3.27.49 \(\int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+(64 x^3-160 x^4+144 x^5-56 x^6+8 x^7) \log (5 e^{-3+x})+e^{e^x+x} (-25-25 e^x) \log ^3(5 e^{-3+x})}{25 \log ^3(5 e^{-3+x})} \, dx\)

Optimal. Leaf size=32 \[ -e^{e^x+x}+\frac {(-2+x)^4 x^4}{25 \log ^2\left (5 e^{-3+x}\right )} \]

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Rubi [B]  time = 1.92, antiderivative size = 103, normalized size of antiderivative = 3.22, number of steps used = 85, number of rules used = 11, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.110, Rules used = {12, 6688, 2282, 2176, 2194, 6742, 2168, 2159, 2158, 2157, 29} \begin {gather*} \frac {x^8}{25 \log ^2\left (5 e^{x-3}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{x-3}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{x-3}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{x-3}\right )}+\frac {16 x^4}{25 \log ^2\left (5 e^{x-3}\right )}+e^{e^x}-e^{e^x} \left (e^x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-32*x^4 + 64*x^5 - 48*x^6 + 16*x^7 - 2*x^8 + (64*x^3 - 160*x^4 + 144*x^5 - 56*x^6 + 8*x^7)*Log[5*E^(-3 +
x)] + E^(E^x + x)*(-25 - 25*E^x)*Log[5*E^(-3 + x)]^3)/(25*Log[5*E^(-3 + x)]^3),x]

[Out]

E^E^x - E^E^x*(1 + E^x) + (16*x^4)/(25*Log[5*E^(-3 + x)]^2) - (32*x^5)/(25*Log[5*E^(-3 + x)]^2) + (24*x^6)/(25
*Log[5*E^(-3 + x)]^2) - (8*x^7)/(25*Log[5*E^(-3 + x)]^2) + x^8/(25*Log[5*E^(-3 + 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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2159

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dis
t[(b*u - a*v)/a, Int[v^(n - 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && Ne
Q[n, 1]

