3.52.79 \(\int \frac {-2 x^3+e^{10} (-3+x^2)+e^5 (-6 x+2 x^3)+e^{2 e^3-2 x} (-2 x+e^5 (6-2 x^2))+e^{e^3-x} (-4 x^2+e^5 (-6+6 x+2 x^2-2 x^3))+(-6 x+2 x^3+e^{2 e^3-2 x} (6-2 x^2)+e^5 (-6+2 x^2)+e^{e^3-x} (-6+6 x+2 x^2-2 x^3)) \log (3-x^2)+(-3+x^2) \log ^2(3-x^2)}{e^{10} (-3+x^2)+e^5 (-6+2 x^2) \log (3-x^2)+(-3+x^2) \log ^2(3-x^2)} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {\left (e^{e^3-x}+x\right )^2}{e^5+\log \left (3-x^2\right )} \]

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Rubi [B]  time = 5.36, antiderivative size = 186, normalized size of antiderivative = 6.20, number of steps used = 45, number of rules used = 18, integrand size = 227, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6741, 6725, 2454, 2389, 2297, 2299, 2178, 2475, 2411, 2353, 2302, 30, 6742, 2390, 6692, 43, 29, 2288} \begin {gather*} \frac {e^{2 e^3-2 x} \left (-e^5 x^2+x^2 \left (-\log \left (3-x^2\right )\right )+3 \log \left (3-x^2\right )+3 e^5\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}-\frac {3-x^2}{\log \left (3-x^2\right )+e^5}+\frac {3}{\log \left (3-x^2\right )+e^5}+\frac {2 e^{e^3-x} \left (-e^5 x^3+3 x \log \left (3-x^2\right )+x^3 \left (-\log \left (3-x^2\right )\right )+3 e^5 x\right )}{\left (3-x^2\right ) \left (\log \left (3-x^2\right )+e^5\right )^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + 3/(E^5 + Log[3 - x^2]) - (3 - x^2)/(E^5 + Log[3 - x^2]) + (E^(2*E^3 - 2*x)*(3*E^5 - E^5*x^2 + 3*Log[3 - x^
2] - x^2*Log[3 - x^2]))/((3 - x^2)*(E^5 + Log[3 - x^2])^2) + (2*E^(E^3 - x)*(3*E^5*x - E^5*x^3 + 3*x*Log[3 - x
^2] - x^3*Log[3 - x^2]))/((3 - x^2)*(E^5 + Log[3 - x^2])^2)

Rule 29

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2178

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2389

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 6692

Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((a_.) + (b_.)*(y_))^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u,
x]}, Dist[q, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y], x] /;  !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n}, x]
&& EqQ[v, y]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6741

