3.32.6 \(\int \frac {-131072-65536 e^4-2 x^2+(-32768-16384 e^4) \log (2)+(-3072-1536 e^4) \log ^2(2)+(-128-64 e^4) \log ^3(2)+(-2-e^4) \log ^4(2)+4 x^2 \log (x)}{(65536 x+16384 x \log (2)+1536 x \log ^2(2)+64 x \log ^3(2)+x \log ^4(2)) \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {2+e^4+\frac {2 x^2}{(16+\log (2))^4}}{\log (x)} \]

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Rubi [A]  time = 0.57, antiderivative size = 26, normalized size of antiderivative = 1.24, number of steps used = 17, number of rules used = 10, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.099, Rules used = {6, 12, 6688, 6742, 2353, 2306, 2309, 2178, 2302, 30} \begin {gather*} \frac {2 x^2}{(16+\log (2))^4 \log (x)}+\frac {2+e^4}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-131072 - 65536*E^4 - 2*x^2 + (-32768 - 16384*E^4)*Log[2] + (-3072 - 1536*E^4)*Log[2]^2 + (-128 - 64*E^4)
*Log[2]^3 + (-2 - E^4)*Log[2]^4 + 4*x^2*Log[x])/((65536*x + 16384*x*Log[2] + 1536*x*Log[2]^2 + 64*x*Log[2]^3 +
x*Log[2]^4)*Log[x]^2),x]

[Out]

(2 + E^4)/Log[x] + (2*x^2)/((16 + Log[2])^4*Log[x])

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 30

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

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 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] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

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 6688

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

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 {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (1536 x \log ^2(2)+64 x \log ^3(2)+x \log ^4(2)+x (65536+16384 \log (2))\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (x \log ^4(2)+x (65536+16384 \log (2))+x \left (1536 \log ^2(2)+64 \log ^3(2)\right )\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (x \left (1536 \log ^2(2)+64 \log ^3(2)\right )+x \left (65536+16384 \log (2)+\log ^4(2)\right )\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{x \left (65536+16384 \log (2)+1536 \log ^2(2)+64 \log ^3(2)+\log ^4(2)\right ) \log ^2(x)} \, dx\\ &=\frac {\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \frac {-e^4 (16+\log (2))^4-2 \left (x^2+(16+\log (2))^4\right )+4 x^2 \log (x)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \left (\frac {-131072-65536 e^4-2 x^2-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)}+\frac {4 x}{\log (x)}\right ) \, dx}{(16+\log (2))^4}\\ &=\frac {\int \frac {-131072-65536 e^4-2 x^2-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}+\frac {4 \int \frac {x}{\log (x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \left (-\frac {2 x}{\log ^2(x)}+\frac {-131072-65536 e^4-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)}\right ) \, dx}{(16+\log (2))^4}+\frac {4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\left (-2-e^4\right ) \int \frac {1}{x \log ^2(x)} \, dx-\frac {2 \int \frac {x}{\log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}+\left (-2-e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )-\frac {4 \int \frac {x}{\log (x)} \, dx}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\frac {2+e^4}{\log (x)}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}-\frac {4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{(16+\log (2))^4}\\ &=\frac {2+e^4}{\log (x)}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 34, normalized size = 1.62 \begin {gather*} \frac {e^4 (16+\log (2))^4+2 \left (x^2+(16+\log (2))^4\right )}{(16+\log (2))^4 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-131072 - 65536*E^4 - 2*x^2 + (-32768 - 16384*E^4)*Log[2] + (-3072 - 1536*E^4)*Log[2]^2 + (-128 - 6
4*E^4)*Log[2]^3 + (-2 - E^4)*Log[2]^4 + 4*x^2*Log[x])/((65536*x + 16384*x*Log[2] + 1536*x*Log[2]^2 + 64*x*Log[
2]^3 + x*Log[2]^4)*Log[x]^2),x]

[Out]

(E^4*(16 + Log[2])^4 + 2*(x^2 + (16 + Log[2])^4))/((16 + Log[2])^4*Log[x])

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fricas [B]  time = 1.04, size = 77, normalized size = 3.67 \begin {gather*} \frac {{\left (e^{4} + 2\right )} \log \relax (2)^{4} + 64 \, {\left (e^{4} + 2\right )} \log \relax (2)^{3} + 1536 \, {\left (e^{4} + 2\right )} \log \relax (2)^{2} + 2 \, x^{2} + 16384 \, {\left (e^{4} + 2\right )} \log \relax (2) + 65536 \, e^{4} + 131072}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)+(-exp(4)-2)*log(2)^4+(-64*exp(4)-128)*log(2)^3+(-1536*exp(4)-3072)*log(2)^2+(-16384*ex
p(4)-32768)*log(2)-65536*exp(4)-2*x^2-131072)/(x*log(2)^4+64*x*log(2)^3+1536*x*log(2)^2+16384*x*log(2)+65536*x
)/log(x)^2,x, algorithm="fricas")

