\(\int \frac {e^{-x} (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2))}{7 x^3} \, dx\) [7868]

Optimal result

Integrand size = 46, antiderivative size = 20 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16 e^{8-x} (3+\log (2))^2}{7 x^2} \]

[Out]

16/7*exp(4)^2/exp(x)*(3+ln(2))^2/x^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2228} \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16 e^{8-x} (3+\log (2))^2}{7 x^2} \]

[In]

Int[(E^8*(-288 - 144*x) + E^8*(-192 - 96*x)*Log[2] + E^8*(-32 - 16*x)*Log[2]^2)/(7*E^x*x^3),x]

[Out]

(16*E^(8 - x)*(3 + Log[2])^2)/(7*x^2)

Rule 12

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

Rule 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{x^3} \, dx \\ & = \frac {16 e^{8-x} (3+\log (2))^2}{7 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16 e^{8-x} (3+\log (2))^2}{7 x^2} \]

[In]

Integrate[(E^8*(-288 - 144*x) + E^8*(-192 - 96*x)*Log[2] + E^8*(-32 - 16*x)*Log[2]^2)/(7*E^x*x^3),x]

[Out]

(16*E^(8 - x)*(3 + Log[2])^2)/(7*x^2)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {16 \left (\ln \left (2\right )^{2}+6 \ln \left (2\right )+9\right ) {\mathrm e}^{8-x}}{7 x^{2}}\)	\(22\)
gosper	\(\frac {16 \,{\mathrm e}^{8} \left (\ln \left (2\right )^{2}+6 \ln \left (2\right )+9\right ) {\mathrm e}^{-x}}{7 x^{2}}\)	\(24\)
norman	\(\frac {\left (\frac {16 \,{\mathrm e}^{8} \ln \left (2\right )^{2}}{7}+\frac {96 \,{\mathrm e}^{8} \ln \left (2\right )}{7}+\frac {144 \,{\mathrm e}^{8}}{7}\right ) {\mathrm e}^{-x}}{x^{2}}\)	\(34\)
parallelrisch	\(\frac {\left (16 \,{\mathrm e}^{8} \ln \left (2\right )^{2}+96 \,{\mathrm e}^{8} \ln \left (2\right )+144 \,{\mathrm e}^{8}\right ) {\mathrm e}^{-x}}{7 x^{2}}\)	\(35\)
default	\(-\frac {288 \,{\mathrm e}^{8} \left (-\frac {{\mathrm e}^{-x}}{2 x^{2}}+\frac {{\mathrm e}^{-x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}-\frac {144 \,{\mathrm e}^{8} \left (-\frac {{\mathrm e}^{-x}}{x}+\operatorname {Ei}_{1}\left (x \right )\right )}{7}-\frac {192 \,{\mathrm e}^{8} \ln \left (2\right ) \left (-\frac {{\mathrm e}^{-x}}{2 x^{2}}+\frac {{\mathrm e}^{-x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}-\frac {32 \,{\mathrm e}^{8} \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{-x}}{2 x^{2}}+\frac {{\mathrm e}^{-x}}{2 x}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}-\frac {96 \,{\mathrm e}^{8} \ln \left (2\right ) \left (-\frac {{\mathrm e}^{-x}}{x}+\operatorname {Ei}_{1}\left (x \right )\right )}{7}-\frac {16 \,{\mathrm e}^{8} \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{-x}}{x}+\operatorname {Ei}_{1}\left (x \right )\right )}{7}\)	\(161\)
meijerg	\(\left (-\frac {16 \,{\mathrm e}^{8} \ln \left (2\right )^{2}}{7}-\frac {96 \,{\mathrm e}^{8} \ln \left (2\right )}{7}-\frac {144 \,{\mathrm e}^{8}}{7}\right ) \left (-\frac {1}{x}+1+\frac {2-2 x}{2 x}-\frac {{\mathrm e}^{-x}}{x}+\operatorname {Ei}_{1}\left (x \right )\right )-\frac {32 \,{\mathrm e}^{8} \ln \left (2\right )^{2} \left (-\frac {1}{2 x^{2}}+\frac {1}{x}-\frac {3}{4}+\frac {9 x^{2}-12 x +6}{12 x^{2}}-\frac {\left (-3 x +3\right ) {\mathrm e}^{-x}}{6 x^{2}}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}-\frac {192 \,{\mathrm e}^{8} \ln \left (2\right ) \left (-\frac {1}{2 x^{2}}+\frac {1}{x}-\frac {3}{4}+\frac {9 x^{2}-12 x +6}{12 x^{2}}-\frac {\left (-3 x +3\right ) {\mathrm e}^{-x}}{6 x^{2}}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}-\frac {288 \,{\mathrm e}^{8} \left (-\frac {1}{2 x^{2}}+\frac {1}{x}-\frac {3}{4}+\frac {9 x^{2}-12 x +6}{12 x^{2}}-\frac {\left (-3 x +3\right ) {\mathrm e}^{-x}}{6 x^{2}}-\frac {\operatorname {Ei}_{1}\left (x \right )}{2}\right )}{7}\)	\(201\)

