x + 2*e^x + 10) + 134*x
*e^(5/2*x + e^x + 15) - 1536*x*e^(3/2*x + 3*e^x + 5) + 816*x*e^(3/2*x + 2*e^x + 10) - 1024*x*e^(1/2*x + 4*e^x)
+ 1888*x*e^(1/2*x + 3*e^x + 5) + 1280*x*e^(-1/2*x + 4*e^x) + 8*e^(9/2*x + 20) + 96*e^(7/2*x + e^x + 15) - 10*
e^(7/2*x + 20) + 384*e^(5/2*x + 2*e^x + 10) - 120*e^(5/2*x + e^x + 15) + 512*e^(3/2*x + 3*e^x + 5) - 480*e^(3/
2*x + 2*e^x + 10) - 640*e^(1/2*x + 3*e^x + 5))/(x^5*e^(1/2*x + 4*e^x) - x^5*e^(-1/2*x + 4*e^x) - 4*x^4*e^(3/2*
x + 3*e^x + 5) - 20*x^4*e^(1/2*x + 4*e^x) + 4*x^4*e^(1/2*x + 3*e^x + 5) + 21*x^4*e^(-1/2*x + 4*e^x) + 6*x^3*e^
(5/2*x + 2*e^x + 10) + 64*x^3*e^(3/2*x + 3*e^x + 5) - 6*x^3*e^(3/2*x + 2*e^x + 10) + 160*x^3*e^(1/2*x + 4*e^x)
- 68*x^3*e^(1/2*x + 3*e^x + 5) - 176*x^3*e^(-1/2*x + 4*e^x) - 4*x^2*e^(7/2*x + e^x + 15) - 72*x^2*e^(5/2*x +
2*e^x + 10) + 4*x^2*e^(5/2*x + e^x + 15) - 384*x^2*e^(3/2*x + 3*e^x + 5) + 78*x^2*e^(3/2*x + 2*e^x + 10) - 640
*x^2*e^(1/2*x + 4*e^x) + 432*x^2*e^(1/2*x + 3*e^x + 5) + 736*x^2*e^(-1/2*x + 4*e^x) + x*e^(9/2*x + 20) + 32*x*
e^(7/2*x + e^x + 15) - x*e^(7/2*x + 20) + 288*x*e^(5/2*x + 2*e^x + 10) - 36*x*e^(5/2*x + e^x + 15) + 1024*x*e^
(3/2*x + 3*e^x + 5) - 336*x*e^(3/2*x + 2*e^x + 10) + 1280*x*e^(1/2*x + 4*e^x) - 1216*x*e^(1/2*x + 3*e^x + 5) -
1536*x*e^(-1/2*x + 4*e^x) - 4*e^(9/2*x + 20) - 64*e^(7/2*x + e^x + 15) + 5*e^(7/2*x + 20) - 384*e^(5/2*x + 2*
e^x + 10) + 80*e^(5/2*x + e^x + 15) - 1024*e^(3/2*x + 3*e^x + 5) + 480*e^(3/2*x + 2*e^x + 10) - 1024*e^(1/2*x
+ 4*e^x) + 1280*e^(1/2*x + 3*e^x + 5) + 1280*e^(-1/2*x + 4*e^x))

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maple [A]  time = 0.18, size = 31, normalized size = 1.11

method	result	size
risch	\(-x -\frac {2}{{\mathrm e}^{{\mathrm e}^{x}-5-x} x -4 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-1}\)	\(31\)
norman	\(\frac {x +8 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}+2 \,{\mathrm e}^{{\mathrm e}^{x}-5-x} x -x^{2} {\mathrm e}^{{\mathrm e}^{x}-5-x}}{{\mathrm e}^{{\mathrm e}^{x}-5-x} x -4 \,{\mathrm e}^{{\mathrm e}^{x}-5-x}-1}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)^2+(-2
*x+8)*exp(exp(x)-5-x)+1),x,method=_RETURNVERBOSE)

[Out]

-x-2/(exp(exp(x)-5-x)*x-4*exp(exp(x)-5-x)-1)

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maxima [A]  time = 0.41, size = 42, normalized size = 1.50 \begin {gather*} \frac {{\left (x e^{5} - 2 \, e^{5}\right )} e^{x} - {\left (x^{2} - 4 \, x\right )} e^{\left (e^{x}\right )}}{{\left (x - 4\right )} e^{\left (e^{x}\right )} - e^{\left (x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8*x-16)*exp(exp(x)-5-x)^2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1)/((x^2-8*x+16)*exp(exp(x)-5-x)
^2+(-2*x+8)*exp(exp(x)-5-x)+1),x, algorithm="maxima")

[Out]

((x*e^5 - 2*e^5)*e^x - (x^2 - 4*x)*e^(e^x))/((x - 4)*e^(e^x) - e^(x + 5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x-2\,x-10}\,\left (x^2-8\,x+16\right )-{\mathrm {e}}^{{\mathrm {e}}^x-x-5}\,\left ({\mathrm {e}}^x\,\left (2\,x-8\right )+2\right )+1}{{\mathrm {e}}^{2\,{\mathrm {e}}^x-2\,x-10}\,\left (x^2-8\,x+16\right )-{\mathrm {e}}^{{\mathrm {e}}^x-x-5}\,\left (2\,x-8\right )+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(x) - 2*x - 10)*(x^2 - 8*x + 16) - exp(exp(x) - x - 5)*(exp(x)*(2*x - 8) + 2) + 1)/(exp(2*exp(x
) - 2*x - 10)*(x^2 - 8*x + 16) - exp(exp(x) - x - 5)*(2*x - 8) + 1),x)

[Out]

int(-(exp(2*exp(x) - 2*x - 10)*(x^2 - 8*x + 16) - exp(exp(x) - x - 5)*(exp(x)*(2*x - 8) + 2) + 1)/(exp(2*exp(x
) - 2*x - 10)*(x^2 - 8*x + 16) - exp(exp(x) - x - 5)*(2*x - 8) + 1), x)

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sympy [A]  time = 0.24, size = 17, normalized size = 0.61 \begin {gather*} - x - \frac {2}{\left (x - 4\right ) e^{- x + e^{x} - 5} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+8*x-16)*exp(exp(x)-5-x)**2+((2*x-8)*exp(x)+2)*exp(exp(x)-5-x)-1)/((x**2-8*x+16)*exp(exp(x)-5
-x)**2+(-2*x+8)*exp(exp(x)-5-x)+1),x)

[Out]

-x - 2/((x - 4)*exp(-x + exp(x) - 5) - 1)

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