\(\int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x (96 x+24 x^2-24 x^3-6 x^4)+(144 x^2+48 x^3+36 x^4+12 x^5+e^x (-96 x-84 x^2-36 x^3-21 x^4-3 x^5)) \log (\frac {16+8 x^2+x^4}{x^2})}{20+5 x^2} \, dx\) [6344]

Optimal result

Integrand size = 120, antiderivative size = 31 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=-x+\frac {3}{5} x^2 (4+x) \left (-e^x+x\right ) \log \left (\left (\frac {4}{x}+x\right )^2\right ) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(31)=62\).

Time = 2.88 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77, number of steps used = 152, number of rules used = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6857, 209, 327, 272, 45, 308, 2608, 2603, 12, 396, 2606, 457, 78, 4940, 2438, 5040, 4964, 2449, 2352, 2605, 455, 2604, 2465, 2441, 266, 2463, 2437, 2338, 2440, 470, 6820, 531, 2225, 2207, 2209, 2227, 2634} \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=-\frac {12}{5} e^x x^2 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-x \]

[In]

Int[(-20 - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + E^x*(96*x + 24*x^2 - 24*x^3 - 6*x^4) + (144*x^2 + 48*x^3 + 36*x
^4 + 12*x^5 + E^x*(-96*x - 84*x^2 - 36*x^3 - 21*x^4 - 3*x^5))*Log[(16 + 8*x^2 + x^4)/x^2])/(20 + 5*x^2),x]

[Out]

-x - (12*E^x*x^2*Log[(4 + x^2)^2/x^2])/5 + (12*x^3*Log[(4 + x^2)^2/x^2])/5 - (3*E^x*x^3*Log[(4 + x^2)^2/x^2])/
5 + (3*x^4*Log[(4 + x^2)^2/x^2])/5

Rule 12

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

Rule 45

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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

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

Rule 272

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

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 396

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

Rule 455

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 470

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

Rule 531

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2209

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

Rule 2225

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 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2437

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 2438

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 2440

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

Rule 2441

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2604

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b
*Log[c*RFx^p])^n/e), x] - Dist[b*n*(p/e), Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2606

Int[Log[(c_.)*(RFx_)^(n_.)]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*L
og[c*RFx^n], x] - Dist[n, Int[SimplifyIntegrand[u*(D[RFx, x]/RFx), x], x], x]] /; FreeQ[{c, d, e, n}, x] && Ra
tionalFunctionQ[RFx, x] &&  !PolynomialQ[RFx, x]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 6820