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

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 2282

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]]]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{\log ^3\left (5 e^{-3+x}\right )} \, dx\\ &=\frac {1}{25} \int \left (-25 e^{e^x+x} \left (1+e^x\right )-\frac {2 (-2+x)^4 x^4}{\log ^3\left (5 e^{-3+x}\right )}+\frac {8 (-2+x)^3 (-1+x) x^3}{\log ^2\left (5 e^{-3+x}\right )}\right ) \, dx\\ &=-\left (\frac {2}{25} \int \frac {(-2+x)^4 x^4}{\log ^3\left (5 e^{-3+x}\right )} \, dx\right )+\frac {8}{25} \int \frac {(-2+x)^3 (-1+x) x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\int e^{e^x+x} \left (1+e^x\right ) \, dx\\ &=-\left (\frac {2}{25} \int \left (\frac {16 x^4}{\log ^3\left (5 e^{-3+x}\right )}-\frac {32 x^5}{\log ^3\left (5 e^{-3+x}\right )}+\frac {24 x^6}{\log ^3\left (5 e^{-3+x}\right )}-\frac {8 x^7}{\log ^3\left (5 e^{-3+x}\right )}+\frac {x^8}{\log ^3\left (5 e^{-3+x}\right )}\right ) \, dx\right )+\frac {8}{25} \int \left (\frac {8 x^3}{\log ^2\left (5 e^{-3+x}\right )}-\frac {20 x^4}{\log ^2\left (5 e^{-3+x}\right )}+\frac {18 x^5}{\log ^2\left (5 e^{-3+x}\right )}-\frac {7 x^6}{\log ^2\left (5 e^{-3+x}\right )}+\frac {x^7}{\log ^2\left (5 e^{-3+x}\right )}\right ) \, dx-\operatorname {Subst}\left (\int e^x (1+x) \, dx,x,e^x\right )\\ &=-e^{e^x} \left (1+e^x\right )-\frac {2}{25} \int \frac {x^8}{\log ^3\left (5 e^{-3+x}\right )} \, dx+\frac {8}{25} \int \frac {x^7}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {16}{25} \int \frac {x^7}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {32}{25} \int \frac {x^4}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {48}{25} \int \frac {x^6}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {56}{25} \int \frac {x^6}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {64}{25} \int \frac {x^5}{\log ^3\left (5 e^{-3+x}\right )} \, dx+\frac {64}{25} \int \frac {x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {144}{25} \int \frac {x^5}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\frac {32}{5} \int \frac {x^4}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {64 x^3}{25 \log \left (5 e^{-3+x}\right )}+\frac {32 x^4}{5 \log \left (5 e^{-3+x}\right )}-\frac {144 x^5}{25 \log \left (5 e^{-3+x}\right )}+\frac {56 x^6}{25 \log \left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log \left (5 e^{-3+x}\right )}-\frac {8}{25} \int \frac {x^7}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {56}{25} \int \frac {x^6}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {56}{25} \int \frac {x^6}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {64}{25} \int \frac {x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\frac {144}{25} \int \frac {x^5}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {32}{5} \int \frac {x^4}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {192}{25} \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {336}{25} \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {128}{5} \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {144}{5} \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {96 x^2}{25}-\frac {128 x^3}{15}+\frac {36 x^4}{5}-\frac {336 x^5}{125}+\frac {28 x^6}{75}+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {56}{25} \int \frac {x^6}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {192}{25} \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {336}{25} \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {128}{5} \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {144}{5} \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {192}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )-\frac {64}{5} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {48}{5} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )-\frac {84}{25} x^4 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {56}{125} x^5 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )-\frac {128}{5} x \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {72}{5} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^2-\frac {112}{25} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {14}{25} x^4 \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {144}{5} x \left (x-\log \left (5 e^{-3+x}\right )\right )^3-\frac {168}{25} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^3+\frac {56}{75} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )^3+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {192}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^2 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )-\frac {336}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )^4+\frac {28}{25} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^4+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {128}{5} \left (x-\log \left (5 e^{-3+x}\right )\right )^3 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {56}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )^5+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {144}{5} \left (x-\log \left (5 e^{-3+x}\right )\right )^4 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {336}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^5 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {56}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^6 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )\\ &=e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.37, size = 198, normalized size = 6.19 \begin {gather*} \frac {(-2+x)^4 x^4-\left (25 e^{e^x+x}+(-2+x)^3 x^2 (-6+7 x)\right ) \log ^2\left (5 e^{-3+x}\right )+6 (-2+x)^2 x \left (4-12 x+7 x^2\right ) \log ^3\left (5 e^{-3+x}\right )-3 \left (16-128 x+240 x^2-160 x^3+35 x^4\right ) \log ^4\left (5 e^{-3+x}\right )+4 \left (-32+120 x-120 x^2+35 x^3\right ) \log ^5\left (5 e^{-3+x}\right )-15 \left (8-16 x+7 x^2\right ) \log ^6\left (5 e^{-3+x}\right )+6 (-8+7 x) \log ^7\left (5 e^{-3+x}\right )-7 \log ^8\left (5 e^{-3+x}\right )}{25 \log ^2\left (5 e^{-3+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32*x^4 + 64*x^5 - 48*x^6 + 16*x^7 - 2*x^8 + (64*x^3 - 160*x^4 + 144*x^5 - 56*x^6 + 8*x^7)*Log[5*E^
(-3 + x)] + E^(E^x + x)*(-25 - 25*E^x)*Log[5*E^(-3 + x)]^3)/(25*Log[5*E^(-3 + x)]^3),x]

[Out]

((-2 + x)^4*x^4 - (25*E^(E^x + x) + (-2 + x)^3*x^2*(-6 + 7*x))*Log[5*E^(-3 + x)]^2 + 6*(-2 + x)^2*x*(4 - 12*x
+ 7*x^2)*Log[5*E^(-3 + x)]^3 - 3*(16 - 128*x + 240*x^2 - 160*x^3 + 35*x^4)*Log[5*E^(-3 + x)]^4 + 4*(-32 + 120*
x - 120*x^2 + 35*x^3)*Log[5*E^(-3 + x)]^5 - 15*(8 - 16*x + 7*x^2)*Log[5*E^(-3 + x)]^6 + 6*(-8 + 7*x)*Log[5*E^(
-3 + x)]^7 - 7*Log[5*E^(-3 + x)]^8)/(25*Log[5*E^(-3 + x)]^2)