[Out]

((e^4 + 2)*log(2)^4 + 64*(e^4 + 2)*log(2)^3 + 1536*(e^4 + 2)*log(2)^2 + 2*x^2 + 16384*(e^4 + 2)*log(2) + 65536
*e^4 + 131072)/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x))

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giac [B]  time = 0.22, size = 99, normalized size = 4.71 \begin {gather*} \frac {e^{4} \log \relax (2)^{4} + 64 \, e^{4} \log \relax (2)^{3} + 2 \, \log \relax (2)^{4} + 1536 \, e^{4} \log \relax (2)^{2} + 128 \, \log \relax (2)^{3} + 2 \, x^{2} + 16384 \, e^{4} \log \relax (2) + 3072 \, \log \relax (2)^{2} + 65536 \, e^{4} + 32768 \, \log \relax (2) + 131072}{\log \relax (2)^{4} \log \relax (x) + 64 \, \log \relax (2)^{3} \log \relax (x) + 1536 \, \log \relax (2)^{2} \log \relax (x) + 16384 \, \log \relax (2) \log \relax (x) + 65536 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)+(-exp(4)-2)*log(2)^4+(-64*exp(4)-128)*log(2)^3+(-1536*exp(4)-3072)*log(2)^2+(-16384*ex
p(4)-32768)*log(2)-65536*exp(4)-2*x^2-131072)/(x*log(2)^4+64*x*log(2)^3+1536*x*log(2)^2+16384*x*log(2)+65536*x
)/log(x)^2,x, algorithm="giac")

[Out]

(e^4*log(2)^4 + 64*e^4*log(2)^3 + 2*log(2)^4 + 1536*e^4*log(2)^2 + 128*log(2)^3 + 2*x^2 + 16384*e^4*log(2) + 3
072*log(2)^2 + 65536*e^4 + 32768*log(2) + 131072)/(log(2)^4*log(x) + 64*log(2)^3*log(x) + 1536*log(2)^2*log(x)
+ 16384*log(2)*log(x) + 65536*log(x))

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maple [B]  time = 0.11, size = 66, normalized size = 3.14

method	result	size
norman	\(\frac {\frac {2 x^{2}}{16+\ln \relax (2)}+8192+\ln \relax (2)^{3} {\mathrm e}^{4}+48 \ln \relax (2)^{2} {\mathrm e}^{4}+2 \ln \relax (2)^{3}+768 \,{\mathrm e}^{4} \ln \relax (2)+96 \ln \relax (2)^{2}+4096 \,{\mathrm e}^{4}+1536 \ln \relax (2)}{\left (16+\ln \relax (2)\right )^{3} \ln \relax (x )}\)	\(66\)
risch	\(\frac {\ln \relax (2)^{4} {\mathrm e}^{4}+64 \ln \relax (2)^{3} {\mathrm e}^{4}+2 \ln \relax (2)^{4}+1536 \ln \relax (2)^{2} {\mathrm e}^{4}+128 \ln \relax (2)^{3}+16384 \,{\mathrm e}^{4} \ln \relax (2)+3072 \ln \relax (2)^{2}+2 x^{2}+65536 \,{\mathrm e}^{4}+32768 \ln \relax (2)+131072}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}\)	\(92\)
default	\(\frac {\ln \relax (2)^{4} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {64 \ln \relax (2)^{3} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {2 \ln \relax (2)^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}-\frac {4 \expIntegralEi \left (1, -2 \ln \relax (x )\right )}{\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536}+\frac {1536 \ln \relax (2)^{2} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {128 \ln \relax (2)^{3}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {16384 \,{\mathrm e}^{4} \ln \relax (2)}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {3072 \ln \relax (2)^{2}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}-\frac {2 \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )}{\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536}+\frac {65536 \,{\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {32768 \ln \relax (2)}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {131072}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*ln(x)+(-exp(4)-2)*ln(2)^4+(-64*exp(4)-128)*ln(2)^3+(-1536*exp(4)-3072)*ln(2)^2+(-16384*exp(4)-32768
)*ln(2)-65536*exp(4)-2*x^2-131072)/(x*ln(2)^4+64*x*ln(2)^3+1536*x*ln(2)^2+16384*x*ln(2)+65536*x)/ln(x)^2,x,met
hod=_RETURNVERBOSE)

[Out]

(2/(16+ln(2))*x^2+8192+ln(2)^3*exp(4)+48*ln(2)^2*exp(4)+2*ln(2)^3+768*exp(4)*ln(2)+96*ln(2)^2+4096*exp(4)+1536
*ln(2))/(16+ln(2))^3/ln(x)