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

Rule 6857

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4}{4+x^2}-\frac {101 x^2}{5 \left (4+x^2\right )}-\frac {24 x^3}{5 \left (4+x^2\right )}+\frac {24 x^4}{5 \left (4+x^2\right )}+\frac {6 x^5}{5 \left (4+x^2\right )}+\frac {144 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {48 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {36 x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}+\frac {12 x^5 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{5 \left (4+x^2\right )}-\frac {3 e^x x \left (-32-8 x+8 x^2+2 x^3+32 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+28 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+12 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+7 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{5 \left (4+x^2\right )}\right ) \, dx \\ & = -\left (\frac {3}{5} \int \frac {e^x x \left (-32-8 x+8 x^2+2 x^3+32 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+28 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+12 x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+7 x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{4+x^2} \, dx\right )+\frac {6}{5} \int \frac {x^5}{4+x^2} \, dx+\frac {12}{5} \int \frac {x^5 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx-4 \int \frac {1}{4+x^2} \, dx-\frac {24}{5} \int \frac {x^3}{4+x^2} \, dx+\frac {24}{5} \int \frac {x^4}{4+x^2} \, dx+\frac {36}{5} \int \frac {x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx+\frac {48}{5} \int \frac {x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx-\frac {101}{5} \int \frac {x^2}{4+x^2} \, dx+\frac {144}{5} \int \frac {x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2} \, dx \\ & = -\frac {101 x}{5}-2 \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3}{5} \int \frac {e^x x \left (2 \left (-16-4 x+4 x^2+x^3\right )+\left (32+28 x+12 x^2+7 x^3+x^4\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right )}{4+x^2} \, dx+\frac {3}{5} \text {Subst}\left (\int \frac {x^2}{4+x} \, dx,x,x^2\right )+\frac {12}{5} \int \left (-4 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {16 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx-\frac {12}{5} \text {Subst}\left (\int \frac {x}{4+x} \, dx,x,x^2\right )+\frac {24}{5} \int \left (-4+x^2+\frac {16}{4+x^2}\right ) \, dx+\frac {36}{5} \int \left (-4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {16 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {48}{5} \int \left (x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {4 x \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {144}{5} \int \left (\log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )}{4+x^2}\right ) \, dx+\frac {404}{5} \int \frac {1}{4+x^2} \, dx \\ & = -\frac {197 x}{5}+\frac {8 x^3}{5}+\frac {192}{5} \tan ^{-1}\left (\frac {x}{2}\right )-\frac {3}{5} \int \left (\frac {2 e^x (-2+x) x (2+x) (4+x)}{4+x^2}+e^x x \left (8+7 x+x^2\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right ) \, dx+\frac {3}{5} \text {Subst}\left (\int \left (-4+x+\frac {16}{4+x}\right ) \, dx,x,x^2\right )+\frac {12}{5} \int x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx-\frac {12}{5} \text {Subst}\left (\int \left (1-\frac {4}{4+x}\right ) \, dx,x,x^2\right )+\frac {36}{5} \int x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx+\frac {384}{5} \int \frac {1}{4+x^2} \, dx \\ & = -\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {8 x^3}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \int \frac {2 x^3 \left (-4+x^2\right )}{4+x^2} \, dx-\frac {3}{5} \int e^x x \left (8+7 x+x^2\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \, dx-\frac {6}{5} \int \frac {e^x (-2+x) x (2+x) (4+x)}{4+x^2} \, dx-\frac {12}{5} \int \frac {2 x^2 \left (-4+x^2\right )}{4+x^2} \, dx \\ & = -\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {8 x^3}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} \int \frac {2 e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \frac {x^3 \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \frac {e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {24}{5} \int \frac {x^2 \left (-4+x^2\right )}{4+x^2} \, dx \\ & = -\frac {197 x}{5}-\frac {24 x^2}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \text {Subst}\left (\int \frac {(-4+x) x}{4+x} \, dx,x,x^2\right )+\frac {6}{5} \int \frac {e^x x (4+x) \left (-4+x^2\right )}{4+x^2} \, dx-\frac {6}{5} \int \left (-8 e^x+4 e^x x+e^x x^2+\frac {32 e^x (1-x)}{4+x^2}\right ) \, dx+\frac {192}{5} \int \frac {x^2}{4+x^2} \, dx \\ & = -x-\frac {24 x^2}{5}+\frac {3 x^4}{10}+\frac {384}{5} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {96}{5} \log \left (4+x^2\right )-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} \text {Subst}\left (\int \left (-8+x+\frac {32}{4+x}\right ) \, dx,x,x^2\right )-\frac {6}{5} \int e^x x^2 \, dx+\frac {6}{5} \int \left (-8 e^x+4 e^x x+e^x x^2+\frac {32 e^x (1-x)}{4+x^2}\right ) \, dx-\frac {24}{5} \int e^x x \, dx+\frac {48 \int e^x \, dx}{5}-\frac {192}{5} \int \frac {e^x (1-x)}{4+x^2} \, dx-\frac {768}{5} \int \frac {1}{4+x^2} \, dx \\ & = \frac {48 e^x}{5}-x-\frac {24 e^x x}{5}-\frac {6 e^x x^2}{5}-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {6}{5} \int e^x x^2 \, dx+\frac {12}{5} \int e^x x \, dx+\frac {24 \int e^x \, dx}{5}+\frac {24}{5} \int e^x x \, dx-\frac {48 \int e^x \, dx}{5}+\frac {192}{5} \int \frac {e^x (1-x)}{4+x^2} \, dx-\frac {192}{5} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{4}\right ) e^x}{2 i-x}-\frac {\left (\frac {1}{2}-\frac {i}{4}\right ) e^x}{2 i+x}\right ) \, dx \\ & = \frac {24 e^x}{5}-x+\frac {12 e^x x}{5}-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\left (-\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i+x} \, dx-\frac {12 \int e^x \, dx}{5}-\frac {12}{5} \int e^x x \, dx-\frac {24 \int e^x \, dx}{5}-\left (\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i-x} \, dx+\frac {192}{5} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{4}\right ) e^x}{2 i-x}-\frac {\left (\frac {1}{2}-\frac {i}{4}\right ) e^x}{2 i+x}\right ) \, dx \\ & = -\frac {12 e^x}{5}-x+\left (\frac {96}{5}+\frac {48 i}{5}\right ) e^{2 i} \text {Ei}(-2 i+x)+\left (\frac {96}{5}-\frac {48 i}{5}\right ) e^{-2 i} \text {Ei}(2 i+x)-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\left (-\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i+x} \, dx+\frac {12 \int e^x \, dx}{5}+\left (\frac {96}{5}+\frac {48 i}{5}\right ) \int \frac {e^x}{2 i-x} \, dx \\ & = -x-\frac {12}{5} e^x x^2 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {1}{5} \left (-5 x+3 x^2 (4+x) \left (-e^x+x\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right ) \]

[In]

Integrate[(-20 - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + E^x*(96*x + 24*x^2 - 24*x^3 - 6*x^4) + (144*x^2 + 48*x^3
+ 36*x^4 + 12*x^5 + E^x*(-96*x - 84*x^2 - 36*x^3 - 21*x^4 - 3*x^5))*Log[(16 + 8*x^2 + x^4)/x^2])/(20 + 5*x^2),
x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(28)=56\).