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fricas [B]  time = 0.72, size = 193, normalized size = 6.03 \begin {gather*} \frac {x^{8} - 2 \, {\left (7 \, x - 60\right )} \log \relax (5)^{7} - 7 \, \log \relax (5)^{8} - 8 \, x^{7} - {\left (7 \, x^{2} - 198 \, x + 876\right )} \log \relax (5)^{6} + 24 \, x^{6} + 2 \, {\left (39 \, x^{2} - 579 \, x + 1772\right )} \log \relax (5)^{5} - 32 \, x^{5} - {\left (345 \, x^{2} - 3614 \, x + 8658\right )} \log \relax (5)^{4} + 16 \, x^{4} + 2 \, {\left (386 \, x^{2} - 3237 \, x + 6516\right )} \log \relax (5)^{3} - 3 \, {\left (307 \, x^{2} - 2214 \, x + 3924\right )} \log \relax (5)^{2} - 135 \, x^{2} - 25 \, {\left (x^{2} + 2 \, {\left (x - 3\right )} \log \relax (5) + \log \relax (5)^{2} - 6 \, x + 9\right )} e^{\left (x + e^{x}\right )} + 18 \, {\left (31 \, x^{2} - 201 \, x + 324\right )} \log \relax (5) + 810 \, x - 1215}{25 \, {\left (x^{2} + 2 \, {\left (x - 3\right )} \log \relax (5) + \log \relax (5)^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*exp(x)-25)*exp(exp(x)+x)*log(5*exp(x-3))^3+(8*x^7-56*x^6+144*x^5-160*x^4+64*x^3)*log(5*ex
p(x-3))-2*x^8+16*x^7-48*x^6+64*x^5-32*x^4)/log(5*exp(x-3))^3,x, algorithm="fricas")

[Out]

1/25*(x^8 - 2*(7*x - 60)*log(5)^7 - 7*log(5)^8 - 8*x^7 - (7*x^2 - 198*x + 876)*log(5)^6 + 24*x^6 + 2*(39*x^2 -
579*x + 1772)*log(5)^5 - 32*x^5 - (345*x^2 - 3614*x + 8658)*log(5)^4 + 16*x^4 + 2*(386*x^2 - 3237*x + 6516)*l
og(5)^3 - 3*(307*x^2 - 2214*x + 3924)*log(5)^2 - 135*x^2 - 25*(x^2 + 2*(x - 3)*log(5) + log(5)^2 - 6*x + 9)*e^
(x + e^x) + 18*(31*x^2 - 201*x + 324)*log(5) + 810*x - 1215)/(x^2 + 2*(x - 3)*log(5) + log(5)^2 - 6*x + 9)

