\(\int \frac {e^{8-4 x} (2 e^{-8+4 x} x^4+e^4 (-15-20 x-24 x^4))}{2 x^4} \, dx\) [7277]

Optimal result

Integrand size = 41, antiderivative size = 21 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=e^{4-4 (-2+x)} \left (3+\frac {5}{2 x^3}\right )+x \]

[Out]

x+(5/2/x^3+3)*exp(4)/exp(4*x-8)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {12, 6820, 2230, 2225, 2208, 2209} \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {5 e^{12-4 x}}{2 x^3}+x+3 e^{12-4 x} \]

[In]

Int[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/(2*x^4),x]

[Out]

3*E^(12 - 4*x) + (5*E^(12 - 4*x))/(2*x^3) + x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \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 )}{x^4} \, dx \\ & = \frac {1}{2} \int \left (2-\frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4}\right ) \, dx \\ & = x-\frac {1}{2} \int \frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4} \, dx \\ & = x-\frac {1}{2} \int \left (24 e^{12-4 x}+\frac {15 e^{12-4 x}}{x^4}+\frac {20 e^{12-4 x}}{x^3}\right ) \, dx \\ & = x-\frac {15}{2} \int \frac {e^{12-4 x}}{x^4} \, dx-10 \int \frac {e^{12-4 x}}{x^3} \, dx-12 \int e^{12-4 x} \, dx \\ & = 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+\frac {5 e^{12-4 x}}{x^2}+x+10 \int \frac {e^{12-4 x}}{x^3} \, dx+20 \int \frac {e^{12-4 x}}{x^2} \, dx \\ & = 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}-\frac {20 e^{12-4 x}}{x}+x-20 \int \frac {e^{12-4 x}}{x^2} \, dx-80 \int \frac {e^{12-4 x}}{x} \, dx \\ & = 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x-80 e^{12} \operatorname {ExpIntegralEi}(-4 x)+80 \int \frac {e^{12-4 x}}{x} \, dx \\ & = 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x \]

[In]

Integrate[(E^(8 - 4*x)*(2*E^(-8 + 4*x)*x^4 + E^4*(-15 - 20*x - 24*x^4)))/(2*x^4),x]

[Out]

3*E^(12 - 4*x) + (5*E^(12 - 4*x))/(2*x^3) + x

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
risch	\(x +\frac {\left (6 x^{3}+5\right ) {\mathrm e}^{-4 x +12}}{2 x^{3}}\)	\(21\)
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	\(x -\frac {{\mathrm e}^{4} \left (-\frac {439 \,{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{6 x^{3}}+\frac {197 \,{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{6 x^{3}}-\frac {12 \,{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{x^{3}}+\frac {8 \,{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{x^{3}}-6 \,{\mathrm e}^{-4 x +8}-\frac {4 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{x^{3}}\right )}{2}\)	\(139\)
derivativedivides	\(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )\)	\(205\)
default	\(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \operatorname {Ei}_{1}\left (4 x \right )}{3}\right )\)	\(205\)
meijerg	\(-\frac {{\mathrm e}^{-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{4 x \left (1-{\mathrm e}^{8}\right )}\right )}{4 \left (1-{\mathrm e}^{8}\right )}-3 \,{\mathrm e}^{4-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{-4 x \,{\mathrm e}^{8}}\right )-160 \,{\mathrm e}^{28-4 x +4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-16}}{32 x^{2}}+\frac {{\mathrm e}^{-8}}{4 x}+\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\ln \left (2\right )+\frac {{\mathrm e}^{-16} \left (144 x^{2} {\mathrm e}^{16}-48 x \,{\mathrm e}^{8}+6\right )}{192 x^{2}}-\frac {{\mathrm e}^{-16-4 x \,{\mathrm e}^{8}} \left (-12 x \,{\mathrm e}^{8}+3\right )}{96 x^{2}}-\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{2}-\frac {\operatorname {Ei}_{1}\left (4 x \,{\mathrm e}^{8}\right )}{2}\right )-480 \,{\mathrm e}^{-4 x +36+4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-24}}{192 x^{3}}+\frac {{\mathrm e}^{-16}}{32 x^{2}}-\frac {{\mathrm e}^{-8}}{8 x}-\frac {37}{36}-\frac {\ln \left (x \right )}{6}-\frac {\ln \left (2\right )}{3}+\frac {{\mathrm e}^{-24} \left (-1408 x^{3} {\mathrm e}^{24}+576 x^{2} {\mathrm e}^{16}-144 x \,{\mathrm e}^{8}+24\right )}{4608 x^{3}}-\frac {{\mathrm e}^{-24-4 x \,{\mathrm e}^{8}} \left (64 x^{2} {\mathrm e}^{16}-16 x \,{\mathrm e}^{8}+8\right )}{1536 x^{3}}+\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{6}+\frac {\operatorname {Ei}_{1}\left (4 x \,{\mathrm e}^{8}\right )}{6}\right )\)	\(268\)

[In]

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,method=_RETURNVERBOSE)

[Out]

x+1/2/x^3*(6*x^3+5)*exp(-4*x+12)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} + {\left (6 \, x^{3} + 5\right )} e^{4}\right )} e^{\left (-4 \, x + 8\right )}}{2 \, x^{3}} \]

[In]

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")

[Out]

1/2*(2*x^4*e^(4*x - 8) + (6*x^3 + 5)*e^4)*e^(-4*x + 8)/x^3

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=x + \frac {\left (6 x^{3} e^{4} + 5 e^{4}\right ) e^{8 - 4 x}}{2 x^{3}} \]

[In]

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)

[Out]

x + (6*x**3*exp(4) + 5*exp(4))*exp(8 - 4*x)/(2*x**3)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=160 \, e^{12} \Gamma \left (-2, 4 \, x\right ) + 480 \, e^{12} \Gamma \left (-3, 4 \, x\right ) + x + 3 \, e^{\left (-4 \, x + 12\right )} \]

[In]

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")

[Out]

160*e^12*gamma(-2, 4*x) + 480*e^12*gamma(-3, 4*x) + x + 3*e^(-4*x + 12)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.38 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=\frac {2 \, {\left (x - 2\right )}^{4} + 6 \, {\left (x - 2\right )}^{3} e^{\left (-4 \, x + 12\right )} + 12 \, {\left (x - 2\right )}^{3} + 36 \, {\left (x - 2\right )}^{2} e^{\left (-4 \, x + 12\right )} + 24 \, {\left (x - 2\right )}^{2} + 72 \, {\left (x - 2\right )} e^{\left (-4 \, x + 12\right )} + 16 \, x + 53 \, e^{\left (-4 \, x + 12\right )} - 32}{2 \, {\left ({\left (x - 2\right )}^{3} + 6 \, {\left (x - 2\right )}^{2} + 12 \, x - 16\right )}} \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{2 x^4} \, dx=x+3\,{\mathrm {e}}^{12-4\,x}+\frac {5\,{\mathrm {e}}^{12-4\,x}}{2\,x^3} \]

[In]

int(-(exp(8 - 4*x)*((exp(4)*(20*x + 24*x^4 + 15))/2 - x^4*exp(4*x - 8)))/x^4,x)

[Out]

x + 3*exp(12 - 4*x) + (5*exp(12 - 4*x))/(2*x^3)