3.95.14 \(\int \frac {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} (32-14 x+2 x^2)+(32-16 x+2 x^2) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} (2048 x^2-1024 x^3+128 x^4)+(-2048 x^2+1024 x^3-128 x^4) \log (16)+(2048 x^2-1024 x^3+128 x^4) \log ^2(16)+e^{\frac {-5+x}{-4+x}} (-2048 x^2+1024 x^3-128 x^4+(4096 x^2-2048 x^3+256 x^4) \log (16))} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{16 x \left (2-4 \left (e^{\frac {x}{x+\frac {x}{-5+x}}}+\log (16)\right )\right )} \]

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Rubi [F]  time = 9.45, 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 {-16+8 x-x^2+e^{\frac {-5+x}{-4+x}} \left (32-14 x+2 x^2\right )+\left (32-16 x+2 x^2\right ) \log (16)}{512 x^2-256 x^3+32 x^4+e^{\frac {2 (-5+x)}{-4+x}} \left (2048 x^2-1024 x^3+128 x^4\right )+\left (-2048 x^2+1024 x^3-128 x^4\right ) \log (16)+\left (2048 x^2-1024 x^3+128 x^4\right ) \log ^2(16)+e^{\frac {-5+x}{-4+x}} \left (-2048 x^2+1024 x^3-128 x^4+\left (4096 x^2-2048 x^3+256 x^4\right ) \log (16)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16 + 8*x - x^2 + E^((-5 + x)/(-4 + x))*(32 - 14*x + 2*x^2) + (32 - 16*x + 2*x^2)*Log[16])/(512*x^2 - 256
*x^3 + 32*x^4 + E^((2*(-5 + x))/(-4 + x))*(2048*x^2 - 1024*x^3 + 128*x^4) + (-2048*x^2 + 1024*x^3 - 128*x^4)*L
og[16] + (2048*x^2 - 1024*x^3 + 128*x^4)*Log[16]^2 + E^((-5 + x)/(-4 + x))*(-2048*x^2 + 1024*x^3 - 128*x^4 + (
4096*x^2 - 2048*x^3 + 256*x^4)*Log[16])),x]

[Out]

((1 - Log[256])*Defer[Int][E^(10/(-4 + x))/((4 - x)^2*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]))^2), x
])/128 + ((1 - Log[256])*Defer[Int][E^(10/(-4 + x))/((4 - x)*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256])
)^2), x])/512 + ((1 - Log[256])*Defer[Int][E^(10/(-4 + x))/(x*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]
))^2), x])/512 + Defer[Int][E^(5/(-4 + x))/((4 - x)^2*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]))), x]/
128 + Defer[Int][E^(5/(-4 + x))/((4 - x)*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]))), x]/512 + Defer[I
nt][E^(5/(-4 + x))/(x^2*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]))), x]/32 + Defer[Int][E^(5/(-4 + x))
/(x*(2*E^(x/(-4 + x)) - E^(5/(-4 + x))*(1 - Log[256]))), x]/512

