\(\int \frac {-10 x^7 \log (2)+(45 x^4+5 x^9) \log ^2(2)+(10 x^5+(-54 x^2-6 x^7) \log (2)) \log (9+x^5)+(9+x^5) \log ^2(9+x^5)}{(9+x^5) \log ^2(2)} \, dx\) [3971]

Optimal result

Integrand size = 77, antiderivative size = 21 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=x \left (-x^2+\frac {\log \left (9+x^5\right )}{\log (2)}\right )^2 \]

[Out]

x*(ln(x^5+9)/ln(2)-x^2)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(568\) vs. \(2(21)=42\).

Time = 6.71 (sec) , antiderivative size = 568, normalized size of antiderivative = 27.05, number of steps used = 223, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {12, 6857, 1850, 327, 299, 648, 632, 210, 642, 31, 2608, 2498, 207, 2521, 2512, 266, 2463, 2441, 2440, 2438, 2437, 2338, 2505, 2500, 2526} \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=-\frac {2 \sqrt [5]{3} \sqrt {10} \log (8) \arctan \left (\frac {3^{2/5} \left (1-\sqrt {5}\right )-4 x}{3^{2/5} \sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {5+\sqrt {5}} \log ^2(2)}+\frac {\sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \log (8) \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (3^{2/5} \left (1+\sqrt {5}\right )-4 x\right )}{2\ 3^{2/5}}\right )}{\log ^2(2)}+\frac {6 \sqrt [5]{3} \sqrt {10} \arctan \left (\frac {3^{2/5} \left (1-\sqrt {5}\right )-4 x}{3^{2/5} \sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{\sqrt {5+\sqrt {5}} \log (2)}-\frac {3 \sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (3^{2/5} \left (1+\sqrt {5}\right )-4 x\right )}{2\ 3^{2/5}}\right )}{\log (2)}+x^5+\frac {x \log ^2\left (x^5+9\right )}{\log ^2(2)}+\frac {10 x^3 \log (8)}{9 \log ^2(2)}-\frac {10 x^3}{3 \log (2)}+\frac {\sqrt [5]{3} \left (1+\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log ^2(2)}+\frac {\sqrt [5]{3} \left (1-\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log ^2(2)}-\frac {3 \sqrt [5]{3} \left (1+\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log (2)}-\frac {3 \sqrt [5]{3} \left (1-\sqrt {5}\right ) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )}{2 \log (2)}-\frac {2 x^3 \log (8) \log \left (x^5+9\right )}{3 \log ^2(2)}-\frac {2 \sqrt [5]{3} \log (8) \log \left (x+3^{2/5}\right )}{\log ^2(2)}+\frac {6 \sqrt [5]{3} \log \left (x+3^{2/5}\right )}{\log (2)} \]

[In]

Int[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x^7)*Log[2])*Log[9 + x^5] + (9 + x^5)
*Log[9 + x^5]^2)/((9 + x^5)*Log[2]^2),x]

[Out]

x^5 - (10*x^3)/(3*Log[2]) + (6*3^(1/5)*Sqrt[10]*ArcTan[(3^(2/5)*(1 - Sqrt[5]) - 4*x)/(3^(2/5)*Sqrt[2*(5 + Sqrt
[5])])])/(Sqrt[5 + Sqrt[5]]*Log[2]) - (3*3^(1/5)*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(3^(2/5)
*(1 + Sqrt[5]) - 4*x))/(2*3^(2/5))])/Log[2] + (10*x^3*Log[8])/(9*Log[2]^2) - (2*3^(1/5)*Sqrt[10]*ArcTan[(3^(2/
5)*(1 - Sqrt[5]) - 4*x)/(3^(2/5)*Sqrt[2*(5 + Sqrt[5])])]*Log[8])/(Sqrt[5 + Sqrt[5]]*Log[2]^2) + (3^(1/5)*Sqrt[
2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(3^(2/5)*(1 + Sqrt[5]) - 4*x))/(2*3^(2/5))]*Log[8])/Log[2]^2 +
(6*3^(1/5)*Log[3^(2/5) + x])/Log[2] - (2*3^(1/5)*Log[8]*Log[3^(2/5) + x])/Log[2]^2 - (3*3^(1/5)*(1 + Sqrt[5])
*Log[3^(4/5) - (3^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/(2*Log[2]) + (3^(1/5)*(1 + Sqrt[5])*Log[8]*Log[3^(4/5) - (3
^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/(2*Log[2]^2) - (3*3^(1/5)*(1 - Sqrt[5])*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])
*x)/2 + x^2])/(2*Log[2]) + (3^(1/5)*(1 - Sqrt[5])*Log[8]*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])*x)/2 + x^2])/(2*
Log[2]^2) - (2*x^3*Log[8]*Log[9 + x^5])/(3*Log[2]^2) + (x*Log[9 + x^5]^2)/Log[2]^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 31