[In]

int(1/7*((-16*x-32)*exp(4)^2*ln(2)^2+(-96*x-192)*exp(4)^2*ln(2)+(-144*x-288)*exp(4)^2)/exp(x)/x^3,x,method=_RE
TURNVERBOSE)

[Out]

16/7/x^2*(ln(2)^2+6*ln(2)+9)*exp(8-x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16 \, {\left (e^{8} \log \left (2\right )^{2} + 6 \, e^{8} \log \left (2\right ) + 9 \, e^{8}\right )} e^{\left (-x\right )}}{7 \, x^{2}} \]

[In]

integrate(1/7*((-16*x-32)*exp(4)^2*log(2)^2+(-96*x-192)*exp(4)^2*log(2)+(-144*x-288)*exp(4)^2)/exp(x)/x^3,x, a
lgorithm="fricas")

[Out]

16/7*(e^8*log(2)^2 + 6*e^8*log(2) + 9*e^8)*e^(-x)/x^2

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {\left (16 e^{8} \log {\left (2 \right )}^{2} + 96 e^{8} \log {\left (2 \right )} + 144 e^{8}\right ) e^{- x}}{7 x^{2}} \]

[In]

integrate(1/7*((-16*x-32)*exp(4)**2*ln(2)**2+(-96*x-192)*exp(4)**2*ln(2)+(-144*x-288)*exp(4)**2)/exp(x)/x**3,x
)

[Out]

(16*exp(8)*log(2)**2 + 96*exp(8)*log(2) + 144*exp(8))*exp(-x)/(7*x**2)

Maxima [C] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16}{7} \, e^{8} \Gamma \left (-1, x\right ) \log \left (2\right )^{2} + \frac {32}{7} \, e^{8} \Gamma \left (-2, x\right ) \log \left (2\right )^{2} + \frac {96}{7} \, e^{8} \Gamma \left (-1, x\right ) \log \left (2\right ) + \frac {192}{7} \, e^{8} \Gamma \left (-2, x\right ) \log \left (2\right ) + \frac {144}{7} \, e^{8} \Gamma \left (-1, x\right ) + \frac {288}{7} \, e^{8} \Gamma \left (-2, x\right ) \]

[In]

integrate(1/7*((-16*x-32)*exp(4)^2*log(2)^2+(-96*x-192)*exp(4)^2*log(2)+(-144*x-288)*exp(4)^2)/exp(x)/x^3,x, a
lgorithm="maxima")

[Out]

16/7*e^8*gamma(-1, x)*log(2)^2 + 32/7*e^8*gamma(-2, x)*log(2)^2 + 96/7*e^8*gamma(-1, x)*log(2) + 192/7*e^8*gam
ma(-2, x)*log(2) + 144/7*e^8*gamma(-1, x) + 288/7*e^8*gamma(-2, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16 \, {\left (e^{\left (-x + 8\right )} \log \left (2\right )^{2} + 6 \, e^{\left (-x + 8\right )} \log \left (2\right ) + 9 \, e^{\left (-x + 8\right )}\right )}}{7 \, x^{2}} \]

[In]

integrate(1/7*((-16*x-32)*exp(4)^2*log(2)^2+(-96*x-192)*exp(4)^2*log(2)+(-144*x-288)*exp(4)^2)/exp(x)/x^3,x, a
lgorithm="giac")

[Out]

16/7*(e^(-x + 8)*log(2)^2 + 6*e^(-x + 8)*log(2) + 9*e^(-x + 8))/x^2

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-x} \left (e^8 (-288-144 x)+e^8 (-192-96 x) \log (2)+e^8 (-32-16 x) \log ^2(2)\right )}{7 x^3} \, dx=\frac {16\,{\mathrm {e}}^{8-x}\,{\left (\ln \left (2\right )+3\right )}^2}{7\,x^2} \]

[In]

int(-(exp(-x)*((exp(8)*(144*x + 288))/7 + (exp(8)*log(2)*(96*x + 192))/7 + (exp(8)*log(2)^2*(16*x + 32))/7))/x
^3,x)

[Out]

(16*exp(8 - x)*(log(2) + 3)^2)/(7*x^2)