3.56.60 \(\int \frac {-45-3 x^2+e^{10} (-15-x^2)+e^x (-15-15 x+x^2+x^3)+e^{e^5} (-9-3 e^{10}+e^x (-3-3 x+x^2))+(-45+6 x+3 x^2+e^{e^5} (-9+e^{10} (-3+x)+e^x (-3+x)+3 x)+e^{10} (-15+2 x+x^2)+e^x (-15+2 x+x^2)) \log (\frac {3+e^{10}+e^x}{-15+e^{e^5} (-3+x)+2 x+x^2})}{-45+6 x+3 x^2+e^{e^5} (-9+e^{10} (-3+x)+e^x (-3+x)+3 x)+e^{10} (-15+2 x+x^2)+e^x (-15+2 x+x^2)} \, dx\)

Optimal. Leaf size=29 \[ x+x \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right ) \]

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Rubi [B]  time = 7.46, antiderivative size = 261, normalized size of antiderivative = 9.00, number of steps used = 30, number of rules used = 12, integrand size = 209, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6688, 6742, 2184, 2190, 2279, 2391, 36, 31, 72, 893, 2548, 1612} \begin {gather*} -\left (2+e^{e^5}\right ) x+e^{e^5} x+3 x+x \log \left (-\frac {e^x+3+e^{10}}{(3-x) \left (x+e^{e^5}+5\right )}\right )-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}-\frac {24 \log (3-x)}{8+e^{e^5}}+3 \log (3-x)-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (x+e^{e^5}+5\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (x+e^{e^5}+5\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (x+e^{e^5}+5\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (x+e^{e^5}+5\right )}{8+e^{e^5}}+\frac {15 \log \left (x+e^{e^5}+5\right )}{8+e^{e^5}}-\left (5+e^{e^5}\right ) \log \left (x+e^{e^5}+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-45 - 3*x^2 + E^10*(-15 - x^2) + E^x*(-15 - 15*x + x^2 + x^3) + E^E^5*(-9 - 3*E^10 + E^x*(-3 - 3*x + x^2)
) + (-45 + 6*x + 3*x^2 + E^E^5*(-9 + E^10*(-3 + x) + E^x*(-3 + x) + 3*x) + E^10*(-15 + 2*x + x^2) + E^x*(-15 +
2*x + x^2))*Log[(3 + E^10 + E^x)/(-15 + E^E^5*(-3 + x) + 2*x + x^2)])/(-45 + 6*x + 3*x^2 + E^E^5*(-9 + E^10*(
-3 + x) + E^x*(-3 + x) + 3*x) + E^10*(-15 + 2*x + x^2) + E^x*(-15 + 2*x + x^2)),x]

[Out]