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giac [B]  time = 0.21, size = 415, normalized size = 12.97 \begin {gather*} \frac {x^{8} e^{\left (2 \, x\right )} - 7 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5)^{6} - 14 \, x e^{\left (2 \, x\right )} \log \relax (5)^{7} - 7 \, e^{\left (2 \, x\right )} \log \relax (5)^{8} - 8 \, x^{7} e^{\left (2 \, x\right )} + 78 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5)^{5} + 198 \, x e^{\left (2 \, x\right )} \log \relax (5)^{6} + 120 \, e^{\left (2 \, x\right )} \log \relax (5)^{7} + 24 \, x^{6} e^{\left (2 \, x\right )} - 345 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5)^{4} - 1158 \, x e^{\left (2 \, x\right )} \log \relax (5)^{5} - 876 \, e^{\left (2 \, x\right )} \log \relax (5)^{6} - 32 \, x^{5} e^{\left (2 \, x\right )} + 772 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5)^{3} + 3614 \, x e^{\left (2 \, x\right )} \log \relax (5)^{4} + 3544 \, e^{\left (2 \, x\right )} \log \relax (5)^{5} + 16 \, x^{4} e^{\left (2 \, x\right )} - 921 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5)^{2} - 6474 \, x e^{\left (2 \, x\right )} \log \relax (5)^{3} - 8658 \, e^{\left (2 \, x\right )} \log \relax (5)^{4} + 558 \, x^{2} e^{\left (2 \, x\right )} \log \relax (5) + 6642 \, x e^{\left (2 \, x\right )} \log \relax (5)^{2} + 13032 \, e^{\left (2 \, x\right )} \log \relax (5)^{3} - 135 \, x^{2} e^{\left (2 \, x\right )} - 25 \, x^{2} e^{\left (3 \, x + e^{x}\right )} - 3618 \, x e^{\left (2 \, x\right )} \log \relax (5) - 50 \, x e^{\left (3 \, x + e^{x}\right )} \log \relax (5) - 11772 \, e^{\left (2 \, x\right )} \log \relax (5)^{2} - 25 \, e^{\left (3 \, x + e^{x}\right )} \log \relax (5)^{2} + 810 \, x e^{\left (2 \, x\right )} + 150 \, x e^{\left (3 \, x + e^{x}\right )} + 5832 \, e^{\left (2 \, x\right )} \log \relax (5) + 150 \, e^{\left (3 \, x + e^{x}\right )} \log \relax (5) - 1215 \, e^{\left (2 \, x\right )} - 225 \, e^{\left (3 \, x + e^{x}\right )}}{25 \, {\left (x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} \log \relax (5) + e^{\left (2 \, x\right )} \log \relax (5)^{2} - 6 \, x e^{\left (2 \, x\right )} - 6 \, e^{\left (2 \, x\right )} \log \relax (5) + 9 \, e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*exp(x)-25)*exp(exp(x)+x)*log(5*exp(x-3))^3+(8*x^7-56*x^6+144*x^5-160*x^4+64*x^3)*log(5*ex
p(x-3))-2*x^8+16*x^7-48*x^6+64*x^5-32*x^4)/log(5*exp(x-3))^3,x, algorithm="giac")

[Out]

1/25*(x^8*e^(2*x) - 7*x^2*e^(2*x)*log(5)^6 - 14*x*e^(2*x)*log(5)^7 - 7*e^(2*x)*log(5)^8 - 8*x^7*e^(2*x) + 78*x
^2*e^(2*x)*log(5)^5 + 198*x*e^(2*x)*log(5)^6 + 120*e^(2*x)*log(5)^7 + 24*x^6*e^(2*x) - 345*x^2*e^(2*x)*log(5)^
4 - 1158*x*e^(2*x)*log(5)^5 - 876*e^(2*x)*log(5)^6 - 32*x^5*e^(2*x) + 772*x^2*e^(2*x)*log(5)^3 + 3614*x*e^(2*x
)*log(5)^4 + 3544*e^(2*x)*log(5)^5 + 16*x^4*e^(2*x) - 921*x^2*e^(2*x)*log(5)^2 - 6474*x*e^(2*x)*log(5)^3 - 865
8*e^(2*x)*log(5)^4 + 558*x^2*e^(2*x)*log(5) + 6642*x*e^(2*x)*log(5)^2 + 13032*e^(2*x)*log(5)^3 - 135*x^2*e^(2*
x) - 25*x^2*e^(3*x + e^x) - 3618*x*e^(2*x)*log(5) - 50*x*e^(3*x + e^x)*log(5) - 11772*e^(2*x)*log(5)^2 - 25*e^
(3*x + e^x)*log(5)^2 + 810*x*e^(2*x) + 150*x*e^(3*x + e^x) + 5832*e^(2*x)*log(5) + 150*e^(3*x + e^x)*log(5) -
1215*e^(2*x) - 225*e^(3*x + e^x))/(x^2*e^(2*x) + 2*x*e^(2*x)*log(5) + e^(2*x)*log(5)^2 - 6*x*e^(2*x) - 6*e^(2*
x)*log(5) + 9*e^(2*x))

