3.34.81 \(\int \frac {(648+792 x+242 x^2) \log (x)+(396 x+242 x^2) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x)))}{9 x} \, dx\)

Optimal. Leaf size=23 \[ \left (6-e^{\log ^2(\log (x))}+\frac {11 x}{3}\right )^2 \log ^2(x) \]

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Rubi [B]  time = 0.46, antiderivative size = 104, normalized size of antiderivative = 4.52, number of steps used = 16, number of rules used = 13, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.141, Rules used = {12, 14, 6742, 43, 2313, 2334, 2301, 2330, 2296, 2295, 2305, 2304, 2288} \begin {gather*} \frac {121}{9} x^2 \log ^2(x)-\frac {121}{9} x^2 \log (x)+\frac {1}{9} \log (x) \left (121 x^2+792 x+648 \log (x)\right )+44 x \log ^2(x)+e^{2 \log ^2(\log (x))} \log ^2(x)-36 \log ^2(x)-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (11 x \log (\log (x))+18 \log (\log (x)))}{3 \log (\log (x))}-88 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((648 + 792*x + 242*x^2)*Log[x] + (396*x + 242*x^2)*Log[x]^2 + E^(2*Log[Log[x]]^2)*(18*Log[x] + 36*Log[x]*
Log[Log[x]]) + E^Log[Log[x]]^2*((-216 - 132*x)*Log[x] - 66*x*Log[x]^2 + (-216 - 132*x)*Log[x]*Log[Log[x]]))/(9
*x),x]

[Out]

-88*x*Log[x] - (121*x^2*Log[x])/9 - 36*Log[x]^2 + E^(2*Log[Log[x]]^2)*Log[x]^2 + 44*x*Log[x]^2 + (121*x^2*Log[
x]^2)/9 + (Log[x]*(792*x + 121*x^2 + 648*Log[x]))/9 - (2*E^Log[Log[x]]^2*Log[x]^2*(18*Log[Log[x]] + 11*x*Log[L
og[x]]))/(3*Log[Log[x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

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 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{x} \, dx\\ &=\frac {1}{9} \int \left (\frac {2 (18+11 x) \log (x) (18+11 x+11 x \log (x))}{x}+\frac {18 e^{2 \log ^2(\log (x))} \log (x) (1+2 \log (\log (x)))}{x}-\frac {6 e^{\log ^2(\log (x))} \log (x) (36+22 x+11 x \log (x)+36 \log (\log (x))+22 x \log (\log (x)))}{x}\right ) \, dx\\ &=\frac {2}{9} \int \frac {(18+11 x) \log (x) (18+11 x+11 x \log (x))}{x} \, dx-\frac {2}{3} \int \frac {e^{\log ^2(\log (x))} \log (x) (36+22 x+11 x \log (x)+36 \log (\log (x))+22 x \log (\log (x)))}{x} \, dx+2 \int \frac {e^{2 \log ^2(\log (x))} \log (x) (1+2 \log (\log (x)))}{x} \, dx\\ &=e^{2 \log ^2(\log (x))} \log ^2(x)-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}+\frac {2}{9} \int \left (\frac {(18+11 x)^2 \log (x)}{x}+11 (18+11 x) \log ^2(x)\right ) \, dx\\ &=e^{2 \log ^2(\log (x))} \log ^2(x)-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}+\frac {2}{9} \int \frac {(18+11 x)^2 \log (x)}{x} \, dx+\frac {22}{9} \int (18+11 x) \log ^2(x) \, dx\\ &=e^{2 \log ^2(\log (x))} \log ^2(x)+\frac {1}{9} \log (x) \left (792 x+121 x^2+648 \log (x)\right )-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}-\frac {2}{9} \int \left (396+\frac {121 x}{2}+\frac {324 \log (x)}{x}\right ) \, dx+\frac {22}{9} \int \left (18 \log ^2(x)+11 x \log ^2(x)\right ) \, dx\\ &=-88 x-\frac {121 x^2}{18}+e^{2 \log ^2(\log (x))} \log ^2(x)+\frac {1}{9} \log (x) \left (792 x+121 x^2+648 \log (x)\right )-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}+\frac {242}{9} \int x \log ^2(x) \, dx+44 \int \log ^2(x) \, dx-72 \int \frac {\log (x)}{x} \, dx\\ &=-88 x-\frac {121 x^2}{18}-36 \log ^2(x)+e^{2 \log ^2(\log (x))} \log ^2(x)+44 x \log ^2(x)+\frac {121}{9} x^2 \log ^2(x)+\frac {1}{9} \log (x) \left (792 x+121 x^2+648 \log (x)\right )-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}-\frac {242}{9} \int x \log (x) \, dx-88 \int \log (x) \, dx\\ &=-88 x \log (x)-\frac {121}{9} x^2 \log (x)-36 \log ^2(x)+e^{2 \log ^2(\log (x))} \log ^2(x)+44 x \log ^2(x)+\frac {121}{9} x^2 \log ^2(x)+\frac {1}{9} \log (x) \left (792 x+121 x^2+648 \log (x)\right )-\frac {2 e^{\log ^2(\log (x))} \log ^2(x) (18 \log (\log (x))+11 x \log (\log (x)))}{3 \log (\log (x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{9} \left (18-3 e^{\log ^2(\log (x))}+11 x\right )^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((648 + 792*x + 242*x^2)*Log[x] + (396*x + 242*x^2)*Log[x]^2 + E^(2*Log[Log[x]]^2)*(18*Log[x] + 36*L
og[x]*Log[Log[x]]) + E^Log[Log[x]]^2*((-216 - 132*x)*Log[x] - 66*x*Log[x]^2 + (-216 - 132*x)*Log[x]*Log[Log[x]
]))/(9*x),x]