3*x + E^E^5*x - (2 + E^E^5)*x + 3*Log[3 - x] - (24*Log[3 - x])/(8 + E^E^5) - (3*E^E^5*Log[3 - x])/(8 + E^E^5)
+ x*Log[-((3 + E^10 + E^x)/((3 - x)*(5 + E^E^5 + x)))] - (5 + E^E^5)*Log[5 + E^E^5 + x] + (15*Log[5 + E^E^5 +
x])/(8 + E^E^5) - (15*(5 + E^E^5)*Log[5 + E^E^5 + x])/(8 + E^E^5) - ((5 + E^E^5)^2*Log[5 + E^E^5 + x])/(8 + E^
E^5) + ((5 + E^E^5)^3*Log[5 + E^E^5 + x])/(8 + E^E^5) - (E^E^5*(37 + 13*E^E^5 + E^(2*E^5))*Log[5 + E^E^5 + x])
/(8 + E^E^5)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, 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} &=\int \frac {45+3 x^2+e^{10} \left (15+x^2\right )-e^x \left (-15-15 x+x^2+x^3\right )-e^{e^5} \left (-9-3 e^{10}+e^x \left (-3-3 x+x^2\right )\right )-\left (3+e^{10}+e^x\right ) (-3+x) \left (5+e^{e^5}+x\right ) \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )}{\left (e^x+3 \left (1+\frac {e^{10}}{3}\right )\right ) (3-x) \left (5+e^{e^5}+x\right )} \, dx\\ &=\int \left (\frac {\left (-3-e^{10}\right ) x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )}+\frac {15 \left (1+\frac {e^{e^5}}{5}\right )+15 \left (1+\frac {e^{e^5}}{5}\right ) x-\left (1+e^{e^5}\right ) x^2-x^3+15 \left (1+\frac {e^{e^5}}{5}\right ) \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )-2 \left (1+\frac {e^{e^5}}{2}\right ) x \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )-x^2 \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )}{(3-x) \left (5+e^{e^5}+x\right )}\right ) \, dx\\ &=\left (-3-e^{10}\right ) \int \frac {x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )} \, dx+\int \frac {15 \left (1+\frac {e^{e^5}}{5}\right )+15 \left (1+\frac {e^{e^5}}{5}\right ) x-\left (1+e^{e^5}\right ) x^2-x^3+15 \left (1+\frac {e^{e^5}}{5}\right ) \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )-2 \left (1+\frac {e^{e^5}}{2}\right ) x \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )-x^2 \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )}{(3-x) \left (5+e^{e^5}+x\right )} \, dx\\ &=-\frac {x^2}{2}+\int \frac {e^x x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )} \, dx+\int \frac {15+15 x-x^2-x^3-e^{e^5} \left (-3-3 x+x^2\right )-(-3+x) \left (5+e^{e^5}+x\right ) \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right )}{(3-x) \left (5+e^{e^5}+x\right )} \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+\frac {e^x}{3+e^{10}}\right )-\int \log \left (1+\frac {e^x}{3 \left (1+\frac {e^{10}}{3}\right )}\right ) \, dx+\int \left (-\frac {15}{(-3+x) \left (5+e^{e^5}+x\right )}-\frac {15 x}{(-3+x) \left (5+e^{e^5}+x\right )}+\frac {x^2}{(-3+x) \left (5+e^{e^5}+x\right )}+\frac {x^3}{(-3+x) \left (5+e^{e^5}+x\right )}+\frac {e^{e^5} \left (-3-3 x+x^2\right )}{(-3+x) \left (5+e^{e^5}+x\right )}+\log \left (\frac {e^x+3 \left (1+\frac {e^{10}}{3}\right )}{(-3+x) \left (5+e^{e^5}+x\right )}\right )\right ) \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+\frac {e^x}{3+e^{10}}\right )-15 \int \frac {1}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx-15 \int \frac {x}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx+e^{e^5} \int \frac {-3-3 x+x^2}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx+\int \frac {x^2}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx+\int \frac {x^3}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx+\int \log \left (\frac {e^x+3 \left (1+\frac {e^{10}}{3}\right )}{(-3+x) \left (5+e^{e^5}+x\right )}\right ) \, dx-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3 \left (1+\frac {e^{10}}{3}\right )}\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {x^2}{2}+x \log \left (1+\frac {e^x}{3+e^{10}}\right )+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )-15 \int \left (\frac {3}{\left (8+e^{e^5}\right ) (-3+x)}+\frac {5+e^{e^5}}{\left (8+e^{e^5}\right ) \left (5+e^{e^5}+x\right )}\right ) \, dx+e^{e^5} \int \left (1-\frac {3}{\left (8+e^{e^5}\right ) (-3+x)}-\frac {37+13 e^{e^5}+e^{2 e^5}}{\left (8+e^{e^5}\right ) \left (5+e^{e^5}+x\right )}\right ) \, dx-\frac {15 \int \frac {1}{-3+x} \, dx}{8+e^{e^5}}+\frac {15 \int \frac {1}{5+e^{e^5}+x} \, dx}{8+e^{e^5}}+\int \left (1+\frac {9}{\left (8+e^{e^5}\right ) (-3+x)}-\frac {\left (5+e^{e^5}\right )^2}{\left (8+e^{e^5}\right ) \left (5+e^{e^5}+x\right )}\right ) \, dx+\int \left (-2 \left (1+\frac {e^{e^5}}{2}\right )+\frac {27}{\left (8+e^{e^5}\right ) (-3+x)}+x+\frac {\left (5+e^{e^5}\right )^3}{\left (8+e^{e^5}\right ) \left (5+e^{e^5}+x\right )}\right ) \, dx-\int \frac {x \left (3 e^{e^5} \left (1+\frac {e^{10}}{3}\right )-e^{e^5+x} (-4+x)+6 \left (1+\frac {e^{10}}{3}\right ) (1+x)-e^x \left (-17+x^2\right )\right )}{\left (e^x+3 \left (1+\frac {e^{10}}{3}\right )\right ) (3-x) \left (5+e^{e^5}+x\right )} \, dx\\ &=x+e^{e^5} x-\left (2+e^{e^5}\right ) x+x \log \left (1+\frac {e^x}{3+e^{10}}\right )-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )-\int \left (\frac {\left (-3-e^{10}\right ) x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )}+\frac {x \left (-17-4 e^{e^5}+e^{e^5} x+x^2\right )}{(-3+x) \left (5+e^{e^5}+x\right )}\right ) \, dx\\ &=x+e^{e^5} x-\left (2+e^{e^5}\right ) x+x \log \left (1+\frac {e^x}{3+e^{10}}\right )-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )-\left (-3-e^{10}\right ) \int \frac {x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )} \, dx-\int \frac {x \left (-17-4 e^{e^5}+e^{e^5} x+x^2\right )}{(-3+x) \left (5+e^{e^5}+x\right )} \, dx\\ &=x+e^{e^5} x-\left (2+e^{e^5}\right ) x+\frac {x^2}{2}+x \log \left (1+\frac {e^x}{3+e^{10}}\right )-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )-\int \frac {e^x x}{e^x+3 \left (1+\frac {e^{10}}{3}\right )} \, dx-\int \left (-2-\frac {3}{-3+x}+x+\frac {5+e^{e^5}}{5+e^{e^5}+x}\right ) \, dx\\ &=3 x+e^{e^5} x-\left (2+e^{e^5}\right ) x+3 \log (3-x)-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )-\left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )+\int \log \left (1+\frac {e^x}{3 \left (1+\frac {e^{10}}{3}\right )}\right ) \, dx\\ &=3 x+e^{e^5} x-\left (2+e^{e^5}\right ) x+3 \log (3-x)-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )-\left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\text {Li}_2\left (-\frac {e^x}{3+e^{10}}\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{3 \left (1+\frac {e^{10}}{3}\right )}\right )}{x} \, dx,x,e^x\right )\\ &=3 x+e^{e^5} x-\left (2+e^{e^5}\right ) x+3 \log (3-x)-\frac {24 \log (3-x)}{8+e^{e^5}}-\frac {3 e^{e^5} \log (3-x)}{8+e^{e^5}}+x \log \left (-\frac {3+e^{10}+e^x}{(3-x) \left (5+e^{e^5}+x\right )}\right )-\left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )+\frac {15 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {15 \left (5+e^{e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {\left (5+e^{e^5}\right )^2 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}+\frac {\left (5+e^{e^5}\right )^3 \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}-\frac {e^{e^5} \left (37+13 e^{e^5}+e^{2 e^5}\right ) \log \left (5+e^{e^5}+x\right )}{8+e^{e^5}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 29, normalized size = 1.00 \begin {gather*} x+x \log \left (\frac {3+e^{10}+e^x}{(-3+x) \left (5+e^{e^5}+x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-45 - 3*x^2 + E^10*(-15 - x^2) + E^x*(-15 - 15*x + x^2 + x^3) + E^E^5*(-9 - 3*E^10 + E^x*(-3 - 3*x
+ x^2)) + (-45 + 6*x + 3*x^2 + E^E^5*(-9 + E^10*(-3 + x) + E^x*(-3 + x) + 3*x) + E^10*(-15 + 2*x + x^2) + E^x*
(-15 + 2*x + x^2))*Log[(3 + E^10 + E^x)/(-15 + E^E^5*(-3 + x) + 2*x + x^2)])/(-45 + 6*x + 3*x^2 + E^E^5*(-9 +
E^10*(-3 + x) + E^x*(-3 + x) + 3*x) + E^10*(-15 + 2*x + x^2) + E^x*(-15 + 2*x + x^2)),x]

