3.74.31 \(\int \frac {2+4 x-4 e^6 x+e^{12} x+(-4 x+2 e^6 x) \log (\frac {4}{x})+x \log ^2(\frac {4}{x})+(4 x-2 e^6 x-2 x \log (\frac {4}{x})) \log (x)+(8 x^2-6 e^6 x^2+e^{12} x^2+(-6 x^2+2 e^6 x^2) \log (\frac {4}{x})+x^2 \log ^2(\frac {4}{x})) \log (\log (2))}{x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac {2}{x}+\left (-2+e^6+\log \left (\frac {4}{x}\right )\right )^2 (\log (x)+x \log (\log (2))) \]

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Rubi [B]  time = 0.39, antiderivative size = 174, normalized size of antiderivative = 6.21, number of steps used = 20, number of rules used = 10, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6, 14, 2301, 2366, 12, 2302, 30, 2346, 2296, 2295} \begin {gather*} -\frac {2}{x}-\frac {1}{3} \log ^3\left (\frac {4}{x}\right )+\left (2-e^6\right ) \log ^2\left (\frac {4}{x}\right )+x \log (\log (2)) \log ^2\left (\frac {4}{x}\right )-\frac {1}{3} \left (-\log \left (\frac {4}{x}\right )-e^6+2\right )^3+\log (x) \left (-\log \left (\frac {4}{x}\right )-e^6+2\right )^2+\left (2-e^6\right )^2 \log (x)-2 \left (3-e^6\right ) x \log (\log (2))+\left (2-e^3\right ) \left (2+e^3\right ) \left (2-e^6\right ) x \log (\log (2))+2 x \log (\log (2))-2 \left (3-e^6\right ) x \log (\log (2)) \log \left (\frac {4}{x}\right )+2 x \log (\log (2)) \log \left (\frac {4}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 4*x - 4*E^6*x + E^12*x + (-4*x + 2*E^6*x)*Log[4/x] + x*Log[4/x]^2 + (4*x - 2*E^6*x - 2*x*Log[4/x])*Lo
g[x] + (8*x^2 - 6*E^6*x^2 + E^12*x^2 + (-6*x^2 + 2*E^6*x^2)*Log[4/x] + x^2*Log[4/x]^2)*Log[Log[2]])/x^2,x]

[Out]

-2/x - (2 - E^6 - Log[4/x])^3/3 + (2 - E^6)*Log[4/x]^2 - Log[4/x]^3/3 + (2 - E^6)^2*Log[x] + (2 - E^6 - Log[4/
x])^2*Log[x] + 2*x*Log[Log[2]] + (2 - E^3)*(2 + E^3)*(2 - E^6)*x*Log[Log[2]] - 2*(3 - E^6)*x*Log[Log[2]] + 2*x
*Log[4/x]*Log[Log[2]] - 2*(3 - E^6)*x*Log[4/x]*Log[Log[2]] + x*Log[4/x]^2*Log[Log[2]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 30

