\(\int \frac {-x+(10+x) \log (10+x)+(20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx\) [2335]

Optimal result

Integrand size = 60, antiderivative size = 17 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=1+\left (x+x^4\right )^2+\frac {x}{\log (10+x)} \]

[Out]

1+(x^4+x)^2+x/ln(x+10)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6820, 14, 2458, 2395, 2334, 2335, 2339, 30, 2436} \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=x^8+2 x^5+x^2+\frac {x+10}{\log (x+10)}-\frac {10}{\log (x+10)} \]

[In]

Int[(-x + (10 + x)*Log[10 + x] + (20*x + 2*x^2 + 100*x^4 + 10*x^5 + 80*x^7 + 8*x^8)*Log[10 + x]^2)/((10 + x)*L
og[10 + x]^2),x]

[Out]

x^2 + 2*x^5 + x^8 - 10/Log[10 + x] + (10 + x)/Log[10 + x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 2334

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 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2339

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 2395

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 2436

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 2458

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \left (2 x \left (1+5 x^3+4 x^6\right )-\frac {x}{(10+x) \log ^2(10+x)}+\frac {1}{\log (10+x)}\right ) \, dx \\ & = 2 \int x \left (1+5 x^3+4 x^6\right ) \, dx-\int \frac {x}{(10+x) \log ^2(10+x)} \, dx+\int \frac {1}{\log (10+x)} \, dx \\ & = 2 \int \left (x+5 x^4+4 x^7\right ) \, dx-\text {Subst}\left (\int \frac {-10+x}{x \log ^2(x)} \, dx,x,10+x\right )+\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,10+x\right ) \\ & = x^2+2 x^5+x^8+\operatorname {LogIntegral}(10+x)-\text {Subst}\left (\int \left (\frac {1}{\log ^2(x)}-\frac {10}{x \log ^2(x)}\right ) \, dx,x,10+x\right ) \\ & = x^2+2 x^5+x^8+\operatorname {LogIntegral}(10+x)+10 \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,10+x\right )-\text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,10+x\right ) \\ & = x^2+2 x^5+x^8+\frac {10+x}{\log (10+x)}+\operatorname {LogIntegral}(10+x)+10 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (10+x)\right )-\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,10+x\right ) \\ & = x^2+2 x^5+x^8-\frac {10}{\log (10+x)}+\frac {10+x}{\log (10+x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=x^2+2 x^5+x^8+\frac {x}{\log (10+x)} \]

[In]

Integrate[(-x + (10 + x)*Log[10 + x] + (20*x + 2*x^2 + 100*x^4 + 10*x^5 + 80*x^7 + 8*x^8)*Log[10 + x]^2)/((10
+ x)*Log[10 + x]^2),x]

[Out]

x^2 + 2*x^5 + x^8 + x/Log[10 + x]

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24

method	result	size
risch	\(x^{8}+2 x^{5}+x^{2}+\frac {x}{\ln \left (x +10\right )}\)	\(21\)
parts	\(x^{8}+2 x^{5}+x^{2}+\frac {x +10}{\ln \left (x +10\right )}-\frac {10}{\ln \left (x +10\right )}\)	\(31\)
parallelrisch	\(-\frac {-20 \ln \left (x +10\right ) x^{8}-40 x^{5} \ln \left (x +10\right )-20 \ln \left (x +10\right ) x^{2}-20 x}{20 \ln \left (x +10\right )}\)	\(40\)
derivativedivides	\(\left (x +10\right )^{8}-80 \left (x +10\right )^{7}+2800 \left (x +10\right )^{6}-55998 \left (x +10\right )^{5}+699900 \left (x +10\right )^{4}-5598000 \left (x +10\right )^{3}[phone] \left (x +10\right )^{2}-79900020 x -799000200+\frac {x +10}{\ln \left (x +10\right )}-\frac {10}{\ln \left (x +10\right )}\)	\(71\)
default	\(\left (x +10\right )^{8}-80 \left (x +10\right )^{7}+2800 \left (x +10\right )^{6}-55998 \left (x +10\right )^{5}+699900 \left (x +10\right )^{4}-5598000 \left (x +10\right )^{3}[phone] \left (x +10\right )^{2}-79900020 x -799000200+\frac {x +10}{\ln \left (x +10\right )}-\frac {10}{\ln \left (x +10\right )}\)	\(71\)

[In]

int(((8*x^8+80*x^7+10*x^5+100*x^4+2*x^2+20*x)*ln(x+10)^2+(x+10)*ln(x+10)-x)/(x+10)/ln(x+10)^2,x,method=_RETURN
VERBOSE)

[Out]

x^8+2*x^5+x^2+x/ln(x+10)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=\frac {{\left (x^{8} + 2 \, x^{5} + x^{2}\right )} \log \left (x + 10\right ) + x}{\log \left (x + 10\right )} \]

[In]

integrate(((8*x^8+80*x^7+10*x^5+100*x^4+2*x^2+20*x)*log(x+10)^2+(x+10)*log(x+10)-x)/(x+10)/log(x+10)^2,x, algo
rithm="fricas")

[Out]

((x^8 + 2*x^5 + x^2)*log(x + 10) + x)/log(x + 10)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=x^{8} + 2 x^{5} + x^{2} + \frac {x}{\log {\left (x + 10 \right )}} \]

[In]

integrate(((8*x**8+80*x**7+10*x**5+100*x**4+2*x**2+20*x)*ln(x+10)**2+(x+10)*ln(x+10)-x)/(x+10)/ln(x+10)**2,x)

[Out]

x**8 + 2*x**5 + x**2 + x/log(x + 10)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=\frac {{\left (x^{8} + 2 \, x^{5} + x^{2}\right )} \log \left (x + 10\right ) + x}{\log \left (x + 10\right )} \]

[In]

integrate(((8*x^8+80*x^7+10*x^5+100*x^4+2*x^2+20*x)*log(x+10)^2+(x+10)*log(x+10)-x)/(x+10)/log(x+10)^2,x, algo
rithm="maxima")

[Out]

((x^8 + 2*x^5 + x^2)*log(x + 10) + x)/log(x + 10)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=x^{8} + 2 \, x^{5} + x^{2} + \frac {x}{\log \left (x + 10\right )} \]

[In]

integrate(((8*x^8+80*x^7+10*x^5+100*x^4+2*x^2+20*x)*log(x+10)^2+(x+10)*log(x+10)-x)/(x+10)/log(x+10)^2,x, algo
rithm="giac")

[Out]

x^8 + 2*x^5 + x^2 + x/log(x + 10)

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-x+(10+x) \log (10+x)+\left (20 x+2 x^2+100 x^4+10 x^5+80 x^7+8 x^8\right ) \log ^2(10+x)}{(10+x) \log ^2(10+x)} \, dx=\frac {x}{\ln \left (x+10\right )}+x^2+2\,x^5+x^8 \]

[In]

int((log(x + 10)^2*(20*x + 2*x^2 + 100*x^4 + 10*x^5 + 80*x^7 + 8*x^8) - x + log(x + 10)*(x + 10))/(log(x + 10)
^2*(x + 10)),x)

[Out]

x/log(x + 10) + x^2 + 2*x^5 + x^8