Integrand size = 139, antiderivative size = 26 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=3+\frac {e^x}{4}-\frac {x^3}{\left (-4+e^{-1+x}-3 x\right )^2} \] Output:
-x^3/(exp(-1+x)-4-3*x)^2+3+1/4*exp(x)
Time = 2.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=\frac {1}{4} \left (e^x-\frac {4 e^2 x^3}{\left (-4 e+e^x-3 e x\right )^2}\right ) \] Input:
Integrate[(48*x^2 + 12*x^3 + E^(-1 + x)*(-12*x^2 + 8*x^3) + E^x*(-64 + E^( -3 + 3*x) + E^(-2 + 2*x)*(-12 - 9*x) - 144*x - 108*x^2 - 27*x^3 + E^(-1 + x)*(48 + 72*x + 27*x^2)))/(-256 + 4*E^(-3 + 3*x) + E^(-2 + 2*x)*(-48 - 36* x) - 576*x - 432*x^2 - 108*x^3 + E^(-1 + x)*(192 + 288*x + 108*x^2)),x]
Output:
(E^x - (4*E^2*x^3)/(-4*E + E^x - 3*E*x)^2)/4
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {12 x^3+48 x^2+e^{x-1} \left (8 x^3-12 x^2\right )+e^x \left (-27 x^3-108 x^2+e^{x-1} \left (27 x^2+72 x+48\right )-144 x+e^{3 x-3}+e^{2 x-2} (-9 x-12)-64\right )}{-108 x^3-432 x^2+e^{x-1} \left (108 x^2+288 x+192\right )-576 x+4 e^{3 x-3}+e^{2 x-2} (-36 x-48)-256} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^3 \left (-12 x^3-48 x^2-e^{x-1} \left (8 x^3-12 x^2\right )-e^x \left (-27 x^3-108 x^2+e^{x-1} \left (27 x^2+72 x+48\right )-144 x+e^{3 x-3}+e^{2 x-2} (-9 x-12)-64\right )\right )}{4 \left (3 e x-e^x+4 e\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} e^3 \int -\frac {12 x^3+48 x^2-4 e^{x-1} \left (3 x^2-2 x^3\right )-e^x \left (27 x^3+108 x^2+144 x-e^{3 x-3}+3 e^{2 x-2} (3 x+4)-3 e^{x-1} \left (9 x^2+24 x+16\right )+64\right )}{\left (3 e x-e^x+4 e\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} e^3 \int \frac {12 x^3+48 x^2-4 e^{x-1} \left (3 x^2-2 x^3\right )-e^x \left (27 x^3+108 x^2+144 x-e^{3 x-3}+3 e^{2 x-2} (3 x+4)-3 e^{x-1} \left (9 x^2+24 x+16\right )+64\right )}{\left (3 e x-e^x+4 e\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} e^3 \int \left (\frac {8 (3 x+1) x^3}{\left (3 e x-e^x+4 e\right )^3}-\frac {4 (2 x-3) x^2}{e \left (3 e x-e^x+4 e\right )^2}-e^{x-3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} e^3 \left (24 \int \frac {x^4}{\left (3 e x-e^x+4 e\right )^3}dx+8 \int \frac {x^3}{\left (3 e x-e^x+4 e\right )^3}dx-\frac {8 \int \frac {x^3}{\left (3 e x-e^x+4 e\right )^2}dx}{e}+\frac {12 \int \frac {x^2}{\left (3 e x-e^x+4 e\right )^2}dx}{e}-e^{x-3}\right )\) |
Input:
Int[(48*x^2 + 12*x^3 + E^(-1 + x)*(-12*x^2 + 8*x^3) + E^x*(-64 + E^(-3 + 3 *x) + E^(-2 + 2*x)*(-12 - 9*x) - 144*x - 108*x^2 - 27*x^3 + E^(-1 + x)*(48 + 72*x + 27*x^2)))/(-256 + 4*E^(-3 + 3*x) + E^(-2 + 2*x)*(-48 - 36*x) - 5 76*x - 432*x^2 - 108*x^3 + E^(-1 + x)*(192 + 288*x + 108*x^2)),x]
Output:
$Aborted
Time = 1.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {{\mathrm e}^{x}}{4}-\frac {x^{3}}{\left ({\mathrm e}^{-1+x}-4-3 x \right )^{2}}\) | \(22\) |
| parallelrisch | \(-\frac {4 x^{3}-9 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{-1+x} {\mathrm e}^{x} x -{\mathrm e}^{x} {\mathrm e}^{-2+2 x}-24 \,{\mathrm e}^{x} x +8 \,{\mathrm e}^{-1+x} {\mathrm e}^{x}-16 \,{\mathrm e}^{x}}{4 \left (9 x^{2}-6 x \,{\mathrm e}^{-1+x}+{\mathrm e}^{-2+2 x}+24 x -8 \,{\mathrm e}^{-1+x}+16\right )}\) | \(83\) |
| norman | \(\frac {\left (-12 \,{\mathrm e}^{4} {\mathrm e}^{x}+18 x^{2} {\mathrm e}^{5}+48 x \,{\mathrm e}^{5}-6 \,{\mathrm e}^{4} {\mathrm e}^{x} x +\frac {{\mathrm e}^{2} {\mathrm e}^{3 x}}{4}-x^{3} {\mathrm e}^{4}-\frac {3 x \,{\mathrm e}^{3} {\mathrm e}^{2 x}}{2}+\frac {9 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{4}+32 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-2}}{\left (3 x \,{\mathrm e}+4 \,{\mathrm e}-{\mathrm e}^{x}\right )^{2}}\) | \(103\) |
| parts | \(-\frac {x^{3}}{\left (3 x -{\mathrm e}^{-1+x}+4\right )^{2}}+\frac {\left (-12 \,{\mathrm e}^{4} {\mathrm e}^{x}+18 x^{2} {\mathrm e}^{5}+48 x \,{\mathrm e}^{5}-6 \,{\mathrm e}^{4} {\mathrm e}^{x} x +\frac {{\mathrm e}^{2} {\mathrm e}^{3 x}}{4}-\frac {3 x \,{\mathrm e}^{3} {\mathrm e}^{2 x}}{2}+\frac {9 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}}{4}+32 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-2}}{\left (3 x \,{\mathrm e}+4 \,{\mathrm e}-{\mathrm e}^{x}\right )^{2}}\) | \(113\) |
Input:
int(((exp(-1+x)^3+(-9*x-12)*exp(-1+x)^2+(27*x^2+72*x+48)*exp(-1+x)-27*x^3- 108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)*exp(-1+x)+12*x^3+48*x^2)/(4*exp(-1 +x)^3+(-36*x-48)*exp(-1+x)^2+(108*x^2+288*x+192)*exp(-1+x)-108*x^3-432*x^2 -576*x-256),x,method=_RETURNVERBOSE)
Output:
1/4*exp(x)-x^3/(exp(-1+x)-4-3*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=-\frac {4 \, x^{3} e^{2} + 2 \, {\left (3 \, x + 4\right )} e^{\left (2 \, x + 1\right )} - {\left (9 \, x^{2} + 24 \, x + 16\right )} e^{\left (x + 2\right )} - e^{\left (3 \, x\right )}}{4 \, {\left ({\left (9 \, x^{2} + 24 \, x + 16\right )} e^{2} - 2 \, {\left (3 \, x + 4\right )} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}\right )}} \] Input:
integrate(((exp(-1+x)^3+(-9*x-12)*exp(-1+x)^2+(27*x^2+72*x+48)*exp(-1+x)-2 7*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)*exp(-1+x)+12*x^3+48*x^2)/(4* exp(-1+x)^3+(-36*x-48)*exp(-1+x)^2+(108*x^2+288*x+192)*exp(-1+x)-108*x^3-4 32*x^2-576*x-256),x, algorithm="fricas")
Output:
-1/4*(4*x^3*e^2 + 2*(3*x + 4)*e^(2*x + 1) - (9*x^2 + 24*x + 16)*e^(x + 2) - e^(3*x))/((9*x^2 + 24*x + 16)*e^2 - 2*(3*x + 4)*e^(x + 1) + e^(2*x))
Timed out. \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(((exp(-1+x)**3+(-9*x-12)*exp(-1+x)**2+(27*x**2+72*x+48)*exp(-1+x )-27*x**3-108*x**2-144*x-64)*exp(x)+(8*x**3-12*x**2)*exp(-1+x)+12*x**3+48* x**2)/(4*exp(-1+x)**3+(-36*x-48)*exp(-1+x)**2+(108*x**2+288*x+192)*exp(-1+ x)-108*x**3-432*x**2-576*x-256),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.46 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=-\frac {4 \, x^{3} e^{2} + 2 \, {\left (3 \, x e + 4 \, e\right )} e^{\left (2 \, x\right )} - {\left (9 \, x^{2} e^{2} + 24 \, x e^{2} + 16 \, e^{2}\right )} e^{x} - e^{\left (3 \, x\right )}}{4 \, {\left (9 \, x^{2} e^{2} + 24 \, x e^{2} - 2 \, {\left (3 \, x e + 4 \, e\right )} e^{x} + 16 \, e^{2} + e^{\left (2 \, x\right )}\right )}} \] Input:
integrate(((exp(-1+x)^3+(-9*x-12)*exp(-1+x)^2+(27*x^2+72*x+48)*exp(-1+x)-2 7*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)*exp(-1+x)+12*x^3+48*x^2)/(4* exp(-1+x)^3+(-36*x-48)*exp(-1+x)^2+(108*x^2+288*x+192)*exp(-1+x)-108*x^3-4 32*x^2-576*x-256),x, algorithm="maxima")
Output:
-1/4*(4*x^3*e^2 + 2*(3*x*e + 4*e)*e^(2*x) - (9*x^2*e^2 + 24*x*e^2 + 16*e^2 )*e^x - e^(3*x))/(9*x^2*e^2 + 24*x*e^2 - 2*(3*x*e + 4*e)*e^x + 16*e^2 + e^ (2*x))
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.96 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=-\frac {4 \, {\left (x - 1\right )}^{3} - 9 \, {\left (x - 1\right )}^{2} e^{x} + 12 \, {\left (x - 1\right )}^{2} + 6 \, {\left (x - 1\right )} e^{\left (2 \, x - 1\right )} - 42 \, {\left (x - 1\right )} e^{x} + 12 \, x - e^{\left (3 \, x - 2\right )} + 14 \, e^{\left (2 \, x - 1\right )} - 49 \, e^{x} - 8}{4 \, {\left (9 \, {\left (x - 1\right )}^{2} - 6 \, {\left (x - 1\right )} e^{\left (x - 1\right )} + 42 \, x + e^{\left (2 \, x - 2\right )} - 14 \, e^{\left (x - 1\right )} + 7\right )}} \] Input:
integrate(((exp(-1+x)^3+(-9*x-12)*exp(-1+x)^2+(27*x^2+72*x+48)*exp(-1+x)-2 7*x^3-108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)*exp(-1+x)+12*x^3+48*x^2)/(4* exp(-1+x)^3+(-36*x-48)*exp(-1+x)^2+(108*x^2+288*x+192)*exp(-1+x)-108*x^3-4 32*x^2-576*x-256),x, algorithm="giac")