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

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 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 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+e^{12} x+\left (4-4 e^6\right ) x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx\\ &=\int \frac {2+\left (4-4 e^6+e^{12}\right ) x+\left (-4 x+2 e^6 x\right ) \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+\left (4 x-2 e^6 x-2 x \log \left (\frac {4}{x}\right )\right ) \log (x)+\left (8 x^2-6 e^6 x^2+e^{12} x^2+\left (-6 x^2+2 e^6 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )\right ) \log (\log (2))}{x^2} \, dx\\ &=\int \left (\frac {2 \left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right ) \log (x)}{x}+\frac {2+4 \left (1+\frac {1}{4} e^6 \left (-4+e^6\right )\right ) x-4 \left (1-\frac {e^6}{2}\right ) x \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+8 \left (1+\frac {1}{8} e^6 \left (-6+e^6\right )\right ) x^2 \log (\log (2))-6 \left (1-\frac {e^6}{3}\right ) x^2 \log \left (\frac {4}{x}\right ) \log (\log (2))+x^2 \log ^2\left (\frac {4}{x}\right ) \log (\log (2))}{x^2}\right ) \, dx\\ &=2 \int \frac {\left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right ) \log (x)}{x} \, dx+\int \frac {2+4 \left (1+\frac {1}{4} e^6 \left (-4+e^6\right )\right ) x-4 \left (1-\frac {e^6}{2}\right ) x \log \left (\frac {4}{x}\right )+x \log ^2\left (\frac {4}{x}\right )+8 \left (1+\frac {1}{8} e^6 \left (-6+e^6\right )\right ) x^2 \log (\log (2))-6 \left (1-\frac {e^6}{3}\right ) x^2 \log \left (\frac {4}{x}\right ) \log (\log (2))+x^2 \log ^2\left (\frac {4}{x}\right ) \log (\log (2))}{x^2} \, dx\\ &=\left (2-e^6-\log \left (\frac {4}{x}\right )\right )^2 \log (x)-2 \int \frac {\left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right )^2}{2 x} \, dx+\int \left (\frac {\log ^2\left (\frac {4}{x}\right ) (1+x \log (\log (2)))}{x}+\frac {2 \log \left (\frac {4}{x}\right ) \left (-2+e^6-\left (3-e^6\right ) x \log (\log (2))\right )}{x}+\frac {2+\left (2-e^6\right )^2 x+\left (8-6 e^6+e^{12}\right ) x^2 \log (\log (2))}{x^2}\right ) \, dx\\ &=\left (2-e^6-\log \left (\frac {4}{x}\right )\right )^2 \log (x)+2 \int \frac {\log \left (\frac {4}{x}\right ) \left (-2+e^6-\left (3-e^6\right ) x \log (\log (2))\right )}{x} \, dx-\int \frac {\left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right )^2}{x} \, dx+\int \frac {\log ^2\left (\frac {4}{x}\right ) (1+x \log (\log (2)))}{x} \, dx+\int \frac {2+\left (2-e^6\right )^2 x+\left (8-6 e^6+e^{12}\right ) x^2 \log (\log (2))}{x^2} \, dx\\ &=\left (2-e^6-\log \left (\frac {4}{x}\right )\right )^2 \log (x)-\left (2 \left (2-e^6\right )\right ) \int \frac {\log \left (\frac {4}{x}\right )}{x} \, dx+\log (\log (2)) \int \log ^2\left (\frac {4}{x}\right ) \, dx-\left (2 \left (3-e^6\right ) \log (\log (2))\right ) \int \log \left (\frac {4}{x}\right ) \, dx+\int \frac {\log ^2\left (\frac {4}{x}\right )}{x} \, dx+\int \left (\frac {2}{x^2}+\frac {\left (-2+e^6\right )^2}{x}+\left (-2+e^3\right ) \left (2+e^3\right ) \left (-2+e^6\right ) \log (\log (2))\right ) \, dx-\operatorname {Subst}\left (\int x^2 \, dx,x,2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right )\\ &=-\frac {2}{x}-\frac {1}{3} \left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right )^3+\left (2-e^6\right ) \log ^2\left (\frac {4}{x}\right )+\left (2-e^6\right )^2 \log (x)+\left (2-e^6-\log \left (\frac {4}{x}\right )\right )^2 \log (x)+\left (2-e^3\right ) \left (2+e^3\right ) \left (2-e^6\right ) x \log (\log (2))-2 \left (3-e^6\right ) x \log (\log (2))-2 \left (3-e^6\right ) x \log \left (\frac {4}{x}\right ) \log (\log (2))+x \log ^2\left (\frac {4}{x}\right ) \log (\log (2))+(2 \log (\log (2))) \int \log \left (\frac {4}{x}\right ) \, dx-\operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (\frac {4}{x}\right )\right )\\ &=-\frac {2}{x}-\frac {1}{3} \left (2 \left (1-\frac {e^6}{2}\right )-\log \left (\frac {4}{x}\right )\right )^3+\left (2-e^6\right ) \log ^2\left (\frac {4}{x}\right )-\frac {1}{3} \log ^3\left (\frac {4}{x}\right )+\left (2-e^6\right )^2 \log (x)+\left (2-e^6-\log \left (\frac {4}{x}\right )\right )^2 \log (x)+2 x \log (\log (2))+\left (2-e^3\right ) \left (2+e^3\right ) \left (2-e^6\right ) x \log (\log (2))-2 \left (3-e^6\right ) x \log (\log (2))+2 x \log \left (\frac {4}{x}\right ) \log (\log (2))-2 \left (3-e^6\right ) x \log \left (\frac {4}{x}\right ) \log (\log (2))+x \log ^2\left (\frac {4}{x}\right ) \log (\log (2))\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 90, normalized size = 3.21 \begin {gather*} -\frac {2}{x}+\left (-2+e^6\right )^2 \log (x)-\left (-2+e^6\right ) \log ^2(x)+4 x \log (\log (2))-4 e^6 x \log (\log (2))+e^{12} x \log (\log (2))+2 \left (-2+e^6\right ) x \log \left (\frac {4}{x}\right ) \log (\log (2))+\log ^2\left (\frac {4}{x}\right ) \left (2-e^6+\log (x)+x \log (\log (2))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*x - 4*E^6*x + E^12*x + (-4*x + 2*E^6*x)*Log[4/x] + x*Log[4/x]^2 + (4*x - 2*E^6*x - 2*x*Log[4/
x])*Log[x] + (8*x^2 - 6*E^6*x^2 + E^12*x^2 + (-6*x^2 + 2*E^6*x^2)*Log[4/x] + x^2*Log[4/x]^2)*Log[Log[2]])/x^2,
x]