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {5+x}{-4+x}} \left (16-7 x+x^2\right )+e^{\frac {10}{-4+x}} (-4+x)^2 (-1+\log (256))}{32 (4-x)^2 x^2 \left (2 e^{\frac {x}{-4+x}}+e^{\frac {5}{-4+x}} (-1+\log (256))\right )^2} \, dx\\ &=\frac {1}{32} \int \frac {2 e^{\frac {5+x}{-4+x}} \left (16-7 x+x^2\right )+e^{\frac {10}{-4+x}} (-4+x)^2 (-1+\log (256))}{(4-x)^2 x^2 \left (2 e^{\frac {x}{-4+x}}+e^{\frac {5}{-4+x}} (-1+\log (256))\right )^2} \, dx\\ &=\frac {1}{32} \int \left (\frac {e^{\frac {5}{-4+x}} \left (16-7 x+x^2\right )}{(4-x)^2 x^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )}+\frac {e^{\frac {10}{-4+x}} (1-\log (256))}{(4-x)^2 x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2}\right ) \, dx\\ &=\frac {1}{32} \int \frac {e^{\frac {5}{-4+x}} \left (16-7 x+x^2\right )}{(4-x)^2 x^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )} \, dx+\frac {1}{32} (1-\log (256)) \int \frac {e^{\frac {10}{-4+x}}}{(4-x)^2 x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2} \, dx\\ &=\frac {1}{32} \int \left (\frac {e^{\frac {5}{-4+x}}}{4 (4-x)^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )}+\frac {e^{\frac {5}{-4+x}}}{16 (4-x) \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )}+\frac {e^{\frac {5}{-4+x}}}{x^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )}+\frac {e^{\frac {5}{-4+x}}}{16 x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )}\right ) \, dx+\frac {1}{32} (1-\log (256)) \int \left (\frac {e^{\frac {10}{-4+x}}}{4 (4-x)^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2}+\frac {e^{\frac {10}{-4+x}}}{16 (4-x) \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2}+\frac {e^{\frac {10}{-4+x}}}{16 x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2}\right ) \, dx\\ &=\frac {1}{512} \int \frac {e^{\frac {5}{-4+x}}}{(4-x) \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )} \, dx+\frac {1}{512} \int \frac {e^{\frac {5}{-4+x}}}{x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )} \, dx+\frac {1}{128} \int \frac {e^{\frac {5}{-4+x}}}{(4-x)^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )} \, dx+\frac {1}{32} \int \frac {e^{\frac {5}{-4+x}}}{x^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )} \, dx+\frac {1}{512} (1-\log (256)) \int \frac {e^{\frac {10}{-4+x}}}{(4-x) \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2} \, dx+\frac {1}{512} (1-\log (256)) \int \frac {e^{\frac {10}{-4+x}}}{x \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2} \, dx+\frac {1}{128} (1-\log (256)) \int \frac {e^{\frac {10}{-4+x}}}{(4-x)^2 \left (2 e^{\frac {x}{-4+x}}-e^{\frac {5}{-4+x}} (1-\log (256))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.39, size = 32, normalized size = 1.03 \begin {gather*} -\frac {e^{\frac {1}{-4+x}}}{32 x \left (2 e+e^{\frac {1}{-4+x}} (-1+\log (256))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 + 8*x - x^2 + E^((-5 + x)/(-4 + x))*(32 - 14*x + 2*x^2) + (32 - 16*x + 2*x^2)*Log[16])/(512*x^2
- 256*x^3 + 32*x^4 + E^((2*(-5 + x))/(-4 + x))*(2048*x^2 - 1024*x^3 + 128*x^4) + (-2048*x^2 + 1024*x^3 - 128*
x^4)*Log[16] + (2048*x^2 - 1024*x^3 + 128*x^4)*Log[16]^2 + E^((-5 + x)/(-4 + x))*(-2048*x^2 + 1024*x^3 - 128*x
^4 + (4096*x^2 - 2048*x^3 + 256*x^4)*Log[16])),x]

[Out]

-1/32*E^(-4 + x)^(-1)/(x*(2*E + E^(-4 + x)^(-1)*(-1 + Log[256])))

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fricas [A]  time = 0.67, size = 26, normalized size = 0.84 \begin {gather*} -\frac {1}{32 \, {\left (2 \, x e^{\left (\frac {x - 5}{x - 4}\right )} + 8 \, x \log \relax (2) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-14*x+32)*exp((x-5)/(x-4))+4*(2*x^2-16*x+32)*log(2)-x^2+8*x-16)/((128*x^4-1024*x^3+2048*x^2)*
exp((x-5)/(x-4))^2+(4*(256*x^4-2048*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((x-5)/(x-4))+16*(128*x
^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+32*x^4-256*x^3+512*x^2),x, algorithm="fri
cas")

[Out]

-1/32/(2*x*e^((x - 5)/(x - 4)) + 8*x*log(2) - x)

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giac [A]  time = 0.41, size = 27, normalized size = 0.87 \begin {gather*} -\frac {1}{32 \, {\left (2 \, x e^{\left (-\frac {x}{4 \, {\left (x - 4\right )}} + \frac {5}{4}\right )} + 8 \, x \log \relax (2) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-14*x+32)*exp((x-5)/(x-4))+4*(2*x^2-16*x+32)*log(2)-x^2+8*x-16)/((128*x^4-1024*x^3+2048*x^2)*
exp((x-5)/(x-4))^2+(4*(256*x^4-2048*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((x-5)/(x-4))+16*(128*x
^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+32*x^4-256*x^3+512*x^2),x, algorithm="gia
c")

[Out]

-1/32/(2*x*e^(-1/4*x/(x - 4) + 5/4) + 8*x*log(2) - x)

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maple [A]  time = 1.58, size = 26, normalized size = 0.84

method	result	size
risch	\(-\frac {1}{32 x \left (-1+2 \,{\mathrm e}^{\frac {x -5}{x -4}}+8 \ln \relax (2)\right )}\)	\(26\)
norman	\(\frac {\frac {1}{8}-\frac {x}{32}}{x \left (x -4\right ) \left (-1+2 \,{\mathrm e}^{\frac {x -5}{x -4}}+8 \ln \relax (2)\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-14*x+32)*exp((x-5)/(x-4))+4*(2*x^2-16*x+32)*ln(2)-x^2+8*x-16)/((128*x^4-1024*x^3+2048*x^2)*exp((x-
5)/(x-4))^2+(4*(256*x^4-2048*x^3+4096*x^2)*ln(2)-128*x^4+1024*x^3-2048*x^2)*exp((x-5)/(x-4))+16*(128*x^4-1024*
x^3+2048*x^2)*ln(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*ln(2)+32*x^4-256*x^3+512*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/32/x/(-1+2*exp((x-5)/(x-4))+8*ln(2))

