\(\int \frac {144 x+(144 x+\log ^2(2)) \log (x)+(144 x+\log ^2(2)) \log (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)})}{144 x+\log ^2(2)} \, dx\) [6363]

Optimal result

Integrand size = 53, antiderivative size = 18 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=x \left (-2+\log (x)+\log \left (x+\frac {144 x^2}{\log ^2(2)}\right )\right ) \]

[Out]

x*(ln(x)-2+ln(144*x^2/ln(2)^2+x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {6820, 45, 2332, 2579, 31, 8} \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=-2 x+x \log \left (x \left (\frac {144 x}{\log ^2(2)}+1\right )\right )+x \log (x) \]

[In]

Int[(144*x + (144*x + Log[2]^2)*Log[x] + (144*x + Log[2]^2)*Log[(144*x^2 + x*Log[2]^2)/Log[2]^2])/(144*x + Log
[2]^2),x]

[Out]

-2*x + x*Log[x] + x*Log[x*(1 + (144*x)/Log[2]^2)]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, 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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2579

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[(a
+ b*x)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b), x] + (Dist[q*r*s*((b*c - a*d)/b), Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

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 (\frac {144 x}{144 x+\log ^2(2)}+\log (x)+\log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right )\right ) \, dx \\ & = 144 \int \frac {x}{144 x+\log ^2(2)} \, dx+\int \log (x) \, dx+\int \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right ) \, dx \\ & = -x+x \log (x)+x \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right )-2 \int 1 \, dx+144 \int \left (\frac {1}{144}-\frac {\log ^2(2)}{144 \left (144 x+\log ^2(2)\right )}\right ) \, dx+\int \frac {1}{1+\frac {144 x}{\log ^2(2)}} \, dx \\ & = -2 x+x \log (x)+x \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=-2 x+x \log (x)+x \log \left (x+\frac {144 x^2}{\log ^2(2)}\right ) \]

[In]

Integrate[(144*x + (144*x + Log[2]^2)*Log[x] + (144*x + Log[2]^2)*Log[(144*x^2 + x*Log[2]^2)/Log[2]^2])/(144*x
+ Log[2]^2),x]

[Out]

-2*x + x*Log[x] + x*Log[x + (144*x^2)/Log[2]^2]

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56

method	result	size
default	\(-2 x \ln \left (\ln \left (2\right )\right )+x \ln \left (x \right )+x \ln \left (x \left (\ln \left (2\right )^{2}+144 x \right )\right )-2 x\)	\(28\)
parts	\(-2 x \ln \left (\ln \left (2\right )\right )+x \ln \left (x \right )+x \ln \left (x \left (\ln \left (2\right )^{2}+144 x \right )\right )-2 x\)	\(28\)
norman	\(x \ln \left (x \right )+x \ln \left (\frac {x \ln \left (2\right )^{2}+144 x^{2}}{\ln \left (2\right )^{2}}\right )-2 x\)	\(29\)
parallelrisch	\(\frac {\ln \left (2\right )^{2} \ln \left (\frac {\ln \left (2\right )^{2}}{144}+x \right )}{72}+\frac {\ln \left (x \right ) \ln \left (2\right )^{2}}{72}-\frac {\ln \left (\frac {x \left (\ln \left (2\right )^{2}+144 x \right )}{\ln \left (2\right )^{2}}\right ) \ln \left (2\right )^{2}}{72}+\frac {\ln \left (2\right )^{2}}{36}+x \ln \left (x \right )+\ln \left (\frac {x \left (\ln \left (2\right )^{2}+144 x \right )}{\ln \left (2\right )^{2}}\right ) x -2 x\)	\(76\)
risch	\(x \ln \left (\ln \left (2\right )^{2}+144 x \right )+2 x \ln \left (x \right )-\frac {i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )^{2}+144 x \right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )}^{2}}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (2\right )^{2}+144 x \right )\right ) {\operatorname {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )}^{3}}{2}-2 x \ln \left (\ln \left (2\right )\right )-2 x\)	\(139\)