[Out]

x + x*Log[(3 + E^10 + E^x)/((-3 + x)*(5 + E^E^5 + x))]

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fricas [A]  time = 0.76, size = 29, normalized size = 1.00 \begin {gather*} x \log \left (\frac {e^{10} + e^{x} + 3}{x^{2} + {\left (x - 3\right )} e^{\left (e^{5}\right )} + 2 \, x - 15}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*
x-45)*log((exp(x)+exp(5)^2+3)/((x-3)*exp(exp(5))+x^2+2*x-15))+((x^2-3*x-3)*exp(x)-3*exp(5)^2-9)*exp(exp(5))+(x
^3+x^2-15*x-15)*exp(x)+(-x^2-15)*exp(5)^2-3*x^2-45)/(((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-
15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*x-45),x, algorithm="fricas")

[Out]

x*log((e^10 + e^x + 3)/(x^2 + (x - 3)*e^(e^5) + 2*x - 15)) + x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{2} + {\left (x^{2} + 15\right )} e^{10} - {\left (x^{3} + x^{2} - 15 \, x - 15\right )} e^{x} - {\left ({\left (x^{2} - 3 \, x - 3\right )} e^{x} - 3 \, e^{10} - 9\right )} e^{\left (e^{5}\right )} - {\left (3 \, x^{2} + {\left (x^{2} + 2 \, x - 15\right )} e^{10} + {\left (x^{2} + 2 \, x - 15\right )} e^{x} + {\left ({\left (x - 3\right )} e^{10} + {\left (x - 3\right )} e^{x} + 3 \, x - 9\right )} e^{\left (e^{5}\right )} + 6 \, x - 45\right )} \log \left (\frac {e^{10} + e^{x} + 3}{x^{2} + {\left (x - 3\right )} e^{\left (e^{5}\right )} + 2 \, x - 15}\right ) + 45}{3 \, x^{2} + {\left (x^{2} + 2 \, x - 15\right )} e^{10} + {\left (x^{2} + 2 \, x - 15\right )} e^{x} + {\left ({\left (x - 3\right )} e^{10} + {\left (x - 3\right )} e^{x} + 3 \, x - 9\right )} e^{\left (e^{5}\right )} + 6 \, x - 45}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*
x-45)*log((exp(x)+exp(5)^2+3)/((x-3)*exp(exp(5))+x^2+2*x-15))+((x^2-3*x-3)*exp(x)-3*exp(5)^2-9)*exp(exp(5))+(x
^3+x^2-15*x-15)*exp(x)+(-x^2-15)*exp(5)^2-3*x^2-45)/(((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-
15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*x-45),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + (x^2 + 15)*e^10 - (x^3 + x^2 - 15*x - 15)*e^x - ((x^2 - 3*x - 3)*e^x - 3*e^10 - 9)*e^(e^5)
- (3*x^2 + (x^2 + 2*x - 15)*e^10 + (x^2 + 2*x - 15)*e^x + ((x - 3)*e^10 + (x - 3)*e^x + 3*x - 9)*e^(e^5) + 6*
x - 45)*log((e^10 + e^x + 3)/(x^2 + (x - 3)*e^(e^5) + 2*x - 15)) + 45)/(3*x^2 + (x^2 + 2*x - 15)*e^10 + (x^2 +
2*x - 15)*e^x + ((x - 3)*e^10 + (x - 3)*e^x + 3*x - 9)*e^(e^5) + 6*x - 45), x)

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maple [A]  time = 1.57, size = 32, normalized size = 1.10

method	result	size
norman	\(x +x \ln \left (\frac {{\mathrm e}^{x}+{\mathrm e}^{10}+3}{\left (x -3\right ) {\mathrm e}^{{\mathrm e}^{5}}+x^{2}+2 x -15}\right )\)	\(32\)
risch	\(x \ln \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )-x \ln \left (x +{\mathrm e}^{{\mathrm e}^{5}}+5\right )-\ln \left (x -3\right ) x -\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -3}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{\left (x -3\right ) \left (x +{\mathrm e}^{{\mathrm e}^{5}}+5\right )}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x -3}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{\left (x -3\right ) \left (x +{\mathrm e}^{{\mathrm e}^{5}}+5\right )}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{x +{\mathrm e}^{{\mathrm e}^{5}}+5}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{\left (x -3\right ) \left (x +{\mathrm e}^{{\mathrm e}^{5}}+5\right )}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+{\mathrm e}^{10}+3\right )}{\left (x -3\right ) \left (x +{\mathrm e}^{{\mathrm e}^{5}}+5\right )}\right )^{3}}{2}+x\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*x-45)*
ln((exp(x)+exp(5)^2+3)/((x-3)*exp(exp(5))+x^2+2*x-15))+((x^2-3*x-3)*exp(x)-3*exp(5)^2-9)*exp(exp(5))+(x^3+x^2-
15*x-15)*exp(x)+(-x^2-15)*exp(5)^2-3*x^2-45)/(((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-15)*exp
(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*x-45),x,method=_RETURNVERBOSE)