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

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 {2 x^3-e^{10} \left (-3+x^2\right )-e^5 \left (-6 x+2 x^3\right )-e^{2 e^3-2 x} \left (-2 x+e^5 \left (6-2 x^2\right )\right )-e^{e^3-x} \left (-4 x^2+e^5 \left (-6+6 x+2 x^2-2 x^3\right )\right )-\left (-6 x+2 x^3+e^{2 e^3-2 x} \left (6-2 x^2\right )+e^5 \left (-6+2 x^2\right )+e^{e^3-x} \left (-6+6 x+2 x^2-2 x^3\right )\right ) \log \left (3-x^2\right )-\left (-3+x^2\right ) \log ^2\left (3-x^2\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\\ &=\int \left (\frac {e^{10}}{\left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^5 x}{\left (e^5+\log \left (3-x^2\right )\right )^2}-\frac {2 x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^5 \log \left (3-x^2\right )}{\left (e^5+\log \left (3-x^2\right )\right )^2}-\frac {6 x \log \left (3-x^2\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 x^3 \log \left (3-x^2\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {\log ^2\left (3-x^2\right )}{\left (e^5+\log \left (3-x^2\right )\right )^2}-\frac {2 e^{2 e^3-2 x} \left (-3 e^5+x+e^5 x^2-3 \log \left (3-x^2\right )+x^2 \log \left (3-x^2\right )\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5-3 e^5 x+2 \left (1-\frac {e^5}{2}\right ) x^2+e^5 x^3+3 \log \left (3-x^2\right )-3 x \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )+x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )+2 \int \frac {x^3 \log \left (3-x^2\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx-2 \int \frac {e^{2 e^3-2 x} \left (-3 e^5+x+e^5 x^2-3 \log \left (3-x^2\right )+x^2 \log \left (3-x^2\right )\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+2 \int \frac {e^{e^3-x} \left (3 e^5-3 e^5 x+2 \left (1-\frac {e^5}{2}\right ) x^2+e^5 x^3+3 \log \left (3-x^2\right )-3 x \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )+x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx-6 \int \frac {x \log \left (3-x^2\right )}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\left (2 e^5\right ) \int \frac {\log \left (3-x^2\right )}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\int \frac {\log ^2\left (3-x^2\right )}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\\ &=\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+2 \int \left (-\frac {e^5 x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )}\right ) \, dx-3 \operatorname {Subst}\left (\int \frac {x}{\left (e^5+x\right )^2} \, dx,x,\log \left (3-x^2\right )\right )+e^5 \operatorname {Subst}\left (\int \frac {1}{\left (e^5+\log (3-x)\right )^2} \, dx,x,x^2\right )+\left (2 e^5\right ) \int \left (-\frac {e^5}{\left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {1}{e^5+\log \left (3-x^2\right )}\right ) \, dx+e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\int \left (1+\frac {e^{10}}{\left (e^5+\log \left (3-x^2\right )\right )^2}-\frac {2 e^5}{e^5+\log \left (3-x^2\right )}\right ) \, dx-\operatorname {Subst}\left (\int \frac {x}{(-3+x) \left (e^5+\log (3-x)\right )^2} \, dx,x,x^2\right )\\ &=x+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+2 \int \frac {x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )} \, dx-3 \operatorname {Subst}\left (\int \left (-\frac {e^5}{\left (e^5+x\right )^2}+\frac {1}{e^5+x}\right ) \, dx,x,\log \left (3-x^2\right )\right )-e^5 \operatorname {Subst}\left (\int \frac {1}{\left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )-\left (2 e^5\right ) \int \frac {x^3}{\left (-3+x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx-\operatorname {Subst}\left (\int \frac {3-x}{x \left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )\\ &=x-\frac {3 e^5}{e^5+\log \left (3-x^2\right )}+\frac {e^5 \left (3-x^2\right )}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}-3 \log \left (e^5+\log \left (3-x^2\right )\right )-e^5 \operatorname {Subst}\left (\int \frac {x}{(-3+x) \left (e^5+\log (3-x)\right )^2} \, dx,x,x^2\right )-e^5 \operatorname {Subst}\left (\int \frac {1}{e^5+\log (x)} \, dx,x,3-x^2\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {x}{(-3+x) \left (e^5+\log (3-x)\right )} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \left (-\frac {1}{\left (e^5+\log (x)\right )^2}+\frac {3}{x \left (e^5+\log (x)\right )^2}\right ) \, dx,x,3-x^2\right )\\ &=x-\frac {3 e^5}{e^5+\log \left (3-x^2\right )}+\frac {e^5 \left (3-x^2\right )}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}-3 \log \left (e^5+\log \left (3-x^2\right )\right )-3 \operatorname {Subst}\left (\int \frac {1}{x \left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )-e^5 \operatorname {Subst}\left (\int \frac {e^x}{e^5+x} \, dx,x,\log \left (3-x^2\right )\right )-e^5 \operatorname {Subst}\left (\int \frac {3-x}{x \left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {1}{\left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )+\operatorname {Subst}\left (\int \frac {3-x}{x \left (e^5+\log (x)\right )} \, dx,x,3-x^2\right )\\ &=x-e^{5-e^5} \text {Ei}\left (e^5+\log \left (3-x^2\right )\right )-\frac {3 e^5}{e^5+\log \left (3-x^2\right )}-\frac {3-x^2}{e^5+\log \left (3-x^2\right )}+\frac {e^5 \left (3-x^2\right )}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}-3 \log \left (e^5+\log \left (3-x^2\right )\right )-3 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,e^5+\log \left (3-x^2\right )\right )-e^5 \operatorname {Subst}\left (\int \left (-\frac {1}{\left (e^5+\log (x)\right )^2}+\frac {3}{x \left (e^5+\log (x)\right )^2}\right ) \, dx,x,3-x^2\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {1}{e^5+\log (x)} \, dx,x,3-x^2\right )+\operatorname {Subst}\left (\int \left (-\frac {1}{e^5+\log (x)}+\frac {3}{x \left (e^5+\log (x)\right )}\right ) \, dx,x,3-x^2\right )\\ &=x-e^{5-e^5} \text {Ei}\left (e^5+\log \left (3-x^2\right )\right )+\frac {3}{e^5+\log \left (3-x^2\right )}-\frac {3 e^5}{e^5+\log \left (3-x^2\right )}-\frac {3-x^2}{e^5+\log \left (3-x^2\right )}+\frac {e^5 \left (3-x^2\right )}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}-3 \log \left (e^5+\log \left (3-x^2\right )\right )+3 \operatorname {Subst}\left (\int \frac {1}{x \left (e^5+\log (x)\right )} \, dx,x,3-x^2\right )+e^5 \operatorname {Subst}\left (\int \frac {1}{\left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )-\left (3 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (e^5+\log (x)\right )^2} \, dx,x,3-x^2\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {e^x}{e^5+x} \, dx,x,\log \left (3-x^2\right )\right )-\operatorname {Subst}\left (\int \frac {1}{e^5+\log (x)} \, dx,x,3-x^2\right )\\ &=x+e^{-e^5} \text {Ei}\left (e^5+\log \left (3-x^2\right )\right )-e^{5-e^5} \text {Ei}\left (e^5+\log \left (3-x^2\right )\right )+\frac {3}{e^5+\log \left (3-x^2\right )}-\frac {3 e^5}{e^5+\log \left (3-x^2\right )}-\frac {3-x^2}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}-3 \log \left (e^5+\log \left (3-x^2\right )\right )+3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^5+\log \left (3-x^2\right )\right )+e^5 \operatorname {Subst}\left (\int \frac {1}{e^5+\log (x)} \, dx,x,3-x^2\right )-\left (3 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,e^5+\log \left (3-x^2\right )\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx-\operatorname {Subst}\left (\int \frac {e^x}{e^5+x} \, dx,x,\log \left (3-x^2\right )\right )\\ &=x-e^{5-e^5} \text {Ei}\left (e^5+\log \left (3-x^2\right )\right )+\frac {3}{e^5+\log \left (3-x^2\right )}-\frac {3-x^2}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+e^5 \operatorname {Subst}\left (\int \frac {e^x}{e^5+x} \, dx,x,\log \left (3-x^2\right )\right )+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\\ &=x+\frac {3}{e^5+\log \left (3-x^2\right )}-\frac {3-x^2}{e^5+\log \left (3-x^2\right )}+\frac {e^{2 e^3-2 x} \left (3 e^5-e^5 x^2+3 \log \left (3-x^2\right )-x^2 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+\frac {2 e^{e^3-x} \left (3 e^5 x-e^5 x^3+3 x \log \left (3-x^2\right )-x^3 \log \left (3-x^2\right )\right )}{\left (3-x^2\right ) \left (e^5+\log \left (3-x^2\right )\right )^2}+2 \left (e^{10} \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\right )-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5+\log \left (3-x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 35, normalized size = 1.17 \begin {gather*} x+\frac {e^{-2 x} \left (e^{e^3}+e^x x\right )^2}{e^5+\log \left (3-x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^3 + E^10*(-3 + x^2) + E^5*(-6*x + 2*x^3) + E^(2*E^3 - 2*x)*(-2*x + E^5*(6 - 2*x^2)) + E^(E^3 -
x)*(-4*x^2 + E^5*(-6 + 6*x + 2*x^2 - 2*x^3)) + (-6*x + 2*x^3 + E^(2*E^3 - 2*x)*(6 - 2*x^2) + E^5*(-6 + 2*x^2)
+ E^(E^3 - x)*(-6 + 6*x + 2*x^2 - 2*x^3))*Log[3 - x^2] + (-3 + x^2)*Log[3 - x^2]^2)/(E^10*(-3 + x^2) + E^5*(-
6 + 2*x^2)*Log[3 - x^2] + (-3 + x^2)*Log[3 - x^2]^2),x]