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maple [C]  time = 0.11, size = 51, normalized size = 1.59

method	result	size
risch	\(-{\mathrm e}^{{\mathrm e}^{x}+x}-\frac {4 \left (x^{8}-8 x^{7}+24 x^{6}-32 x^{5}+16 x^{4}\right )}{25 \left (2 i \ln \relax (5)+2 i \ln \left ({\mathrm e}^{x}\right )-6 i\right )^{2}}\)	\(51\)
default	\(\frac {42 x}{25}+\frac {x^{6}}{25}-\frac {2 x^{5}}{25}+\frac {3 x^{4}}{25}+\frac {4 x^{3}}{25}+\frac {13 x^{2}}{25}-{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}+\frac {108 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{6}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}-\frac {16 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{5}}{\ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}+\frac {886 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{4}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}-\frac {48 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{3}}{\ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}+\frac {972 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}-\frac {432 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}+\frac {\left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{8}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}-\frac {16 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{7}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}-\frac {1944 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}-\frac {8 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{7}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}+\frac {112 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{6}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}-\frac {648 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{5}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}+\frac {80 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{4}}{\ln \left (5 \,{\mathrm e}^{x -3}\right )}-\frac {3544 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{3}}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}-\frac {2 x^{5} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25}+2 x \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{4}-\frac {156 x \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{3}}{25}+\frac {228 x \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{25}-\frac {158 x \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25}-\frac {4 x^{3} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{3}}{25}+\frac {x^{2} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{4}}{5}-\frac {28 x^{2} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{3}}{25}+\frac {54 x^{2} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{25}-\frac {44 x^{2} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25}-\frac {2 x^{4} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25}+\frac {3 x^{4} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{25}+\frac {144 \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{\ln \left (5 \,{\mathrm e}^{x -3}\right )}+\frac {12 x^{3} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{2}}{25}-\frac {12 x^{3} \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )}{25}-\frac {6 x \left (\ln \left (5 \,{\mathrm e}^{x -3}\right )-x +3\right )^{5}}{25}+\frac {432}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )}+\frac {81}{25 \ln \left (5 \,{\mathrm e}^{x -3}\right )^{2}}\)	\(695\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*((-25*exp(x)-25)*exp(exp(x)+x)*ln(5*exp(x-3))^3+(8*x^7-56*x^6+144*x^5-160*x^4+64*x^3)*ln(5*exp(x-3))-
2*x^8+16*x^7-48*x^6+64*x^5-32*x^4)/ln(5*exp(x-3))^3,x,method=_RETURNVERBOSE)

[Out]

-exp(exp(x)+x)-4/25*(x^8-8*x^7+24*x^6-32*x^5+16*x^4)/(2*I*ln(5)+2*I*ln(exp(x))-6*I)^2

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maxima [B]  time = 0.76, size = 1477, normalized size = 46.16 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*exp(x)-25)*exp(exp(x)+x)*log(5*exp(x-3))^3+(8*x^7-56*x^6+144*x^5-160*x^4+64*x^3)*log(5*ex
p(x-3))-2*x^8+16*x^7-48*x^6+64*x^5-32*x^4)/log(5*exp(x-3))^3,x, algorithm="maxima")

[Out]