[Out]

-2/x + (-2 + E^6)^2*Log[x] - (-2 + E^6)*Log[x]^2 + 4*x*Log[Log[2]] - 4*E^6*x*Log[Log[2]] + E^12*x*Log[Log[2]]
+ 2*(-2 + E^6)*x*Log[4/x]*Log[Log[2]] + Log[4/x]^2*(2 - E^6 + Log[x] + x*Log[Log[2]])

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fricas [B]  time = 0.62, size = 128, normalized size = 4.57 \begin {gather*} -\frac {x \log \left (\frac {4}{x}\right )^{3} + 2 \, {\left (x e^{6} - x \log \relax (2) - 2 \, x\right )} \log \left (\frac {4}{x}\right )^{2} + {\left (x e^{12} - 4 \, x e^{6} - 4 \, {\left (x e^{6} - 2 \, x\right )} \log \relax (2) + 4 \, x\right )} \log \left (\frac {4}{x}\right ) - {\left (x^{2} \log \left (\frac {4}{x}\right )^{2} + x^{2} e^{12} - 4 \, x^{2} e^{6} + 4 \, x^{2} + 2 \, {\left (x^{2} e^{6} - 2 \, x^{2}\right )} \log \left (\frac {4}{x}\right )\right )} \log \left (\log \relax (2)\right ) + 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6*x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*
x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x*log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^
2,x, algorithm="fricas")

[Out]

-(x*log(4/x)^3 + 2*(x*e^6 - x*log(2) - 2*x)*log(4/x)^2 + (x*e^12 - 4*x*e^6 - 4*(x*e^6 - 2*x)*log(2) + 4*x)*log
(4/x) - (x^2*log(4/x)^2 + x^2*e^12 - 4*x^2*e^6 + 4*x^2 + 2*(x^2*e^6 - 2*x^2)*log(4/x))*log(log(2)) + 2)/x