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maxima [A]  time = 0.52, size = 30, normalized size = 0.97 \begin {gather*} -\frac {e^{\left (\frac {1}{x - 4}\right )}}{32 \, {\left (x {\left (8 \, \log \relax (2) - 1\right )} e^{\left (\frac {1}{x - 4}\right )} + 2 \, x e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-14*x+32)*exp((x-5)/(x-4))+4*(2*x^2-16*x+32)*log(2)-x^2+8*x-16)/((128*x^4-1024*x^3+2048*x^2)*
exp((x-5)/(x-4))^2+(4*(256*x^4-2048*x^3+4096*x^2)*log(2)-128*x^4+1024*x^3-2048*x^2)*exp((x-5)/(x-4))+16*(128*x
^4-1024*x^3+2048*x^2)*log(2)^2+4*(-128*x^4+1024*x^3-2048*x^2)*log(2)+32*x^4-256*x^3+512*x^2),x, algorithm="max
ima")

[Out]

-1/32*e^(1/(x - 4))/(x*(8*log(2) - 1)*e^(1/(x - 4)) + 2*x*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x+4\,\ln \relax (2)\,\left (2\,x^2-16\,x+32\right )+{\mathrm {e}}^{\frac {x-5}{x-4}}\,\left (2\,x^2-14\,x+32\right )-x^2-16}{{\mathrm {e}}^{\frac {x-5}{x-4}}\,\left (4\,\ln \relax (2)\,\left (256\,x^4-2048\,x^3+4096\,x^2\right )-2048\,x^2+1024\,x^3-128\,x^4\right )+16\,{\ln \relax (2)}^2\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )+{\mathrm {e}}^{\frac {2\,\left (x-5\right )}{x-4}}\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )-4\,\ln \relax (2)\,\left (128\,x^4-1024\,x^3+2048\,x^2\right )+512\,x^2-256\,x^3+32\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 4*log(2)*(2*x^2 - 16*x + 32) + exp((x - 5)/(x - 4))*(2*x^2 - 14*x + 32) - x^2 - 16)/(exp((x - 5)/(x
- 4))*(4*log(2)*(4096*x^2 - 2048*x^3 + 256*x^4) - 2048*x^2 + 1024*x^3 - 128*x^4) + 16*log(2)^2*(2048*x^2 - 10
24*x^3 + 128*x^4) + exp((2*(x - 5))/(x - 4))*(2048*x^2 - 1024*x^3 + 128*x^4) - 4*log(2)*(2048*x^2 - 1024*x^3 +
128*x^4) + 512*x^2 - 256*x^3 + 32*x^4),x)

[Out]

int((8*x + 4*log(2)*(2*x^2 - 16*x + 32) + exp((x - 5)/(x - 4))*(2*x^2 - 14*x + 32) - x^2 - 16)/(exp((x - 5)/(x
- 4))*(4*log(2)*(4096*x^2 - 2048*x^3 + 256*x^4) - 2048*x^2 + 1024*x^3 - 128*x^4) + 16*log(2)^2*(2048*x^2 - 10
24*x^3 + 128*x^4) + exp((2*(x - 5))/(x - 4))*(2048*x^2 - 1024*x^3 + 128*x^4) - 4*log(2)*(2048*x^2 - 1024*x^3 +
128*x^4) + 512*x^2 - 256*x^3 + 32*x^4), x)

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sympy [A]  time = 0.33, size = 24, normalized size = 0.77 \begin {gather*} - \frac {1}{64 x e^{\frac {x - 5}{x - 4}} - 32 x + 256 x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-14*x+32)*exp((x-5)/(x-4))+4*(2*x**2-16*x+32)*ln(2)-x**2+8*x-16)/((128*x**4-1024*x**3+2048*x
**2)*exp((x-5)/(x-4))**2+(4*(256*x**4-2048*x**3+4096*x**2)*ln(2)-128*x**4+1024*x**3-2048*x**2)*exp((x-5)/(x-4)
)+16*(128*x**4-1024*x**3+2048*x**2)*ln(2)**2+4*(-128*x**4+1024*x**3-2048*x**2)*ln(2)+32*x**4-256*x**3+512*x**2
),x)

[Out]

-1/(64*x*exp((x - 5)/(x - 4)) - 32*x + 256*x*log(2))

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