-(e^x - 1)*e^(e^x) - 1/375*(5*x^8 - 8*x^7*(log(5) - 3) + 225*log(5)^8 + 14*(log(5)^2 - 6*log(5) + 9)*x^6 - 540
0*log(5)^7 - 28*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^5 + 56700*log(5)^6 + 70*(log(5)^4 - 12*log(5)^3 + 5
4*log(5)^2 - 108*log(5) + 81)*x^4 - 340200*log(5)^5 - 280*(log(5)^5 - 15*log(5)^4 + 90*log(5)^3 - 270*log(5)^2
+ 405*log(5) - 243)*x^3 + 1275750*log(5)^4 - 1035*(log(5)^6 - 18*log(5)^5 + 135*log(5)^4 - 540*log(5)^3 + 121
5*log(5)^2 - 1458*log(5) + 729)*x^2 - 3061800*log(5)^3 - 390*(log(5)^7 - 21*log(5)^6 + 189*log(5)^5 - 945*log(
5)^4 + 2835*log(5)^3 - 5103*log(5)^2 + 5103*log(5) - 2187)*x + 4592700*log(5)^2 - 3936600*log(5) + 1476225)/(x
^2 + 2*x*(log(5) - 3) + log(5)^2 - 6*log(5) + 9) + 4/125*(4*x^7 - 7*x^6*(log(5) - 3) - 130*log(5)^7 + 14*(log(
5)^2 - 6*log(5) + 9)*x^5 + 2730*log(5)^6 - 35*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^4 - 24570*log(5)^5 +
140*(log(5)^4 - 12*log(5)^3 + 54*log(5)^2 - 108*log(5) + 81)*x^3 + 122850*log(5)^4 + 500*(log(5)^5 - 15*log(5)
^4 + 90*log(5)^3 - 270*log(5)^2 + 405*log(5) - 243)*x^2 - 368550*log(5)^3 + 160*(log(5)^6 - 18*log(5)^5 + 135*
log(5)^4 - 540*log(5)^3 + 1215*log(5)^2 - 1458*log(5) + 729)*x + 663390*log(5)^2 - 663390*log(5) + 284310)/(x^
2 + 2*x*(log(5) - 3) + log(5)^2 - 6*log(5) + 9) - 12/25*(x^6 - 2*x^5*(log(5) - 3) + 22*log(5)^6 + 5*(log(5)^2
- 6*log(5) + 9)*x^4 - 396*log(5)^5 - 20*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^3 + 2970*log(5)^4 - 68*(log
(5)^4 - 12*log(5)^3 + 54*log(5)^2 - 108*log(5) + 81)*x^2 - 11880*log(5)^3 - 16*(log(5)^5 - 15*log(5)^4 + 90*lo
g(5)^3 - 270*log(5)^2 + 405*log(5) - 243)*x + 26730*log(5)^2 - 32076*log(5) + 16038)/(x^2 + 2*x*(log(5) - 3) +
log(5)^2 - 6*log(5) + 9) + 32/75*(2*x^5 - 5*x^4*(log(5) - 3) - 27*log(5)^5 + 20*(log(5)^2 - 6*log(5) + 9)*x^3
+ 405*log(5)^4 + 63*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^2 - 2430*log(5)^3 + 6*(log(5)^4 - 12*log(5)^3
+ 54*log(5)^2 - 108*log(5) + 81)*x + 7290*log(5)^2 - 10935*log(5) + 6561)/(x^2 + 2*x*(log(5) - 3) + log(5)^2 -
6*log(5) + 9) - 16/25*(x^4 - 4*x^3*(log(5) - 3) + 7*log(5)^4 - 11*(log(5)^2 - 6*log(5) + 9)*x^2 - 84*log(5)^3
+ 2*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x + 378*log(5)^2 - 756*log(5) + 567)/(x^2 + 2*x*(log(5) - 3) + l
og(5)^2 - 6*log(5) + 9) + 2/375*(10*x^7 - 14*x^6*(log(5) - 3) + 60*log(5)^7 + 21*(log(5)^2 - 6*log(5) + 9)*x^5
- 1260*log(5)^6 - 35*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^4 + 11340*log(5)^5 + 70*(log(5)^4 - 12*log(5)
^3 + 54*log(5)^2 - 108*log(5) + 81)*x^3 - 56700*log(5)^4 - 210*(log(5)^5 - 15*log(5)^4 + 90*log(5)^3 - 270*log
(5)^2 + 405*log(5) - 243)*x^2 + 170100*log(5)^3 - 360*(log(5)^6 - 18*log(5)^5 + 135*log(5)^4 - 540*log(5)^3 +
1215*log(5)^2 - 1458*log(5) + 729)*x - 306180*log(5)^2 + 306180*log(5) - 131220)/(x + log(5) - 3) - 28/125*(2*
x^6 - 3*x^5*(log(5) - 3) - 10*log(5)^6 + 5*(log(5)^2 - 6*log(5) + 9)*x^4 + 180*log(5)^5 - 10*(log(5)^3 - 9*log
(5)^2 + 27*log(5) - 27)*x^3 - 1350*log(5)^4 + 30*(log(5)^4 - 12*log(5)^3 + 54*log(5)^2 - 108*log(5) + 81)*x^2
+ 5400*log(5)^3 + 50*(log(5)^5 - 15*log(5)^4 + 90*log(5)^3 - 270*log(5)^2 + 405*log(5) - 243)*x - 12150*log(5)
^2 + 14580*log(5) - 7290)/(x + log(5) - 3) + 12/25*(3*x^5 - 5*x^4*(log(5) - 3) + 12*log(5)^5 + 10*(log(5)^2 -
6*log(5) + 9)*x^3 - 180*log(5)^4 - 30*(log(5)^3 - 9*log(5)^2 + 27*log(5) - 27)*x^2 + 1080*log(5)^3 - 48*(log(5
)^4 - 12*log(5)^3 + 54*log(5)^2 - 108*log(5) + 81)*x - 3240*log(5)^2 + 4860*log(5) - 2916)/(x + log(5) - 3) -
32/15*(x^4 - 2*x^3*(log(5) - 3) - 3*log(5)^4 + 6*(log(5)^2 - 6*log(5) + 9)*x^2 + 36*log(5)^3 + 9*(log(5)^3 - 9
*log(5)^2 + 27*log(5) - 27)*x - 162*log(5)^2 + 324*log(5) - 243)/(x + log(5) - 3) + 32/25*(x^3 - 3*x^2*(log(5)
- 3) + 2*log(5)^3 - 4*(log(5)^2 - 6*log(5) + 9)*x - 18*log(5)^2 + 54*log(5) - 54)/(x + log(5) - 3) - e^(e^x)