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giac [B]  time = 0.24, size = 186, normalized size = 6.64 \begin {gather*} \frac {4 \, x^{2} e^{6} \log \relax (2) \log \left (\log \relax (2)\right ) + 4 \, x^{2} \log \relax (2)^{2} \log \left (\log \relax (2)\right ) - 2 \, x^{2} e^{6} \log \relax (x) \log \left (\log \relax (2)\right ) - 4 \, x^{2} \log \relax (2) \log \relax (x) \log \left (\log \relax (2)\right ) + x^{2} \log \relax (x)^{2} \log \left (\log \relax (2)\right ) + 4 \, x e^{6} \log \relax (2) \log \relax (x) + 4 \, x \log \relax (2)^{2} \log \relax (x) - 2 \, x e^{6} \log \relax (x)^{2} - 4 \, x \log \relax (2) \log \relax (x)^{2} + x \log \relax (x)^{3} + x^{2} e^{12} \log \left (\log \relax (2)\right ) - 4 \, x^{2} e^{6} \log \left (\log \relax (2)\right ) - 8 \, x^{2} \log \relax (2) \log \left (\log \relax (2)\right ) + 4 \, x^{2} \log \relax (x) \log \left (\log \relax (2)\right ) + x e^{12} \log \relax (x) - 4 \, x e^{6} \log \relax (x) - 8 \, x \log \relax (2) \log \relax (x) + 4 \, x \log \relax (x)^{2} + 4 \, x^{2} \log \left (\log \relax (2)\right ) + 4 \, x \log \relax (x) - 2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6*x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*
x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x*log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^
2,x, algorithm="giac")

[Out]

(4*x^2*e^6*log(2)*log(log(2)) + 4*x^2*log(2)^2*log(log(2)) - 2*x^2*e^6*log(x)*log(log(2)) - 4*x^2*log(2)*log(x
)*log(log(2)) + x^2*log(x)^2*log(log(2)) + 4*x*e^6*log(2)*log(x) + 4*x*log(2)^2*log(x) - 2*x*e^6*log(x)^2 - 4*
x*log(2)*log(x)^2 + x*log(x)^3 + x^2*e^12*log(log(2)) - 4*x^2*e^6*log(log(2)) - 8*x^2*log(2)*log(log(2)) + 4*x
^2*log(x)*log(log(2)) + x*e^12*log(x) - 4*x*e^6*log(x) - 8*x*log(2)*log(x) + 4*x*log(x)^2 + 4*x^2*log(log(2))
+ 4*x*log(x) - 2)/x

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maple [B]  time = 0.75, size = 164, normalized size = 5.86

method	result	size
risch	\(\ln \relax (x )^{3}+\left (x \ln \left (\ln \relax (2)\right )+4-2 \,{\mathrm e}^{6}-4 \ln \relax (2)\right ) \ln \relax (x )^{2}-\left (2 \ln \left (\ln \relax (2)\right ) {\mathrm e}^{6}+4 \ln \left (\ln \relax (2)\right ) \ln \relax (2)-4 \ln \left (\ln \relax (2)\right )\right ) x \ln \relax (x )+\frac {-8-32 x \ln \relax (2) \ln \relax (x )+16 x^{2} \ln \left (\ln \relax (2)\right )+16 x \ln \relax (x )+16 \,{\mathrm e}^{6} \ln \left (\ln \relax (2)\right ) x^{2} \ln \relax (2)+4 \ln \relax (x ) {\mathrm e}^{12} x +16 \ln \relax (x ) \ln \relax (2)^{2} x +4 \,{\mathrm e}^{12} \ln \left (\ln \relax (2)\right ) x^{2}-16 \,{\mathrm e}^{6} \ln \left (\ln \relax (2)\right ) x^{2}-16 x \,{\mathrm e}^{6} \ln \relax (x )+16 x^{2} \ln \left (\ln \relax (2)\right ) \ln \relax (2)^{2}-32 x^{2} \ln \left (\ln \relax (2)\right ) \ln \relax (2)+16 \ln \relax (x ) {\mathrm e}^{6} \ln \relax (2) x}{4 x}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*ln(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*ln(4/x)+x^2*exp(3)^4-6*x^2*exp(3)^2+8*x^2)*ln(ln(2))+(-2*x*ln(4/x)-
2*x*exp(3)^2+4*x)*ln(x)+x*ln(4/x)^2+(2*x*exp(3)^2-4*x)*ln(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^2,x,method=_RE
TURNVERBOSE)

