\(\int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx\) [7886]

Optimal result

Integrand size = 106, antiderivative size = 22 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{\left (-3+2 e^{2 x+e \left (e^2+\log (5)\right )}\right )^4} \]

[Out]

16/(2*exp((ln(5)+exp(2))*exp(1)+2*x)-3)^4

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2306, 2320, 32} \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{\left (3-2\ 5^e e^{2 x+e^3}\right )^4} \]

[In]

Int[(-256*E^(E^3 + 2*x + E*Log[5]))/(-243 + 810*E^(E^3 + 2*x + E*Log[5]) - 1080*E^(2*E^3 + 4*x + 2*E*Log[5]) +
720*E^(3*E^3 + 6*x + 3*E*Log[5]) - 240*E^(4*E^3 + 8*x + 4*E*Log[5]) + 32*E^(5*E^3 + 10*x + 5*E*Log[5])),x]

[Out]

16/(3 - 2*5^E*E^(E^3 + 2*x))^4

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2306

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps \begin{align*} \text {integral}& = -\left (256 \int \frac {e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx\right ) \\ & = -\left (256 \int \frac {5^e e^{e^3+2 x}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx\right ) \\ & = -\left (\left (256\ 5^e\right ) \int \frac {e^{e^3+2 x}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx\right ) \\ & = -\left (\left (128\ 5^e\right ) \text {Subst}\left (\int \frac {e^{e^3}}{\left (-3+2\ 5^e e^{e^3} x\right )^5} \, dx,x,e^{2 x}\right )\right ) \\ & = -\left (\left (128\ 5^e e^{e^3}\right ) \text {Subst}\left (\int \frac {1}{\left (-3+2\ 5^e e^{e^3} x\right )^5} \, dx,x,e^{2 x}\right )\right ) \\ & = \frac {16}{\left (3-2\ 5^e e^{e^3+2 x}\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{\left (-3+2\ 5^e e^{e^3+2 x}\right )^4} \]

[In]

Integrate[(-256*E^(E^3 + 2*x + E*Log[5]))/(-243 + 810*E^(E^3 + 2*x + E*Log[5]) - 1080*E^(2*E^3 + 4*x + 2*E*Log
[5]) + 720*E^(3*E^3 + 6*x + 3*E*Log[5]) - 240*E^(4*E^3 + 8*x + 4*E*Log[5]) + 32*E^(5*E^3 + 10*x + 5*E*Log[5]))
,x]

[Out]

16/(-3 + 2*5^E*E^(E^3 + 2*x))^4

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {16}{\left (2 \,5^{{\mathrm e}} {\mathrm e}^{2 x +{\mathrm e}^{3}}-3\right )^{4}}\)	\(20\)
derivativedivides	\(\frac {16}{\left (2 \,{\mathrm e}^{{\mathrm e} \ln \left (5\right )+{\mathrm e} \,{\mathrm e}^{2}+2 x}-3\right )^{4}}\)	\(24\)
default	\(\frac {16}{\left (2 \,{\mathrm e}^{{\mathrm e} \ln \left (5\right )+{\mathrm e} \,{\mathrm e}^{2}+2 x}-3\right )^{4}}\)	\(24\)
norman	\(\frac {16}{\left (2 \,{\mathrm e}^{{\mathrm e} \ln \left (5\right )+{\mathrm e} \,{\mathrm e}^{2}+2 x}-3\right )^{4}}\)	\(24\)
parallelrisch	\(\frac {16}{16 \,{\mathrm e}^{4 \,{\mathrm e} \ln \left (5\right )+4 \,{\mathrm e}^{3}+8 x}-96 \,{\mathrm e}^{3 \,{\mathrm e} \ln \left (5\right )+3 \,{\mathrm e}^{3}+6 x}+216 \,{\mathrm e}^{2 \,{\mathrm e} \ln \left (5\right )+2 \,{\mathrm e}^{3}+4 x}-216 \,{\mathrm e}^{{\mathrm e} \ln \left (5\right )+{\mathrm e} \,{\mathrm e}^{2}+2 x}+81}\)	\(81\)

[In]

int(-256*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)/(32*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)^5-240*exp(exp(1)*ln(5)+ex
p(1)*exp(2)+2*x)^4+720*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)^3-1080*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)^2+810*ex
p(exp(1)*ln(5)+exp(1)*exp(2)+2*x)-243),x,method=_RETURNVERBOSE)

[Out]

16/(2*5^exp(1)*exp(2*x+exp(3))-3)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (21) = 42\).

Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{16 \, e^{\left (4 \, e \log \left (5\right ) + 8 \, x + 4 \, e^{3}\right )} - 96 \, e^{\left (3 \, e \log \left (5\right ) + 6 \, x + 3 \, e^{3}\right )} + 216 \, e^{\left (2 \, e \log \left (5\right ) + 4 \, x + 2 \, e^{3}\right )} - 216 \, e^{\left (e \log \left (5\right ) + 2 \, x + e^{3}\right )} + 81} \]

[In]

integrate(-256*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)/(32*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^5-240*exp(exp(1)*
log(5)+exp(1)*exp(2)+2*x)^4+720*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^3-1080*exp(exp(1)*log(5)+exp(1)*exp(2)+2*
x)^2+810*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)-243),x, algorithm="fricas")

[Out]

16/(16*e^(4*e*log(5) + 8*x + 4*e^3) - 96*e^(3*e*log(5) + 6*x + 3*e^3) + 216*e^(2*e*log(5) + 4*x + 2*e^3) - 216
*e^(e*log(5) + 2*x + e^3) + 81)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{- 216 \cdot 5^{e} e^{2 x + e^{3}} + 216 \cdot 5^{2 e} e^{4 x + 2 e^{3}} - 96 \cdot 5^{3 e} e^{6 x + 3 e^{3}} + 16 \cdot 5^{4 e} e^{8 x + 4 e^{3}} + 81} \]

[In]

integrate(-256*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)/(32*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)**5-240*exp(exp(1)*l
n(5)+exp(1)*exp(2)+2*x)**4+720*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)**3-1080*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)
**2+810*exp(exp(1)*ln(5)+exp(1)*exp(2)+2*x)-243),x)

[Out]

16/(-216*5**E*exp(2*x + exp(3)) + 216*5**(2*E)*exp(4*x + 2*exp(3)) - 96*5**(3*E)*exp(6*x + 3*exp(3)) + 16*5**(
4*E)*exp(8*x + 4*exp(3)) + 81)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (21) = 42\).

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{16 \, e^{\left (4 \, e \log \left (5\right ) + 8 \, x + 4 \, e^{3}\right )} - 96 \, e^{\left (3 \, e \log \left (5\right ) + 6 \, x + 3 \, e^{3}\right )} + 216 \, e^{\left (2 \, e \log \left (5\right ) + 4 \, x + 2 \, e^{3}\right )} - 216 \, e^{\left (e \log \left (5\right ) + 2 \, x + e^{3}\right )} + 81} \]

[In]

integrate(-256*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)/(32*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^5-240*exp(exp(1)*
log(5)+exp(1)*exp(2)+2*x)^4+720*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^3-1080*exp(exp(1)*log(5)+exp(1)*exp(2)+2*
x)^2+810*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)-243),x, algorithm="maxima")

[Out]

16/(16*e^(4*e*log(5) + 8*x + 4*e^3) - 96*e^(3*e*log(5) + 6*x + 3*e^3) + 216*e^(2*e*log(5) + 4*x + 2*e^3) - 216
*e^(e*log(5) + 2*x + e^3) + 81)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=\frac {16}{{\left (2 \, e^{\left (e \log \left (5\right ) + 2 \, x + e^{3}\right )} - 3\right )}^{4}} \]

[In]

integrate(-256*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)/(32*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^5-240*exp(exp(1)*
log(5)+exp(1)*exp(2)+2*x)^4+720*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)^3-1080*exp(exp(1)*log(5)+exp(1)*exp(2)+2*
x)^2+810*exp(exp(1)*log(5)+exp(1)*exp(2)+2*x)-243),x, algorithm="giac")

[Out]

16/(2*e^(e*log(5) + 2*x + e^3) - 3)^4

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.91 \[ \int -\frac {256 e^{e^3+2 x+e \log (5)}}{-243+810 e^{e^3+2 x+e \log (5)}-1080 e^{2 e^3+4 x+2 e \log (5)}+720 e^{3 e^3+6 x+3 e \log (5)}-240 e^{4 e^3+8 x+4 e \log (5)}+32 e^{5 e^3+10 x+5 e \log (5)}} \, dx=-\frac {\frac {128\,5^{2\,\mathrm {e}}\,{\mathrm {e}}^{4\,x+2\,{\mathrm {e}}^3}}{3}-\frac {128\,5^{\mathrm {e}}\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^3}\,\left (12\,5^{2\,\mathrm {e}}\,{\mathrm {e}}^{4\,x+2\,{\mathrm {e}}^3}-2\,5^{3\,\mathrm {e}}\,{\mathrm {e}}^{6\,x+3\,{\mathrm {e}}^3}+27\right )}{81}}{{\left (2\,5^{\mathrm {e}}\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^3}-3\right )}^4} \]

[In]

int(-(256*exp(2*x + exp(3) + exp(1)*log(5)))/(810*exp(2*x + exp(3) + exp(1)*log(5)) - 1080*exp(4*x + 2*exp(3)
+ 2*exp(1)*log(5)) + 720*exp(6*x + 3*exp(3) + 3*exp(1)*log(5)) - 240*exp(8*x + 4*exp(3) + 4*exp(1)*log(5)) + 3
2*exp(10*x + 5*exp(3) + 5*exp(1)*log(5)) - 243),x)

[Out]

-((128*5^(2*exp(1))*exp(4*x + 2*exp(3)))/3 - (128*5^exp(1)*exp(2*x + exp(3))*(12*5^(2*exp(1))*exp(4*x + 2*exp(
3)) - 2*5^(3*exp(1))*exp(6*x + 3*exp(3)) + 27))/81)/(2*5^exp(1)*exp(2*x + exp(3)) - 3)^4