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mupad [B]  time = 1.66, size = 586, normalized size = 18.31 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((32*x^4)/25 - (64*x^5)/25 + (48*x^6)/25 - (16*x^7)/25 + (2*x^8)/25 - (log(5*exp(x - 3))*(64*x^3 - 160*x^
4 + 144*x^5 - 56*x^6 + 8*x^7))/25 + (log(5*exp(x - 3))^3*exp(x + exp(x))*(25*exp(x) + 25))/25)/log(5*exp(x - 3
))^3,x)

[Out]

x^2*((3*(log(5) - 3)^2*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(log(5) - 3) - 264/25
))/2 - (16*log(5))/5 + (((2*log(5))/5 + 2/5)*(log(5) - 3)^3)/2 - (3*(log(5) - 3)*((144*log(5))/25 - (6*(log(5)
- 3)^3)/25 + 3*(log(5) - 3)*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(log(5) - 3) -
264/25) + 3*((2*log(5))/5 + 2/5)*(log(5) - 3)^2 - 528/25))/2 + 256/25) - x*(3*(log(5) - 3)^2*((144*log(5))/25
- (6*(log(5) - 3)^3)/25 + 3*(log(5) - 3)*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(lo
g(5) - 3) - 264/25) + 3*((2*log(5))/5 + 2/5)*(log(5) - 3)^2 - 528/25) - (64*log(5))/25 + 3*(log(5) - 3)*(3*(lo
g(5) - 3)^2*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(log(5) - 3) - 264/25) - (32*log
(5))/5 + ((2*log(5))/5 + 2/5)*(log(5) - 3)^3 - 3*(log(5) - 3)*((144*log(5))/25 - (6*(log(5) - 3)^3)/25 + 3*(lo
g(5) - 3)*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(log(5) - 3) - 264/25) + 3*((2*log
(5))/5 + 2/5)*(log(5) - 3)^2 - 528/25) + 512/25) - (log(5) - 3)^3*((56*log(5))/25 + (18*(log(5) - 3)^2)/25 - 3
*((2*log(5))/5 + 2/5)*(log(5) - 3) - 264/25) + 192/25) - x^4*((14*log(5))/25 + (9*(log(5) - 3)^2)/50 - (3*((2*
log(5))/5 + 2/5)*(log(5) - 3))/4 - 66/25) - x^5*((2*log(5))/25 + 2/25) - exp(x + exp(x)) - (48*(log(5) - 3)^4
+ 128*(log(5) - 3)^5 + 120*(log(5) - 3)^6 + 48*(log(5) - 3)^7 + 7*(log(5) - 3)^8 + x*(64*(log(5) - 3)^3 + 160*
(log(5) - 3)^4 + 144*(log(5) - 3)^5 + 56*(log(5) - 3)^6 + 8*(log(5) - 3)^7))/(25*(log(5) - 3)^2 + 50*x*(log(5)
- 3) + 25*x^2) + x^6/25 + x^3*((48*log(5))/25 - (2*(log(5) - 3)^3)/25 + (log(5) - 3)*((56*log(5))/25 + (18*(l
og(5) - 3)^2)/25 - 3*((2*log(5))/5 + 2/5)*(log(5) - 3) - 264/25) + ((2*log(5))/5 + 2/5)*(log(5) - 3)^2 - 176/2
5)

