3.64.63 \(\int \frac {144 x+(144 x+\log ^2(2)) \log (x)+(144 x+\log ^2(2)) \log (\frac {144 x^2+x \log ^2(2)}{\log ^2(2)})}{144 x+\log ^2(2)} \, dx\)

Optimal. Leaf size=18 \[ x \left (-2+\log (x)+\log \left (x+\frac {144 x^2}{\log ^2(2)}\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 6, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {6688, 43, 2295, 2487, 31, 8} \begin {gather*} -2 x+x \log \left (x \left (\frac {144 x}{\log ^2(2)}+1\right )\right )+x \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 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 2295

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

Rule 2487

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.22 \begin {gather*} -2 x+x \log (x)+x \log \left (x+\frac {144 x^2}{\log ^2(2)}\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.68, size = 28, normalized size = 1.56 \begin {gather*} x \log \relax (x) + x \log \left (\frac {x \log \relax (2)^{2} + 144 \, x^{2}}{\log \relax (2)^{2}}\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.28, size = 25, normalized size = 1.39 \begin {gather*} -2 \, x {\left (\log \left (\log \relax (2)\right ) + 1\right )} + x \log \left (\log \relax (2)^{2} + 144 \, x\right ) + 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.70, size = 28, normalized size = 1.56

method	result	size
default	\(-2 x \ln \left (\ln \relax (2)\right )+x \ln \relax (x )+x \ln \left (x \left (\ln \relax (2)^{2}+144 x \right )\right )-2 x\)	\(28\)
norman	\(x \ln \relax (x )+x \ln \left (\frac {x \ln \relax (2)^{2}+144 x^{2}}{\ln \relax (2)^{2}}\right )-2 x\)	\(29\)
risch	\(x \ln \left (\ln \relax (2)^{2}+144 x \right )+2 x \ln \relax (x )-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (2)^{2}+144 x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)^{2}+144 x \right )\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)^{2}+144 x \right )\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (2)^{2}+144 x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)^{2}+144 x \right )\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x \left (\ln \relax (2)^{2}+144 x \right )\right )^{3}}{2}-2 x \ln \left (\ln \relax (2)\right )-2 x\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 0.51, size = 51, normalized size = 2.83 \begin {gather*} -\frac {1}{144} \, \log \relax (2)^{2} \log \left (\log \relax (2)^{2} + 144 \, x\right ) - x {\left (2 \, \log \left (\log \relax (2)\right ) + 3\right )} + \frac {1}{144} \, {\left (\log \relax (2)^{2} + 144 \, x\right )} \log \left (\log \relax (2)^{2} + 144 \, x\right ) + 2 \, x \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 4.39, size = 29, normalized size = 1.61 \begin {gather*} x\,\ln \left (144\,x^2+{\ln \relax (2)}^2\,x\right )-2\,x-2\,x\,\ln \left (\ln \relax (2)\right )+x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.77, size = 54, normalized size = 3.00 \begin {gather*} x \log {\relax (x )} - 2 x + \left (x + \frac {\log {\relax (2 )}^{2}}{864}\right ) \log {\left (\frac {144 x^{2} + x \log {\relax (2 )}^{2}}{\log {\relax (2 )}^{2}} \right )} - \frac {\log {\relax (2 )}^{2} \log {\left (144 x^{2} + x \log {\relax (2 )}^{2} \right )}}{864} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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