3.36.55 \(\int \frac {-144 x+(431+144 x+144 \log (2)) \log (\frac {1}{9} (431+144 x+144 \log (2)))}{(431+144 x+144 \log (2)) \log ^2(\frac {1}{9} (431+144 x+144 \log (2)))} \, dx\)

Optimal. Leaf size=14 \[ \frac {x}{\log \left (16 \left (\frac {431}{144}+x+\log (2)\right )\right )} \]

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Rubi [B]  time = 0.41, antiderivative size = 58, normalized size of antiderivative = 4.14, number of steps used = 14, number of rules used = 11, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.196, Rules used = {6741, 6688, 6742, 2411, 12, 2353, 2297, 2298, 2302, 30, 2389} \begin {gather*} \frac {144 x+431+144 \log (2)}{144 \log \left (16 x+\frac {1}{9} (431+144 \log (2))\right )}-\frac {431+144 \log (2)}{144 \log \left (16 x+\frac {1}{9} (431+144 \log (2))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-144*x + (431 + 144*x + 144*Log[2])*Log[(431 + 144*x + 144*Log[2])/9])/((431 + 144*x + 144*Log[2])*Log[(4
31 + 144*x + 144*Log[2])/9]^2),x]

[Out]

-1/144*(431 + 144*Log[2])/Log[16*x + (431 + 144*Log[2])/9] + (431 + 144*x + 144*Log[2])/(144*Log[16*x + (431 +
144*Log[2])/9])

Rule 12

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

Rule 30

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

Rule 2297

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 2298

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

Rule 2302

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 2353

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 2389

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 2411

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 6688

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

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-144 x+(431+144 x+144 \log (2)) \log \left (\frac {1}{9} (431+144 x+144 \log (2))\right )}{(431+144 x+144 \log (2)) \log ^2\left (16 x+\frac {1}{9} (431+144 \log (2))\right )} \, dx\\ &=\int \frac {-\frac {144 x}{431+144 x+144 \log (2)}+\log \left (\frac {431}{9}+16 x+16 \log (2)\right )}{\log ^2\left (\frac {431}{9}+16 x+16 \log (2)\right )} \, dx\\ &=\int \left (-\frac {144 x}{(431+144 x+144 \log (2)) \log ^2\left (\frac {431}{9}+16 x+16 \log (2)\right )}+\frac {1}{\log \left (\frac {431}{9}+16 x+16 \log (2)\right )}\right ) \, dx\\ &=-\left (144 \int \frac {x}{(431+144 x+144 \log (2)) \log ^2\left (\frac {431}{9}+16 x+16 \log (2)\right )} \, dx\right )+\int \frac {1}{\log \left (\frac {431}{9}+16 x+16 \log (2)\right )} \, dx\\ &=\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )-9 \operatorname {Subst}\left (\int \frac {\frac {x}{16}+\frac {1}{16} \left (-\frac {431}{9}-16 \log (2)\right )}{9 x \log ^2(x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )\\ &=\frac {1}{16} \text {li}\left (16 x+\frac {1}{9} (431+144 \log (2))\right )-\operatorname {Subst}\left (\int \frac {\frac {x}{16}+\frac {1}{16} \left (-\frac {431}{9}-16 \log (2)\right )}{x \log ^2(x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )\\ &=\frac {1}{16} \text {li}\left (16 x+\frac {1}{9} (431+144 \log (2))\right )-\operatorname {Subst}\left (\int \left (\frac {1}{16 \log ^2(x)}+\frac {-431-144 \log (2)}{144 x \log ^2(x)}\right ) \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )\\ &=\frac {1}{16} \text {li}\left (16 x+\frac {1}{9} (431+144 \log (2))\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )-\frac {1}{144} (-431-144 \log (2)) \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )\\ &=\frac {431+144 x+144 \log (2)}{144 \log \left (16 x+\frac {1}{9} (431+144 \log (2))\right )}+\frac {1}{16} \text {li}\left (16 x+\frac {1}{9} (431+144 \log (2))\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {431}{9}+16 x+16 \log (2)\right )-\frac {1}{144} (-431-144 \log (2)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {431}{9}+16 x+16 \log (2)\right )\right )\\ &=-\frac {431+144 \log (2)}{144 \log \left (16 x+\frac {1}{9} (431+144 \log (2))\right )}+\frac {431+144 x+144 \log (2)}{144 \log \left (16 x+\frac {1}{9} (431+144 \log (2))\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 16, normalized size = 1.14 \begin {gather*} \frac {x}{\log \left (\frac {431}{9}+16 x+16 \log (2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-144*x + (431 + 144*x + 144*Log[2])*Log[(431 + 144*x + 144*Log[2])/9])/((431 + 144*x + 144*Log[2])*
Log[(431 + 144*x + 144*Log[2])/9]^2),x]