Time = 0.73 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.87

method	result	size
parallelrisch	\(\frac {3 \ln \left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{4}}{5}-\frac {3 \ln \left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) {\mathrm e}^{x} x^{3}}{5}+\frac {12 \ln \left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{3}}{5}-\frac {12 \ln \left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) {\mathrm e}^{x} x^{2}}{5}-x\)	\(89\)
risch	\(\text {Expression too large to display}\)	\(1157\)

[In]

int((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x^2)*ln((x^4+8*x^2+16)/x^2)+(-6*x^4-2
4*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x,method=_RETURNVERBOSE)

[Out]

3/5*ln((x^4+8*x^2+16)/x^2)*x^4-3/5*ln((x^4+8*x^2+16)/x^2)*exp(x)*x^3+12/5*ln((x^4+8*x^2+16)/x^2)*x^3-12/5*ln((
x^4+8*x^2+16)/x^2)*exp(x)*x^2-x

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {3}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x}\right )} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \]

[In]

integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x^2)*log((x^4+8*x^2+16)/x^2)+(-
6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=- x + \left (\frac {3 x^{4}}{5} + \frac {12 x^{3}}{5}\right ) \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} + \frac {\left (- 3 x^{3} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} - 12 x^{2} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )}\right ) e^{x}}{5} \]

[In]

integrate((((-3*x**5-21*x**4-36*x**3-84*x**2-96*x)*exp(x)+12*x**5+36*x**4+48*x**3+144*x**2)*ln((x**4+8*x**2+16
)/x**2)+(-6*x**4-24*x**3+24*x**2+96*x)*exp(x)+6*x**5+24*x**4-24*x**3-101*x**2-20)/(5*x**2+20),x)

[Out]

-x + (3*x**4/5 + 12*x**3/5)*log((x**4 + 8*x**2 + 16)/x**2) + (-3*x**3*log((x**4 + 8*x**2 + 16)/x**2) - 12*x**2
*log((x**4 + 8*x**2 + 16)/x**2))*exp(x)/5

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (28) = 56\).

Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {6}{5} \, {\left (x^{3} + 4 \, x^{2}\right )} e^{x} \log \left (x\right ) + \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x} - 16\right )} \log \left (x^{2} + 4\right ) - \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3}\right )} \log \left (x\right ) - x + \frac {96}{5} \, \log \left (x^{2} + 4\right ) \]

[In]

integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x^2)*log((x^4+8*x^2+16)/x^2)+(-
6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).

Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {3}{5} \, x^{4} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {3}{5} \, x^{3} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) + \frac {12}{5} \, x^{3} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {12}{5} \, x^{2} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \]

[In]

integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x^2)*log((x^4+8*x^2+16)/x^2)+(-
6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="giac")

[Out]

3/5*x^4*log((x^4 + 8*x^2 + 16)/x^2) - 3/5*x^3*e^x*log((x^4 + 8*x^2 + 16)/x^2) + 12/5*x^3*log((x^4 + 8*x^2 + 16
)/x^2) - 12/5*x^2*e^x*log((x^4 + 8*x^2 + 16)/x^2) - x

Mupad [B] (verification not implemented)

Time = 11.63 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\ln \left (\frac {x^4+8\,x^2+16}{x^2}\right )\,\left (\frac {12\,x^3}{5}-{\mathrm {e}}^x\,\left (\frac {3\,x^3}{5}+\frac {12\,x^2}{5}\right )+\frac {3\,x^4}{5}\right )-x \]

[In]

int((exp(x)*(96*x + 24*x^2 - 24*x^3 - 6*x^4) - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + log((8*x^2 + x^4 + 16)/x^2)
*(144*x^2 - exp(x)*(96*x + 84*x^2 + 36*x^3 + 21*x^4 + 3*x^5) + 48*x^3 + 36*x^4 + 12*x^5) - 20)/(5*x^2 + 20),x)

[Out]

log((8*x^2 + x^4 + 16)/x^2)*((12*x^3)/5 - exp(x)*((12*x^2)/5 + (3*x^3)/5) + (3*x^4)/5) - x