[Out]

ln(x)^3+(x*ln(ln(2))+4-2*exp(6)-4*ln(2))*ln(x)^2-(2*ln(ln(2))*exp(6)+4*ln(ln(2))*ln(2)-4*ln(ln(2)))*x*ln(x)+1/
4*(-8-32*x*ln(2)*ln(x)+16*x^2*ln(ln(2))+16*x*ln(x)+16*exp(6)*ln(ln(2))*x^2*ln(2)+4*ln(x)*exp(12)*x+16*ln(x)*ln
(2)^2*x+4*exp(12)*ln(ln(2))*x^2-16*exp(6)*ln(ln(2))*x^2-16*x*exp(6)*ln(x)+16*x^2*ln(ln(2))*ln(2)^2-32*x^2*ln(l
n(2))*ln(2)+16*ln(x)*exp(6)*ln(2)*x)/x

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maxima [B]  time = 0.37, size = 151, normalized size = 5.39 \begin {gather*} x \log \left (\frac {4}{x}\right )^{2} \log \left (\log \relax (2)\right ) - e^{6} \log \relax (x)^{2} - \frac {1}{3} \, \log \relax (x)^{3} - \log \relax (x)^{2} \log \left (\frac {4}{x}\right ) - e^{6} \log \left (\frac {4}{x}\right )^{2} - \frac {1}{3} \, \log \left (\frac {4}{x}\right )^{3} + x e^{12} \log \left (\log \relax (2)\right ) + 2 \, {\left (x \log \left (\frac {4}{x}\right ) + x\right )} e^{6} \log \left (\log \relax (2)\right ) - 6 \, x e^{6} \log \left (\log \relax (2)\right ) + e^{12} \log \relax (x) - 4 \, e^{6} \log \relax (x) + 2 \, \log \relax (x)^{2} + 2 \, \log \left (\frac {4}{x}\right )^{2} - 4 \, {\left (x \log \left (\frac {4}{x}\right ) + x\right )} \log \left (\log \relax (2)\right ) + 8 \, x \log \left (\log \relax (2)\right ) - \frac {2}{x} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(4/x)^2+(2*x^2*exp(3)^2-6*x^2)*log(4/x)+x^2*exp(3)^4-6*x^2*exp(3)^2+8*x^2)*log(log(2))+(-2*
x*log(4/x)-2*x*exp(3)^2+4*x)*log(x)+x*log(4/x)^2+(2*x*exp(3)^2-4*x)*log(4/x)+x*exp(3)^4-4*x*exp(3)^2+4*x+2)/x^
2,x, algorithm="maxima")

[Out]

x*log(4/x)^2*log(log(2)) - e^6*log(x)^2 - 1/3*log(x)^3 - log(x)^2*log(4/x) - e^6*log(4/x)^2 - 1/3*log(4/x)^3 +
x*e^12*log(log(2)) + 2*(x*log(4/x) + x)*e^6*log(log(2)) - 6*x*e^6*log(log(2)) + e^12*log(x) - 4*e^6*log(x) +
2*log(x)^2 + 2*log(4/x)^2 - 4*(x*log(4/x) + x)*log(log(2)) + 8*x*log(log(2)) - 2/x + 4*log(x)

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mupad [B]  time = 4.92, size = 174, normalized size = 6.21 \begin {gather*} 4\,\ln \relax (x)+4\,{\ln \relax (2)}^2\,\ln \relax (x)-4\,\ln \left (\frac {1}{x}\right )\,\ln \relax (x)-4\,{\mathrm {e}}^6\,\ln \relax (x)+{\mathrm {e}}^{12}\,\ln \relax (x)+4\,x\,\ln \left (\ln \relax (2)\right )-8\,\ln \relax (2)\,\ln \relax (x)-\frac {2}{x}+{\ln \left (\frac {1}{x}\right )}^2\,\ln \relax (x)-4\,x\,{\mathrm {e}}^6\,\ln \left (\ln \relax (2)\right )+4\,{\mathrm {e}}^6\,\ln \relax (2)\,\ln \relax (x)+x\,{\mathrm {e}}^{12}\,\ln \left (\ln \relax (2)\right )-8\,x\,\ln \relax (2)\,\ln \left (\ln \relax (2)\right )+x\,{\ln \left (\frac {1}{x}\right )}^2\,\ln \left (\ln \relax (2)\right )+4\,x\,{\ln \relax (2)}^2\,\ln \left (\ln \relax (2)\right )+2\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^6\,\ln \relax (x)-4\,x\,\ln \left (\frac {1}{x}\right )\,\ln \left (\ln \relax (2)\right )+4\,\ln \left (\frac {1}{x}\right )\,\ln \relax (2)\,\ln \relax (x)+2\,x\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^6\,\ln \left (\ln \relax (2)\right )+4\,x\,\ln \left (\frac {1}{x}\right )\,\ln \relax (2)\,\ln \left (\ln \relax (2)\right )+4\,x\,{\mathrm {e}}^6\,\ln \relax (2)\,\ln \left (\ln \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - log(x)*(2*x*exp(6) - 4*x + 2*x*log(4/x)) - 4*x*exp(6) + x*exp(12) + log(log(2))*(log(4/x)*(2*x^2*ex
p(6) - 6*x^2) + x^2*log(4/x)^2 - 6*x^2*exp(6) + x^2*exp(12) + 8*x^2) - log(4/x)*(4*x - 2*x*exp(6)) + x*log(4/x
)^2 + 2)/x^2,x)