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

Rule 207

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; (r/
(a*n))*Int[1/(r + s*x), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 299

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(
2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; (-(-r)^(m + 1)/(a*n*s^m))*Int[1/(r + s*x), x] + Dist[2*(r^(m + 1)/(a*n*s^m)
), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1]
&& PosQ[a/b]

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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1850

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)
/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

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 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 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 2498

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

Rule 2500

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

Rule 2505

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

Rule 2512

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

Rule 2521

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2526

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

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 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}& = \frac {\int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)} \\ & = \frac {\int \left (\frac {5 x^4 \log (2) \left (-2 x^3+x^5 \log (2)+\log (512)\right )}{9+x^5}-\frac {2 x^2 \left (-5 x^3+27 \log (2)+x^5 \log (8)\right ) \log \left (9+x^5\right )}{9+x^5}+\log ^2\left (9+x^5\right )\right ) \, dx}{\log ^2(2)} \\ & = \frac {\int \log ^2\left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {2 \int \frac {x^2 \left (-5 x^3+27 \log (2)+x^5 \log (8)\right ) \log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}+\frac {5 \int \frac {x^4 \left (-2 x^3+x^5 \log (2)+\log (512)\right )}{9+x^5} \, dx}{\log (2)} \\ & = x^5+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}-\frac {2 \int \left (-5 \log \left (9+x^5\right )+\frac {45 \log \left (9+x^5\right )}{9+x^5}+x^2 \log (8) \log \left (9+x^5\right )\right ) \, dx}{\log ^2(2)}-\frac {10 \int \frac {x^5 \log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}+\frac {\int -\frac {10 x^7}{9+x^5} \, dx}{\log (2)} \\ & = x^5+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}+\frac {10 \int \log \left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {10 \int \left (\log \left (9+x^5\right )-\frac {9 \log \left (9+x^5\right )}{9+x^5}\right ) \, dx}{\log ^2(2)}-\frac {90 \int \frac {\log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}-\frac {10 \int \frac {x^7}{9+x^5} \, dx}{\log (2)}-\frac {(2 \log (8)) \int x^2 \log \left (9+x^5\right ) \, dx}{\log ^2(2)} \\ & = x^5-\frac {10 x^3}{3 \log (2)}+\frac {10 x \log \left (9+x^5\right )}{\log ^2(2)}-\frac {2 x^3 \log (8) \log \left (9+x^5\right )}{3 \log ^2(2)}+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}-\frac {10 \int \log \left (9+x^5\right ) \, dx}{\log ^2(2)}-\frac {50 \int \frac {x^5}{9+x^5} \, dx}{\log ^2(2)}+\frac {90 \int \frac {\log \left (9+x^5\right )}{9+x^5} \, dx}{\log ^2(2)}-\frac {90 \int \left (-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+\sqrt [5]{-1} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{2/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+(-1)^{3/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{4/5} x\right )}\right ) \, dx}{\log ^2(2)}+\frac {90 \int \frac {x^2}{9+x^5} \, dx}{\log (2)}+\frac {(10 \log (8)) \int \frac {x^7}{9+x^5} \, dx}{3 \log ^2(2)} \\ & = x^5-\frac {50 x}{\log ^2(2)}-\frac {10 x^3}{3 \log (2)}+\frac {10 x^3 \log (8)}{9 \log ^2(2)}-\frac {2 x^3 \log (8) \log \left (9+x^5\right )}{3 \log ^2(2)}+\frac {x \log ^2\left (9+x^5\right )}{\log ^2(2)}+\frac {50 \int \frac {x^5}{9+x^5} \, dx}{\log ^2(2)}+\frac {90 \int \left (-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+\sqrt [5]{-1} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{2/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}+(-1)^{3/5} x\right )}-\frac {\log \left (9+x^5\right )}{15\ 3^{3/5} \left (-3^{2/5}-(-1)^{4/5} x\right )}\right ) \, dx}{\log ^2(2)}+\frac {450 \int \frac {1}{9+x^5} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}+\sqrt [5]{-1} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-(-1)^{2/5} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}+(-1)^{3/5} x} \, dx}{\log ^2(2)}+\frac {\left (2\ 3^{2/5}\right ) \int \frac {\log \left (9+x^5\right )}{-3^{2/5}-(-1)^{4/5} x} \, dx}{\log ^2(2)}+\frac {\left (6 \sqrt [5]{3}\right ) \int \frac {1}{3^{2/5}+x} \, dx}{\log (2)}+\frac {\left (12 \sqrt [5]{3}\right ) \int \frac {\frac {1}{4} 3^{2/5} \left (-1-\sqrt {5}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) x}{3^{4/5}-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+x^2} \, dx}{\log (2)}+\frac {\left (12 \sqrt [5]{3}\right ) \int \frac {\frac {1}{4} 3^{2/5} \left (-1+\sqrt {5}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) x}{3^{4/5}-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+x^2} \, dx}{\log (2)}-\frac {(30 \log (8)) \int \frac {x^2}{9+x^5} \, dx}{\log ^2(2)} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x \left (x^4 \log ^2(2)-x^2 \log (4) \log \left (9+x^5\right )+\log ^2\left (9+x^5\right )\right )}{\log ^2(2)} \]