[Out]

x+x*ln((exp(x)+exp(5)^2+3)/((x-3)*exp(exp(5))+x^2+2*x-15))

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maxima [A]  time = 0.47, size = 28, normalized size = 0.97 \begin {gather*} -x \log \left (x + e^{\left (e^{5}\right )} + 5\right ) - x \log \left (x - 3\right ) + x \log \left (e^{10} + e^{x} + 3\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*
x-45)*log((exp(x)+exp(5)^2+3)/((x-3)*exp(exp(5))+x^2+2*x-15))+((x^2-3*x-3)*exp(x)-3*exp(5)^2-9)*exp(exp(5))+(x
^3+x^2-15*x-15)*exp(x)+(-x^2-15)*exp(5)^2-3*x^2-45)/(((x-3)*exp(x)+(x-3)*exp(5)^2+3*x-9)*exp(exp(5))+(x^2+2*x-
15)*exp(x)+(x^2+2*x-15)*exp(5)^2+3*x^2+6*x-45),x, algorithm="maxima")

[Out]

-x*log(x + e^(e^5) + 5) - x*log(x - 3) + x*log(e^10 + e^x + 3) + x

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mupad [B]  time = 4.57, size = 29, normalized size = 1.00 \begin {gather*} x\,\left (\ln \left (\frac {{\mathrm {e}}^{10}+{\mathrm {e}}^x+3}{2\,x+{\mathrm {e}}^{{\mathrm {e}}^5}\,\left (x-3\right )+x^2-15}\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 - log((exp(10) + exp(x) + 3)/(2*x + exp(exp(5))*(x - 3) + x^2 - 15))*(6*x + exp(exp(5))*(3*x + exp
(x)*(x - 3) + exp(10)*(x - 3) - 9) + exp(x)*(2*x + x^2 - 15) + exp(10)*(2*x + x^2 - 15) + 3*x^2 - 45) + exp(x)
*(15*x - x^2 - x^3 + 15) + exp(exp(5))*(3*exp(10) + exp(x)*(3*x - x^2 + 3) + 9) + exp(10)*(x^2 + 15) + 45)/(6*
x + exp(exp(5))*(3*x + exp(x)*(x - 3) + exp(10)*(x - 3) - 9) + exp(x)*(2*x + x^2 - 15) + exp(10)*(2*x + x^2 -
15) + 3*x^2 - 45),x)

[Out]

x*(log((exp(10) + exp(x) + 3)/(2*x + exp(exp(5))*(x - 3) + x^2 - 15)) + 1)

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sympy [B]  time = 2.79, size = 88, normalized size = 3.03 \begin {gather*} x + \left (x + \frac {1}{3} + \frac {e^{e^{5}}}{6}\right ) \log {\left (\frac {e^{x} + 3 + e^{10}}{x^{2} + 2 x + \left (x - 3\right ) e^{e^{5}} - 15} \right )} - \frac {\left (2 + e^{e^{5}}\right ) \log {\left (e^{x} + 3 + e^{10} \right )}}{6} + \frac {\left (2 + e^{e^{5}}\right ) \log {\left (x^{2} + x \left (2 + e^{e^{5}}\right ) - 3 e^{e^{5}} - 15 \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(x)+(x-3)*exp(5)**2+3*x-9)*exp(exp(5))+(x**2+2*x-15)*exp(x)+(x**2+2*x-15)*exp(5)**2+3*x*
*2+6*x-45)*ln((exp(x)+exp(5)**2+3)/((x-3)*exp(exp(5))+x**2+2*x-15))+((x**2-3*x-3)*exp(x)-3*exp(5)**2-9)*exp(ex
p(5))+(x**3+x**2-15*x-15)*exp(x)+(-x**2-15)*exp(5)**2-3*x**2-45)/(((x-3)*exp(x)+(x-3)*exp(5)**2+3*x-9)*exp(exp
(5))+(x**2+2*x-15)*exp(x)+(x**2+2*x-15)*exp(5)**2+3*x**2+6*x-45),x)

[Out]

x + (x + 1/3 + exp(exp(5))/6)*log((exp(x) + 3 + exp(10))/(x**2 + 2*x + (x - 3)*exp(exp(5)) - 15)) - (2 + exp(e
xp(5)))*log(exp(x) + 3 + exp(10))/6 + (2 + exp(exp(5)))*log(x**2 + x*(2 + exp(exp(5))) - 3*exp(exp(5)) - 15)/6

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