[Out]

x + (E^E^3 + E^x*x)^2/(E^(2*x)*(E^5 + Log[3 - x^2]))

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fricas [A]  time = 0.65, size = 51, normalized size = 1.70 \begin {gather*} \frac {x^{2} + x e^{5} + 2 \, x e^{\left (-x + e^{3}\right )} + x \log \left (-x^{2} + 3\right ) + e^{\left (-2 \, x + 2 \, e^{3}\right )}}{e^{5} + \log \left (-x^{2} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*ex
p(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6)*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-
x+exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+(2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-
3)*exp(5)^2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.20, size = 82, normalized size = 2.73 \begin {gather*} \frac {x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (3 \, x\right )} \log \left (-x^{2} + 3\right ) + x^{2} e^{\left (3 \, x + 5\right )} + 2 \, x^{2} e^{\left (2 \, x + e^{3}\right )} + x e^{\left (x + 2 \, e^{3}\right )}}{x e^{\left (3 \, x\right )} \log \left (-x^{2} + 3\right ) + x e^{\left (3 \, x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*ex
p(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6)*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-
x+exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+(2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-
3)*exp(5)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.39, size = 40, normalized size = 1.33

method	result	size
risch	\(x +\frac {x^{2}+2 x \,{\mathrm e}^{-x +{\mathrm e}^{3}}+{\mathrm e}^{-2 x +2 \,{\mathrm e}^{3}}}{{\mathrm e}^{5}+\ln \left (-x^{2}+3\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-3)*ln(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*exp(5)+2*
x^3-6*x)*ln(-x^2+3)+((-2*x^2+6)*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-x+exp(3)
)+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*ln(-x^2+3)^2+(2*x^2-6)*exp(5)*ln(-x^2+3)+(x^2-3)*exp(5)^
2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.41, size = 64, normalized size = 2.13 \begin {gather*} \frac {x e^{\left (2 \, x\right )} \log \left (-x^{2} + 3\right ) + {\left (x^{2} + x e^{5}\right )} e^{\left (2 \, x\right )} + 2 \, x e^{\left (x + e^{3}\right )} + e^{\left (2 \, e^{3}\right )}}{e^{\left (2 \, x\right )} \log \left (-x^{2} + 3\right ) + e^{\left (2 \, x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-3)*log(-x^2+3)^2+((-2*x^2+6)*exp(-x+exp(3))^2+(-2*x^3+2*x^2+6*x-6)*exp(-x+exp(3))+(2*x^2-6)*ex
p(5)+2*x^3-6*x)*log(-x^2+3)+((-2*x^2+6)*exp(5)-2*x)*exp(-x+exp(3))^2+((-2*x^3+2*x^2+6*x-6)*exp(5)-4*x^2)*exp(-
x+exp(3))+(x^2-3)*exp(5)^2+(2*x^3-6*x)*exp(5)-2*x^3)/((x^2-3)*log(-x^2+3)^2+(2*x^2-6)*exp(5)*log(-x^2+3)+(x^2-
3)*exp(5)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.97, size = 95, normalized size = 3.17 \begin {gather*} x+\frac {3\,{\mathrm {e}}^5-\ln \left (3-x^2\right )\,\left (x^2-3\right )-x^2\,{\mathrm {e}}^5+x^2}{\ln \left (3-x^2\right )+{\mathrm {e}}^5}+x^2+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^3-2\,x}}{\ln \left (3-x^2\right )+{\mathrm {e}}^5}+\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^3-x}}{\ln \left (3-x^2\right )+{\mathrm {e}}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(3) - x)*(exp(5)*(6*x + 2*x^2 - 2*x^3 - 6) - 4*x^2) - exp(2*exp(3) - 2*x)*(2*x + exp(5)*(2*x^2 - 6
)) - exp(5)*(6*x - 2*x^3) + log(3 - x^2)*(exp(exp(3) - x)*(6*x + 2*x^2 - 2*x^3 - 6) - 6*x - exp(2*exp(3) - 2*x
)*(2*x^2 - 6) + exp(5)*(2*x^2 - 6) + 2*x^3) - 2*x^3 + log(3 - x^2)^2*(x^2 - 3) + exp(10)*(x^2 - 3))/(log(3 - x
^2)^2*(x^2 - 3) + exp(10)*(x^2 - 3) + log(3 - x^2)*exp(5)*(2*x^2 - 6)),x)

[Out]

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

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sympy [B]  time = 0.53, size = 82, normalized size = 2.73 \begin {gather*} \frac {x^{2}}{\log {\left (3 - x^{2} \right )} + e^{5}} + x + \frac {\left (2 x \log {\left (3 - x^{2} \right )} + 2 x e^{5}\right ) e^{- x + e^{3}} + \left (\log {\left (3 - x^{2} \right )} + e^{5}\right ) e^{- 2 x + 2 e^{3}}}{\log {\left (3 - x^{2} \right )}^{2} + 2 e^{5} \log {\left (3 - x^{2} \right )} + e^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-3)*ln(-x**2+3)**2+((-2*x**2+6)*exp(-x+exp(3))**2+(-2*x**3+2*x**2+6*x-6)*exp(-x+exp(3))+(2*x**
2-6)*exp(5)+2*x**3-6*x)*ln(-x**2+3)+((-2*x**2+6)*exp(5)-2*x)*exp(-x+exp(3))**2+((-2*x**3+2*x**2+6*x-6)*exp(5)-
4*x**2)*exp(-x+exp(3))+(x**2-3)*exp(5)**2+(2*x**3-6*x)*exp(5)-2*x**3)/((x**2-3)*ln(-x**2+3)**2+(2*x**2-6)*exp(
5)*ln(-x**2+3)+(x**2-3)*exp(5)**2),x)

[Out]

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

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