[In]

Integrate[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x^7)*Log[2])*Log[9 + x^5] + (9
+ x^5)*Log[9 + x^5]^2)/((9 + x^5)*Log[2]^2),x]

[Out]

(x*(x^4*Log[2]^2 - x^2*Log[4]*Log[9 + x^5] + Log[9 + x^5]^2))/Log[2]^2

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62

method	result	size
risch	\(x^{5}-\frac {2 x^{3} \ln \left (x^{5}+9\right )}{\ln \left (2\right )}+\frac {x \ln \left (x^{5}+9\right )^{2}}{\ln \left (2\right )^{2}}\)	\(34\)
parallelrisch	\(\frac {x^{5} \ln \left (2\right )^{2}-2 \ln \left (2\right ) x^{3} \ln \left (x^{5}+9\right )+x \ln \left (x^{5}+9\right )^{2}-18 \ln \left (2\right )^{2}}{\ln \left (2\right )^{2}}\)	\(44\)

[In]

int(((x^5+9)*ln(x^5+9)^2+((-6*x^7-54*x^2)*ln(2)+10*x^5)*ln(x^5+9)+(5*x^9+45*x^4)*ln(2)^2-10*x^7*ln(2))/(x^5+9)
/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

x^5-2/ln(2)*x^3*ln(x^5+9)+1/ln(2)^2*x*ln(x^5+9)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x^{5} \log \left (2\right )^{2} - 2 \, x^{3} \log \left (2\right ) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \left (2\right )^{2}} \]

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="fricas")

[Out]

(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=x^{5} - \frac {2 x^{3} \log {\left (x^{5} + 9 \right )}}{\log {\left (2 \right )}} + \frac {x \log {\left (x^{5} + 9 \right )}^{2}}{\log {\left (2 \right )}^{2}} \]

[In]

integrate(((x**5+9)*ln(x**5+9)**2+((-6*x**7-54*x**2)*ln(2)+10*x**5)*ln(x**5+9)+(5*x**9+45*x**4)*ln(2)**2-10*x*
*7*ln(2))/(x**5+9)/ln(2)**2,x)

[Out]

x**5 - 2*x**3*log(x**5 + 9)/log(2) + x*log(x**5 + 9)**2/log(2)**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imaginary; found sqrt(sqrt(5)-5)

Giac [A] (verification not implemented)

none

Time = 0.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x^{5} \log \left (2\right )^{2} - 2 \, x^{3} \log \left (2\right ) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \left (2\right )^{2}} \]

[In]

integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^9+45*x^4)*log(2)^2-10*x^7*log(
2))/(x^5+9)/log(2)^2,x, algorithm="giac")

[Out]

(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x\,{\left (\ln \left (x^5+9\right )-x^2\,\ln \left (2\right )\right )}^2}{{\ln \left (2\right )}^2} \]

[In]

int(-(log(x^5 + 9)*(log(2)*(54*x^2 + 6*x^7) - 10*x^5) - log(x^5 + 9)^2*(x^5 + 9) + 10*x^7*log(2) - log(2)^2*(4
5*x^4 + 5*x^9))/(log(2)^2*(x^5 + 9)),x)

[Out]

(x*(log(x^5 + 9) - x^2*log(2))^2)/log(2)^2