Integrand size = 41, antiderivative size = 21 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=e^{4-4 (-2+x)} \left (3+\frac {5}{2 x^3}\right )+x \] Output:
x+(5/2/x^3+3)*exp(4)/exp(4*x-8)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x \] Input:
Integrate[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/( 2*x^4),x]
Output:
3*E^(12 - 4*x) + (5*E^(12 - 4*x))/(2*x^3) + x
Time = 0.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {27, 7293, 2009}
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.
\(\displaystyle \int \frac {e^{8-4 x} \left (2 e^{4 x-8} x^4+e^4 \left (-24 x^4-20 x-15\right )\right )}{2 x^4} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {e^{8-4 x} \left (2 e^{4 x-8} x^4-e^4 \left (24 x^4+20 x+15\right )\right )}{x^4}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (2-\frac {e^{12-4 x} \left (24 x^4+20 x+15\right )}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {5 e^{12-4 x}}{x^3}+2 x+6 e^{12-4 x}\right )\) |
Input:
Int[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/(2*x^4) ,x]
Output:
(6*E^(12 - 4*x) + (5*E^(12 - 4*x))/x^3 + 2*x)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x +\frac {\left (6 x^{3}+5\right ) {\mathrm e}^{12-4 x}}{2 x^{3}}\) | \(21\) |
norman | \(\frac {\left (x^{4} {\mathrm e}^{4 x -8}+3 x^{3} {\mathrm e}^{4}+\frac {5 \,{\mathrm e}^{4}}{2}\right ) {\mathrm e}^{-4 x +8}}{x^{3}}\) | \(35\) |
parallelrisch | \(\frac {\left (2 x^{4} {\mathrm e}^{4 x -8}+6 x^{3} {\mathrm e}^{4}+5 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4 x +8}}{2 x^{3}}\) | \(37\) |
parts | \(x -\frac {{\mathrm e}^{4} \left (-\frac {439 \,{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{6 x^{3}}+\frac {197 \,{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{6 x^{3}}-\frac {12 \,{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{x^{3}}+\frac {8 \,{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{x^{3}}-6 \,{\mathrm e}^{-4 x +8}-\frac {4 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{x^{3}}\right )}{2}\) | \(139\) |
orering | \(\frac {\left (x -\frac {1}{4}\right ) \left (2 x^{4} {\mathrm e}^{4 x -8}+\left (-24 x^{4}-20 x -15\right ) {\mathrm e}^{4}\right ) {\mathrm e}^{-4 x +8}}{2 x^{4}}+\frac {\left (96 x^{5}+80 x^{2}+60 x -15\right ) x \left (\frac {\left (8 x^{3} {\mathrm e}^{4 x -8}+8 x^{4} {\mathrm e}^{4 x -8}+\left (-96 x^{3}-20\right ) {\mathrm e}^{4}\right ) {\mathrm e}^{-4 x +8}}{2 x^{4}}-\frac {2 \left (2 x^{4} {\mathrm e}^{4 x -8}+\left (-24 x^{4}-20 x -15\right ) {\mathrm e}^{4}\right ) {\mathrm e}^{-4 x +8}}{x^{5}}-\frac {2 \left (2 x^{4} {\mathrm e}^{4 x -8}+\left (-24 x^{4}-20 x -15\right ) {\mathrm e}^{4}\right ) {\mathrm e}^{-4 x +8}}{x^{4}}\right )}{384 x^{5}+320 x^{2}+480 x +240}\) | \(201\) |
derivativedivides | \(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )\) | \(205\) |
default | \(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \operatorname {expIntegral}_{1}\left (4 x \right )}{3}\right )\) | \(205\) |
meijerg | \(-\frac {{\mathrm e}^{-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{4 x \left (-{\mathrm e}^{8}+1\right )}\right )}{4 \left (-{\mathrm e}^{8}+1\right )}-3 \,{\mathrm e}^{4-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{-4 x \,{\mathrm e}^{8}}\right )-160 \,{\mathrm e}^{28-4 x +4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-16}}{32 x^{2}}+\frac {{\mathrm e}^{-8}}{4 x}+\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\ln \left (2\right )+\frac {{\mathrm e}^{-16} \left (144 x^{2} {\mathrm