[Out]

x/Log[431/9 + 16*x + 16*Log[2]]

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fricas [A]  time = 1.11, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (16 \, x + 16 \, \log \relax (2) + \frac {431}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*log(2)+144*x+431)*log(16*log(2)+16*x+431/9)-144*x)/(144*log(2)+144*x+431)/log(16*log(2)+16*x+4
31/9)^2,x, algorithm="fricas")

[Out]

x/log(16*x + 16*log(2) + 431/9)

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giac [A]  time = 0.17, size = 22, normalized size = 1.57 \begin {gather*} -\frac {x}{2 \, \log \relax (3) - \log \left (144 \, x + 144 \, \log \relax (2) + 431\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*log(2)+144*x+431)*log(16*log(2)+16*x+431/9)-144*x)/(144*log(2)+144*x+431)/log(16*log(2)+16*x+4
31/9)^2,x, algorithm="giac")

[Out]

-x/(2*log(3) - log(144*x + 144*log(2) + 431))

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maple [A]  time = 0.08, size = 15, normalized size = 1.07

method	result	size
norman	\(\frac {x}{\ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}\)	\(15\)
risch	\(\frac {x}{\ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}\)	\(15\)
derivativedivides	\(\frac {16 \ln \relax (2)+16 x +\frac {431}{9}}{16 \ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}-\frac {\ln \relax (2)}{\ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}-\frac {431}{144 \ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}\)	\(55\)
default	\(\frac {16 \ln \relax (2)+16 x +\frac {431}{9}}{16 \ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}-\frac {\ln \relax (2)}{\ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}-\frac {431}{144 \ln \left (16 \ln \relax (2)+16 x +\frac {431}{9}\right )}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((144*ln(2)+144*x+431)*ln(16*ln(2)+16*x+431/9)-144*x)/(144*ln(2)+144*x+431)/ln(16*ln(2)+16*x+431/9)^2,x,me
thod=_RETURNVERBOSE)

[Out]

x/ln(16*ln(2)+16*x+431/9)

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maxima [A]  time = 0.75, size = 22, normalized size = 1.57 \begin {gather*} -\frac {x}{2 \, \log \relax (3) - \log \left (144 \, x + 144 \, \log \relax (2) + 431\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*log(2)+144*x+431)*log(16*log(2)+16*x+431/9)-144*x)/(144*log(2)+144*x+431)/log(16*log(2)+16*x+4
31/9)^2,x, algorithm="maxima")

[Out]

-x/(2*log(3) - log(144*x + 144*log(2) + 431))

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mupad [B]  time = 2.41, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{\ln \left (16\,x+16\,\ln \relax (2)+\frac {431}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(144*x - log(16*x + 16*log(2) + 431/9)*(144*x + 144*log(2) + 431))/(log(16*x + 16*log(2) + 431/9)^2*(144*
x + 144*log(2) + 431)),x)

[Out]

x/log(16*x + 16*log(2) + 431/9)

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sympy [A]  time = 0.11, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{\log {\left (16 x + 16 \log {\relax (2 )} + \frac {431}{9} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((144*ln(2)+144*x+431)*ln(16*ln(2)+16*x+431/9)-144*x)/(144*ln(2)+144*x+431)/ln(16*ln(2)+16*x+431/9)*
*2,x)

[Out]

x/log(16*x + 16*log(2) + 431/9)

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