[Out]

((18 - 3*E^Log[Log[x]]^2 + 11*x)^2*Log[x]^2)/9

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fricas [B]  time = 0.61, size = 47, normalized size = 2.04 \begin {gather*} -\frac {2}{3} \, {\left (11 \, x + 18\right )} e^{\left (\log \left (\log \relax (x)\right )^{2}\right )} \log \relax (x)^{2} + \frac {1}{9} \, {\left (121 \, x^{2} + 396 \, x + 324\right )} \log \relax (x)^{2} + e^{\left (2 \, \log \left (\log \relax (x)\right )^{2}\right )} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-132*x-216)*log(x)*log(log(x))-66*x*lo
g(x)^2+(-132*x-216)*log(x))*exp(log(log(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algor
ithm="fricas")

[Out]

-2/3*(11*x + 18)*e^(log(log(x))^2)*log(x)^2 + 1/9*(121*x^2 + 396*x + 324)*log(x)^2 + e^(2*log(log(x))^2)*log(x
)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (11 \, {\left (11 \, x^{2} + 18 \, x\right )} \log \relax (x)^{2} + 9 \, {\left (2 \, \log \relax (x) \log \left (\log \relax (x)\right ) + \log \relax (x)\right )} e^{\left (2 \, \log \left (\log \relax (x)\right )^{2}\right )} - 3 \, {\left (11 \, x \log \relax (x)^{2} + 2 \, {\left (11 \, x + 18\right )} \log \relax (x) \log \left (\log \relax (x)\right ) + 2 \, {\left (11 \, x + 18\right )} \log \relax (x)\right )} e^{\left (\log \left (\log \relax (x)\right )^{2}\right )} + {\left (121 \, x^{2} + 396 \, x + 324\right )} \log \relax (x)\right )}}{9 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-132*x-216)*log(x)*log(log(x))-66*x*lo
g(x)^2+(-132*x-216)*log(x))*exp(log(log(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algor
ithm="giac")

[Out]

integrate(2/9*(11*(11*x^2 + 18*x)*log(x)^2 + 9*(2*log(x)*log(log(x)) + log(x))*e^(2*log(log(x))^2) - 3*(11*x*l
og(x)^2 + 2*(11*x + 18)*log(x)*log(log(x)) + 2*(11*x + 18)*log(x))*e^(log(log(x))^2) + (121*x^2 + 396*x + 324)
*log(x))/x, x)