Output:
-1/4*(4*(x - 1)^3 - 9*(x - 1)^2*e^x + 12*(x - 1)^2 + 6*(x - 1)*e^(2*x - 1) - 42*(x - 1)*e^x + 12*x - e^(3*x - 2) + 14*e^(2*x - 1) - 49*e^x - 8)/(9*( x - 1)^2 - 6*(x - 1)*e^(x - 1) + 42*x + e^(2*x - 2) - 14*e^(x - 1) + 7)
Timed out. \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (144\,x-{\mathrm {e}}^{3\,x-3}-{\mathrm {e}}^{x-1}\,\left (27\,x^2+72\,x+48\right )+{\mathrm {e}}^{2\,x-2}\,\left (9\,x+12\right )+108\,x^2+27\,x^3+64\right )+{\mathrm {e}}^{x-1}\,\left (12\,x^2-8\,x^3\right )-48\,x^2-12\,x^3}{576\,x-4\,{\mathrm {e}}^{3\,x-3}-{\mathrm {e}}^{x-1}\,\left (108\,x^2+288\,x+192\right )+{\mathrm {e}}^{2\,x-2}\,\left (36\,x+48\right )+432\,x^2+108\,x^3+256} \,d x \] Input:
int((exp(x)*(144*x - exp(3*x - 3) - exp(x - 1)*(72*x + 27*x^2 + 48) + exp( 2*x - 2)*(9*x + 12) + 108*x^2 + 27*x^3 + 64) + exp(x - 1)*(12*x^2 - 8*x^3) - 48*x^2 - 12*x^3)/(576*x - 4*exp(3*x - 3) - exp(x - 1)*(288*x + 108*x^2 + 192) + exp(2*x - 2)*(36*x + 48) + 432*x^2 + 108*x^3 + 256),x)
Output:
int((exp(x)*(144*x - exp(3*x - 3) - exp(x - 1)*(72*x + 27*x^2 + 48) + exp( 2*x - 2)*(9*x + 12) + 108*x^2 + 27*x^3 + 64) + exp(x - 1)*(12*x^2 - 8*x^3) - 48*x^2 - 12*x^3)/(576*x - 4*exp(3*x - 3) - exp(x - 1)*(288*x + 108*x^2 + 192) + exp(2*x - 2)*(36*x + 48) + 432*x^2 + 108*x^3 + 256), x)
Time = 0.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.31 \[ \int \frac {48 x^2+12 x^3+e^{-1+x} \left (-12 x^2+8 x^3\right )+e^x \left (-64+e^{-3+3 x}+e^{-2+2 x} (-12-9 x)-144 x-108 x^2-27 x^3+e^{-1+x} \left (48+72 x+27 x^2\right )\right )}{-256+4 e^{-3+3 x}+e^{-2+2 x} (-48-36 x)-576 x-432 x^2-108 x^3+e^{-1+x} \left (192+288 x+108 x^2\right )} \, dx=\frac {e^{3 x}-6 e^{2 x} e x -4 e^{2 x} e +9 e^{x} e^{2} x^{2}-16 e^{x} e^{2}+36 e^{3} x^{2}+96 e^{3} x +64 e^{3}-4 e^{2} x^{3}}{4 e^{2 x}-24 e^{x} e x -32 e^{x} e +36 e^{2} x^{2}+96 e^{2} x +64 e^{2}} \] Input:
int(((exp(-1+x)^3+(-9*x-12)*exp(-1+x)^2+(27*x^2+72*x+48)*exp(-1+x)-27*x^3- 108*x^2-144*x-64)*exp(x)+(8*x^3-12*x^2)*exp(-1+x)+12*x^3+48*x^2)/(4*exp(-1 +x)^3+(-36*x-48)*exp(-1+x)^2+(108*x^2+288*x+192)*exp(-1+x)-108*x^3-432*x^2 -576*x-256),x)
Output:
(e**(3*x) - 6*e**(2*x)*e*x - 4*e**(2*x)*e + 9*e**x*e**2*x**2 - 16*e**x*e** 2 + 36*e**3*x**2 + 96*e**3*x + 64*e**3 - 4*e**2*x**3)/(4*(e**(2*x) - 6*e** x*e*x - 8*e**x*e + 9*e**2*x**2 + 24*e**2*x + 16*e**2))