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sympy [B]  time = 1.96, size = 291, normalized size = 9.09 \begin {gather*} \frac {x^{6}}{25} + x^{5} \left (- \frac {2 \log {\relax (5 )}}{25} - \frac {2}{25}\right ) + x^{4} \left (- \frac {2 \log {\relax (5 )}}{25} + \frac {3}{25} + \frac {3 \log {\relax (5 )}^{2}}{25}\right ) + x^{3} \left (- \frac {12 \log {\relax (5 )}}{25} - \frac {4 \log {\relax (5 )}^{3}}{25} + \frac {4}{25} + \frac {12 \log {\relax (5 )}^{2}}{25}\right ) + x^{2} \left (- \frac {28 \log {\relax (5 )}^{3}}{25} - \frac {44 \log {\relax (5 )}}{25} + \frac {13}{25} + \frac {\log {\relax (5 )}^{4}}{5} + \frac {54 \log {\relax (5 )}^{2}}{25}\right ) + x \left (- \frac {156 \log {\relax (5 )}^{3}}{25} - \frac {158 \log {\relax (5 )}}{25} - \frac {6 \log {\relax (5 )}^{5}}{25} + \frac {42}{25} + 2 \log {\relax (5 )}^{4} + \frac {228 \log {\relax (5 )}^{2}}{25}\right ) - e^{x + e^{x}} + \frac {x \left (- 3544 \log {\relax (5 )}^{3} - 648 \log {\relax (5 )}^{5} - 1944 \log {\relax (5 )} - 8 \log {\relax (5 )}^{7} + 432 + 112 \log {\relax (5 )}^{6} + 3600 \log {\relax (5 )}^{2} + 2000 \log {\relax (5 )}^{4}\right ) - 8658 \log {\relax (5 )}^{4} - 11772 \log {\relax (5 )}^{2} - 876 \log {\relax (5 )}^{6} - 1215 - 7 \log {\relax (5 )}^{8} + 120 \log {\relax (5 )}^{7} + 5832 \log {\relax (5 )} + 3544 \log {\relax (5 )}^{5} + 13032 \log {\relax (5 )}^{3}}{25 x^{2} + x \left (-150 + 50 \log {\relax (5 )}\right ) - 150 \log {\relax (5 )} + 25 \log {\relax (5 )}^{2} + 225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*exp(x)-25)*exp(exp(x)+x)*ln(5*exp(x-3))**3+(8*x**7-56*x**6+144*x**5-160*x**4+64*x**3)*ln(
5*exp(x-3))-2*x**8+16*x**7-48*x**6+64*x**5-32*x**4)/ln(5*exp(x-3))**3,x)

[Out]

x**6/25 + x**5*(-2*log(5)/25 - 2/25) + x**4*(-2*log(5)/25 + 3/25 + 3*log(5)**2/25) + x**3*(-12*log(5)/25 - 4*l
og(5)**3/25 + 4/25 + 12*log(5)**2/25) + x**2*(-28*log(5)**3/25 - 44*log(5)/25 + 13/25 + log(5)**4/5 + 54*log(5
)**2/25) + x*(-156*log(5)**3/25 - 158*log(5)/25 - 6*log(5)**5/25 + 42/25 + 2*log(5)**4 + 228*log(5)**2/25) - e
xp(x + exp(x)) + (x*(-3544*log(5)**3 - 648*log(5)**5 - 1944*log(5) - 8*log(5)**7 + 432 + 112*log(5)**6 + 3600*
log(5)**2 + 2000*log(5)**4) - 8658*log(5)**4 - 11772*log(5)**2 - 876*log(5)**6 - 1215 - 7*log(5)**8 + 120*log(
5)**7 + 5832*log(5) + 3544*log(5)**5 + 13032*log(5)**3)/(25*x**2 + x*(-150 + 50*log(5)) - 150*log(5) + 25*log(
5)**2 + 225)

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