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maple [B]  time = 0.04, size = 48, normalized size = 2.09

method	result	size
risch	\(\frac {\left (121 x^{2}+396 x +324\right ) \ln \relax (x )^{2}}{9}+\ln \relax (x )^{2} {\mathrm e}^{2 \ln \left (\ln \relax (x )\right )^{2}}-\frac {2 \ln \relax (x )^{2} \left (11 x +18\right ) {\mathrm e}^{\ln \left (\ln \relax (x )\right )^{2}}}{3}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((36*ln(x)*ln(ln(x))+18*ln(x))*exp(ln(ln(x))^2)^2+((-132*x-216)*ln(x)*ln(ln(x))-66*x*ln(x)^2+(-132*x-2
16)*ln(x))*exp(ln(ln(x))^2)+(242*x^2+396*x)*ln(x)^2+(242*x^2+792*x+648)*ln(x))/x,x,method=_RETURNVERBOSE)

[Out]

1/9*(121*x^2+396*x+324)*ln(x)^2+ln(x)^2*exp(2*ln(ln(x))^2)-2/3*ln(x)^2*(11*x+18)*exp(ln(ln(x))^2)

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maxima [B]  time = 1.14, size = 87, normalized size = 3.78 \begin {gather*} -\frac {2}{3} \, {\left (11 \, x + 18\right )} e^{\left (\log \left (\log \relax (x)\right )^{2}\right )} \log \relax (x)^{2} + \frac {121}{18} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + \frac {121}{9} \, x^{2} \log \relax (x) + e^{\left (2 \, \log \left (\log \relax (x)\right )^{2}\right )} \log \relax (x)^{2} + 44 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x - \frac {121}{18} \, x^{2} + 88 \, x \log \relax (x) + 36 \, \log \relax (x)^{2} - 88 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-132*x-216)*log(x)*log(log(x))-66*x*lo
g(x)^2+(-132*x-216)*log(x))*exp(log(log(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algor
ithm="maxima")

[Out]

-2/3*(11*x + 18)*e^(log(log(x))^2)*log(x)^2 + 121/18*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 121/9*x^2*log(x) + e^(2
*log(log(x))^2)*log(x)^2 + 44*(log(x)^2 - 2*log(x) + 2)*x - 121/18*x^2 + 88*x*log(x) + 36*log(x)^2 - 88*x

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mupad [B]  time = 2.24, size = 21, normalized size = 0.91 \begin {gather*} \frac {{\ln \relax (x)}^2\,{\left (11\,x-3\,{\mathrm {e}}^{{\ln \left (\ln \relax (x)\right )}^2}+18\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(2*log(log(x))^2)*(18*log(x) + 36*log(log(x))*log(x)))/9 + (log(x)^2*(396*x + 242*x^2))/9 - (exp(log(
log(x))^2)*(66*x*log(x)^2 + log(x)*(132*x + 216) + log(log(x))*log(x)*(132*x + 216)))/9 + (log(x)*(792*x + 242
*x^2 + 648))/9)/x,x)

[Out]

(log(x)^2*(11*x - 3*exp(log(log(x))^2) + 18)^2)/9

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sympy [B]  time = 10.23, size = 58, normalized size = 2.52 \begin {gather*} \frac {\left (- 22 x \log {\relax (x )}^{2} - 36 \log {\relax (x )}^{2}\right ) e^{\log {\left (\log {\relax (x )} \right )}^{2}}}{3} + \left (\frac {121 x^{2}}{9} + 44 x + 36\right ) \log {\relax (x )}^{2} + e^{2 \log {\left (\log {\relax (x )} \right )}^{2}} \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((36*ln(x)*ln(ln(x))+18*ln(x))*exp(ln(ln(x))**2)**2+((-132*x-216)*ln(x)*ln(ln(x))-66*x*ln(x)**2+
(-132*x-216)*ln(x))*exp(ln(ln(x))**2)+(242*x**2+396*x)*ln(x)**2+(242*x**2+792*x+648)*ln(x))/x,x)

[Out]

(-22*x*log(x)**2 - 36*log(x)**2)*exp(log(log(x))**2)/3 + (121*x**2/9 + 44*x + 36)*log(x)**2 + exp(2*log(log(x)
)**2)*log(x)**2

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