3.87.98 \(\int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x (192-112 x+16 x^2)}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x (-128+64 x-8 x^2)} \, dx\)

Optimal. Leaf size=35 \[ 3+e^2-x+\log \left (\frac {5}{4-\left (-\frac {e^x}{4}+(4-x)^2\right )^2+x}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 2.33, 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 {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(80 - 3*E^(2*x) + 1040*x - 768*x^2 + 192*x^3 - 16*x^4 + E^x*(192 - 112*x + 16*x^2))/(4032 + E^(2*x) - 4112
*x + 1536*x^2 - 256*x^3 + 16*x^4 + E^x*(-128 + 64*x - 8*x^2)),x]

[Out]

-3*x + 12176*Defer[Int][(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x^2 - 256*x^3 + 16*x^
4)^(-1), x] - 192*Defer[Int][E^x/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x^2 - 256*x^
3 + 16*x^4), x] - 11296*Defer[Int][x/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x^2 - 25
6*x^3 + 16*x^4), x] + 80*Defer[Int][(E^x*x)/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x
^2 - 256*x^3 + 16*x^4), x] + 3840*Defer[Int][x^2/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*
E^x*x^2 - 256*x^3 + 16*x^4), x] - 8*Defer[Int][(E^x*x^2)/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*
x^2 - 8*E^x*x^2 - 256*x^3 + 16*x^4), x] - 576*Defer[Int][x^3/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1
536*x^2 - 8*E^x*x^2 - 256*x^3 + 16*x^4), x] + 32*Defer[Int][x^4/(4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x
+ 1536*x^2 - 8*E^x*x^2 - 256*x^3 + 16*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3-\frac {8 \left (-1522+24 e^x+1412 x-10 e^x x-480 x^2+e^x x^2+72 x^3-4 x^4\right )}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}\right ) \, dx\\ &=-3 x-8 \int \frac {-1522+24 e^x+1412 x-10 e^x x-480 x^2+e^x x^2+72 x^3-4 x^4}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx\\ &=-3 x-8 \int \left (-\frac {1522}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}+\frac {24 e^x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}+\frac {1412 x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}-\frac {10 e^x x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}-\frac {480 x^2}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}+\frac {e^x x^2}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}+\frac {72 x^3}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}-\frac {4 x^4}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4}\right ) \, dx\\ &=-3 x-8 \int \frac {e^x x^2}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx+32 \int \frac {x^4}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx+80 \int \frac {e^x x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx-192 \int \frac {e^x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx-576 \int \frac {x^3}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx+3840 \int \frac {x^2}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx-11296 \int \frac {x}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx+12176 \int \frac {1}{4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 51, normalized size = 1.46 \begin {gather*} -x-\log \left (4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80 - 3*E^(2*x) + 1040*x - 768*x^2 + 192*x^3 - 16*x^4 + E^x*(192 - 112*x + 16*x^2))/(4032 + E^(2*x)
- 4112*x + 1536*x^2 - 256*x^3 + 16*x^4 + E^x*(-128 + 64*x - 8*x^2)),x]

[Out]

-x - Log[4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x^2 - 256*x^3 + 16*x^4]

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 43, normalized size = 1.23 \begin {gather*} -x - \log \left (16 \, x^{4} - 256 \, x^{3} + 1536 \, x^{2} - 8 \, {\left (x^{2} - 8 \, x + 16\right )} e^{x} - 4112 \, x + e^{\left (2 \, x\right )} + 4032\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+80)/(exp(x)^2+(-8*x^2+64*x-128)
*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032),x, algorithm="fricas")

[Out]

-x - log(16*x^4 - 256*x^3 + 1536*x^2 - 8*(x^2 - 8*x + 16)*e^x - 4112*x + e^(2*x) + 4032)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 47, normalized size = 1.34 \begin {gather*} -x - \log \left (16 \, x^{4} - 256 \, x^{3} - 8 \, x^{2} e^{x} + 1536 \, x^{2} + 64 \, x e^{x} - 4112 \, x + e^{\left (2 \, x\right )} - 128 \, e^{x} + 4032\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+80)/(exp(x)^2+(-8*x^2+64*x-128)
*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032),x, algorithm="giac")

[Out]

-x - log(16*x^4 - 256*x^3 - 8*x^2*e^x + 1536*x^2 + 64*x*e^x - 4112*x + e^(2*x) - 128*e^x + 4032)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 45, normalized size = 1.29

method	result	size
risch	\(-x -\ln \left ({\mathrm e}^{2 x}+\left (-8 x^{2}+64 x -128\right ) {\mathrm e}^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032\right )\)	\(45\)
norman	\(-x -\ln \left (16 x^{4}-8 \,{\mathrm e}^{x} x^{2}-256 x^{3}+{\mathrm e}^{2 x}+64 \,{\mathrm e}^{x} x +1536 x^{2}-128 \,{\mathrm e}^{x}-4112 x +4032\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+80)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x
)+16*x^4-256*x^3+1536*x^2-4112*x+4032),x,method=_RETURNVERBOSE)

[Out]

-x-ln(exp(2*x)+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 43, normalized size = 1.23 \begin {gather*} -x - \log \left (16 \, x^{4} - 256 \, x^{3} + 1536 \, x^{2} - 8 \, {\left (x^{2} - 8 \, x + 16\right )} e^{x} - 4112 \, x + e^{\left (2 \, x\right )} + 4032\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+80)/(exp(x)^2+(-8*x^2+64*x-128)
*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032),x, algorithm="maxima")

[Out]

-x - log(16*x^4 - 256*x^3 + 1536*x^2 - 8*(x^2 - 8*x + 16)*e^x - 4112*x + e^(2*x) + 4032)

________________________________________________________________________________________

mupad [B]  time = 5.79, size = 47, normalized size = 1.34 \begin {gather*} -x-\ln \left (\frac {{\mathrm {e}}^{2\,x}}{16}-257\,x-8\,{\mathrm {e}}^x-\frac {x^2\,{\mathrm {e}}^x}{2}+4\,x\,{\mathrm {e}}^x+96\,x^2-16\,x^3+x^4+252\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1040*x - 3*exp(2*x) + exp(x)*(16*x^2 - 112*x + 192) - 768*x^2 + 192*x^3 - 16*x^4 + 80)/(exp(2*x) - 4112*x
- exp(x)*(8*x^2 - 64*x + 128) + 1536*x^2 - 256*x^3 + 16*x^4 + 4032),x)

[Out]

- x - log(exp(2*x)/16 - 257*x - 8*exp(x) - (x^2*exp(x))/2 + 4*x*exp(x) + 96*x^2 - 16*x^3 + x^4 + 252)

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 42, normalized size = 1.20 \begin {gather*} - x - \log {\left (16 x^{4} - 256 x^{3} + 1536 x^{2} - 4112 x + \left (- 8 x^{2} + 64 x - 128\right ) e^{x} + e^{2 x} + 4032 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(x)**2+(16*x**2-112*x+192)*exp(x)-16*x**4+192*x**3-768*x**2+1040*x+80)/(exp(x)**2+(-8*x**2+64
*x-128)*exp(x)+16*x**4-256*x**3+1536*x**2-4112*x+4032),x)

[Out]

-x - log(16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + (-8*x**2 + 64*x - 128)*exp(x) + exp(2*x) + 4032)

________________________________________________________________________________________