e}^{16}-48 x \,{\mathrm e}^{8}+6\right )}{192 x^{2}}-\frac {{\mathrm e}^{-16-4 x \,{\mathrm e}^{8}} \left (-12 x \,{\mathrm e}^{8}+3\right )}{96 x^{2}}-\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{2}-\frac {\operatorname {expIntegral}_{1}\left (4 x \,{\mathrm e}^{8}\right )}{2}\right )-480 \,{\mathrm e}^{36-4 x +4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-24}}{192 x^{3}}+\frac {{\mathrm e}^{-16}}{32 x^{2}}-\frac {{\mathrm e}^{-8}}{8 x}-\frac {37}{36}-\frac {\ln \left (x \right )}{6}-\frac {\ln \left (2\right )}{3}+\frac {{\mathrm e}^{-24} \left (-1408 x^{3} {\mathrm e}^{24}+576 x^{2} {\mathrm e}^{16}-144 x \,{\mathrm e}^{8}+24\right )}{4608 x^{3}}-\frac {{\mathrm e}^{-24-4 x \,{\mathrm e}^{8}} \left (64 x^{2} {\mathrm e}^{16}-16 x \,{\mathrm e}^{8}+8\right )}{1536 x^{3}}+\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{6}+\frac {\operatorname {expIntegral}_{1}\left (4 x \,{\mathrm e}^{8}\right )}{6}\right )\) | \(268\) |
Input:
int(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x,metho d=_RETURNVERBOSE)
Output:
x+1/2/x^3*(6*x^3+5)*exp(12-4*x)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} + {\left (6 \, x^{3} + 5\right )} e^{4}\right )} e^{\left (-4 \, x + 8\right )}}{2 \, x^{3}} \] Input:
integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x , algorithm="fricas")
Output:
1/2*(2*x^4*e^(4*x - 8) + (6*x^3 + 5)*e^4)*e^(-4*x + 8)/x^3
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=x + \frac {\left (6 x^{3} e^{4} + 5 e^{4}\right ) e^{8 - 4 x}}{2 x^{3}} \] Input:
integrate(1/2*(2*x**4*exp(4*x-8)+(-24*x**4-20*x-15)*exp(4))/x**4/exp(4*x-8 ),x)
Output:
x + (6*x**3*exp(4) + 5*exp(4))*exp(8 - 4*x)/(2*x**3)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=160 \, e^{12} \Gamma \left (-2, 4 \, x\right ) + 480 \, e^{12} \Gamma \left (-3, 4 \, x\right ) + x + 3 \, e^{\left (-4 \, x + 12\right )} \] Input:
integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x , algorithm="maxima")
Output:
160*e^12*gamma(-2, 4*x) + 480*e^12*gamma(-3, 4*x) + x + 3*e^(-4*x + 12)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (17) = 34\).
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.38 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {2 \, {\left (x - 2\right )}^{4} + 6 \, {\left (x - 2\right )}^{3} e^{\left (-4 \, x + 12\right )} + 12 \, {\left (x - 2\right )}^{3} + 36 \, {\left (x - 2\right )}^{2} e^{\left (-4 \, x + 12\right )} + 24 \, {\left (x - 2\right )}^{2} + 72 \, {\left (x - 2\right )} e^{\left (-4 \, x + 12\right )} + 16 \, x + 53 \, e^{\left (-4 \, x + 12\right )} - 32}{2 \, {\left ({\left (x - 2\right )}^{3} + 6 \, {\left (x - 2\right )}^{2} + 12 \, x - 16\right )}} \] Input:
integrate(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x , algorithm="giac")
Output:
1/2*(2*(x - 2)^4 + 6*(x - 2)^3*e^(-4*x + 12) + 12*(x - 2)^3 + 36*(x - 2)^2 *e^(-4*x + 12) + 24*(x - 2)^2 + 72*(x - 2)*e^(-4*x + 12) + 16*x + 53*e^(-4 *x + 12) - 32)/((x - 2)^3 + 6*(x - 2)^2 + 12*x - 16)
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=x+3\,{\mathrm {e}}^{12-4\,x}+\frac {5\,{\mathrm {e}}^{12-4\,x}}{2\,x^3} \] Input:
int(-(exp(8 - 4*x)*((exp(4)*(20*x + 24*x^4 + 15))/2 - x^4*exp(4*x - 8)))/x ^4,x)
Output:
x + 3*exp(12 - 4*x) + (5*exp(12 - 4*x))/(2*x^3)
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {2 e^{4 x} x^{4}+6 e^{12} x^{3}+5 e^{12}}{2 e^{4 x} x^{3}} \] Input:
int(1/2*(2*x^4*exp(4*x-8)+(-24*x^4-20*x-15)*exp(4))/x^4/exp(4*x-8),x)
Output:
(2*e**(4*x)*x**4 + 6*e**12*x**3 + 5*e**12)/(2*e**(4*x)*x**3)