[In]

int(((ln(2)^2+144*x)*ln((x*ln(2)^2+144*x^2)/ln(2)^2)+(ln(2)^2+144*x)*ln(x)+144*x)/(ln(2)^2+144*x),x,method=_RE
TURNVERBOSE)

[Out]

-2*x*ln(ln(2))+x*ln(x)+x*ln(x*(ln(2)^2+144*x))-2*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=x \log \left (x\right ) + x \log \left (\frac {x \log \left (2\right )^{2} + 144 \, x^{2}}{\log \left (2\right )^{2}}\right ) - 2 \, x \]

[In]

integrate(((log(2)^2+144*x)*log((x*log(2)^2+144*x^2)/log(2)^2)+(log(2)^2+144*x)*log(x)+144*x)/(log(2)^2+144*x)
,x, algorithm="fricas")

[Out]

x*log(x) + x*log((x*log(2)^2 + 144*x^2)/log(2)^2) - 2*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).

Time = 0.46 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.00 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=x \log {\left (x \right )} - 2 x + \left (x + \frac {\log {\left (2 \right )}^{2}}{864}\right ) \log {\left (\frac {144 x^{2} + x \log {\left (2 \right )}^{2}}{\log {\left (2 \right )}^{2}} \right )} - \frac {\log {\left (2 \right )}^{2} \log {\left (144 x^{2} + x \log {\left (2 \right )}^{2} \right )}}{864} \]

[In]

integrate(((ln(2)**2+144*x)*ln((x*ln(2)**2+144*x**2)/ln(2)**2)+(ln(2)**2+144*x)*ln(x)+144*x)/(ln(2)**2+144*x),
x)

[Out]

x*log(x) - 2*x + (x + log(2)**2/864)*log((144*x**2 + x*log(2)**2)/log(2)**2) - log(2)**2*log(144*x**2 + x*log(
2)**2)/864

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (18) = 36\).

Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=-\frac {1}{144} \, \log \left (2\right )^{2} \log \left (\log \left (2\right )^{2} + 144 \, x\right ) - x {\left (2 \, \log \left (\log \left (2\right )\right ) + 3\right )} + \frac {1}{144} \, {\left (\log \left (2\right )^{2} + 144 \, x\right )} \log \left (\log \left (2\right )^{2} + 144 \, x\right ) + 2 \, x \log \left (x\right ) + x \]

[In]

integrate(((log(2)^2+144*x)*log((x*log(2)^2+144*x^2)/log(2)^2)+(log(2)^2+144*x)*log(x)+144*x)/(log(2)^2+144*x)
,x, algorithm="maxima")

[Out]

-1/144*log(2)^2*log(log(2)^2 + 144*x) - x*(2*log(log(2)) + 3) + 1/144*(log(2)^2 + 144*x)*log(log(2)^2 + 144*x)
+ 2*x*log(x) + x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=-2 \, x {\left (\log \left (\log \left (2\right )\right ) + 1\right )} + x \log \left (\log \left (2\right )^{2} + 144 \, x\right ) + 2 \, x \log \left (x\right ) \]

[In]

integrate(((log(2)^2+144*x)*log((x*log(2)^2+144*x^2)/log(2)^2)+(log(2)^2+144*x)*log(x)+144*x)/(log(2)^2+144*x)
,x, algorithm="giac")

[Out]

-2*x*(log(log(2)) + 1) + x*log(log(2)^2 + 144*x) + 2*x*log(x)

Mupad [B] (verification not implemented)

Time = 13.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {144 x+\left (144 x+\log ^2(2)\right ) \log (x)+\left (144 x+\log ^2(2)\right ) \log \left (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)}\right )}{144 x+\log ^2(2)} \, dx=x\,\ln \left (144\,x^2+{\ln \left (2\right )}^2\,x\right )-2\,x-2\,x\,\ln \left (\ln \left (2\right )\right )+x\,\ln \left (x\right ) \]

[In]

int((144*x + log((x*log(2)^2 + 144*x^2)/log(2)^2)*(144*x + log(2)^2) + log(x)*(144*x + log(2)^2))/(144*x + log
(2)^2),x)

[Out]

x*log(x*log(2)^2 + 144*x^2) - 2*x - 2*x*log(log(2)) + x*log(x)