[Out]

4*log(x) + 4*log(2)^2*log(x) - 4*log(1/x)*log(x) - 4*exp(6)*log(x) + exp(12)*log(x) + 4*x*log(log(2)) - 8*log(
2)*log(x) - 2/x + log(1/x)^2*log(x) - 4*x*exp(6)*log(log(2)) + 4*exp(6)*log(2)*log(x) + x*exp(12)*log(log(2))
- 8*x*log(2)*log(log(2)) + x*log(1/x)^2*log(log(2)) + 4*x*log(2)^2*log(log(2)) + 2*log(1/x)*exp(6)*log(x) - 4*
x*log(1/x)*log(log(2)) + 4*log(1/x)*log(2)*log(x) + 2*x*log(1/x)*exp(6)*log(log(2)) + 4*x*log(1/x)*log(2)*log(
log(2)) + 4*x*exp(6)*log(2)*log(log(2))

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sympy [B]  time = 0.66, size = 144, normalized size = 5.14 \begin {gather*} x \left (e^{12} \log {\left (\log {\relax (2 )} \right )} + 4 e^{6} \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} + 4 \log {\left (\log {\relax (2 )} \right )} + 4 \log {\relax (2 )}^{2} \log {\left (\log {\relax (2 )} \right )} - 8 \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} - 4 e^{6} \log {\left (\log {\relax (2 )} \right )}\right ) + \left (4 x \log {\left (\log {\relax (2 )} \right )} - 4 x \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} - 2 x e^{6} \log {\left (\log {\relax (2 )} \right )}\right ) \log {\relax (x )} + \left (x \log {\left (\log {\relax (2 )} \right )} - 2 e^{6} - 4 \log {\relax (2 )} + 4\right ) \log {\relax (x )}^{2} + \log {\relax (x )}^{3} + \left (-2 + 2 \log {\relax (2 )} + e^{6}\right )^{2} \log {\relax (x )} - \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2*ln(4/x)**2+(2*x**2*exp(3)**2-6*x**2)*ln(4/x)+x**2*exp(3)**4-6*x**2*exp(3)**2+8*x**2)*ln(ln(2)
)+(-2*x*ln(4/x)-2*x*exp(3)**2+4*x)*ln(x)+x*ln(4/x)**2+(2*x*exp(3)**2-4*x)*ln(4/x)+x*exp(3)**4-4*x*exp(3)**2+4*
x+2)/x**2,x)

[Out]

x*(exp(12)*log(log(2)) + 4*exp(6)*log(2)*log(log(2)) + 4*log(log(2)) + 4*log(2)**2*log(log(2)) - 8*log(2)*log(
log(2)) - 4*exp(6)*log(log(2))) + (4*x*log(log(2)) - 4*x*log(2)*log(log(2)) - 2*x*exp(6)*log(log(2)))*log(x) +
(x*log(log(2)) - 2*exp(6) - 4*log(2) + 4)*log(x)**2 + log(x)**3 + (-2 + 2*log(2) + exp(6))**2*log(x) - 2/x

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