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maxima [B]  time = 0.61, size = 371, normalized size = 17.67 \begin {gather*} \frac {e^{4} \log \relax (2)^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {64 \, e^{4} \log \relax (2)^{3}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {2 \, \log \relax (2)^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {1536 \, e^{4} \log \relax (2)^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {128 \, \log \relax (2)^{3}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {2 \, x^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {16384 \, e^{4} \log \relax (2)}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {3072 \, \log \relax (2)^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {65536 \, e^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {32768 \, \log \relax (2)}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {131072}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(x)+(-exp(4)-2)*log(2)^4+(-64*exp(4)-128)*log(2)^3+(-1536*exp(4)-3072)*log(2)^2+(-16384*ex
p(4)-32768)*log(2)-65536*exp(4)-2*x^2-131072)/(x*log(2)^4+64*x*log(2)^3+1536*x*log(2)^2+16384*x*log(2)+65536*x
)/log(x)^2,x, algorithm="maxima")

[Out]

e^4*log(2)^4/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 64*e^4*log(2)^3/((log(
2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 2*log(2)^4/((log(2)^4 + 64*log(2)^3 + 153
6*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 1536*e^4*log(2)^2/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 1638
4*log(2) + 65536)*log(x)) + 128*log(2)^3/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(
x)) + 2*x^2/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 16384*e^4*log(2)/((log(
2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 3072*log(2)^2/((log(2)^4 + 64*log(2)^3 +
1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) + 65536*e^4/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log
(2) + 65536)*log(x)) + 32768*log(2)/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x)) +
131072/((log(2)^4 + 64*log(2)^3 + 1536*log(2)^2 + 16384*log(2) + 65536)*log(x))

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mupad [B]  time = 1.85, size = 73, normalized size = 3.48 \begin {gather*} \frac {2\,x^2+65536\,{\mathrm {e}}^4+32768\,\ln \relax (2)+16384\,{\mathrm {e}}^4\,\ln \relax (2)+1536\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+64\,{\mathrm {e}}^4\,{\ln \relax (2)}^3+{\mathrm {e}}^4\,{\ln \relax (2)}^4+3072\,{\ln \relax (2)}^2+128\,{\ln \relax (2)}^3+2\,{\ln \relax (2)}^4+131072}{\ln \relax (x)\,{\left (\ln \relax (2)+16\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(65536*exp(4) - 4*x^2*log(x) + log(2)^3*(64*exp(4) + 128) + log(2)^2*(1536*exp(4) + 3072) + 2*x^2 + log(2
)*(16384*exp(4) + 32768) + log(2)^4*(exp(4) + 2) + 131072)/(log(x)^2*(65536*x + 16384*x*log(2) + 1536*x*log(2)
^2 + 64*x*log(2)^3 + x*log(2)^4)),x)

[Out]

(65536*exp(4) + 32768*log(2) + 16384*exp(4)*log(2) + 1536*exp(4)*log(2)^2 + 64*exp(4)*log(2)^3 + exp(4)*log(2)
^4 + 3072*log(2)^2 + 128*log(2)^3 + 2*log(2)^4 + 2*x^2 + 131072)/(log(x)*(log(2) + 16)^4)

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sympy [B]  time = 0.15, size = 102, normalized size = 4.86 \begin {gather*} \frac {2 x^{2} + 2 \log {\relax (2 )}^{4} + e^{4} \log {\relax (2 )}^{4} + 128 \log {\relax (2 )}^{3} + 64 e^{4} \log {\relax (2 )}^{3} + 3072 \log {\relax (2 )}^{2} + 32768 \log {\relax (2 )} + 1536 e^{4} \log {\relax (2 )}^{2} + 131072 + 16384 e^{4} \log {\relax (2 )} + 65536 e^{4}}{\left (\log {\relax (2 )}^{4} + 64 \log {\relax (2 )}^{3} + 1536 \log {\relax (2 )}^{2} + 16384 \log {\relax (2 )} + 65536\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2*ln(x)+(-exp(4)-2)*ln(2)**4+(-64*exp(4)-128)*ln(2)**3+(-1536*exp(4)-3072)*ln(2)**2+(-16384*ex
p(4)-32768)*ln(2)-65536*exp(4)-2*x**2-131072)/(x*ln(2)**4+64*x*ln(2)**3+1536*x*ln(2)**2+16384*x*ln(2)+65536*x)
/ln(x)**2,x)

[Out]

(2*x**2 + 2*log(2)**4 + exp(4)*log(2)**4 + 128*log(2)**3 + 64*exp(4)*log(2)**3 + 3072*log(2)**2 + 32768*log(2)
+ 1536*exp(4)*log(2)**2 + 131072 + 16384*exp(4)*log(2) + 65536*exp(4))/((log(2)**4 + 64*log(2)**3 + 1536*log(
2)**2 + 16384*log(2) + 65536)*log(x))

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