3.37.71 \(\int \frac {-36+12 x+(-8+12 x-4 x^2) \log (4)+e^{2 x} (3+6 x-2 x^2 \log (4))+e^x (3-3 x+3 x^2+(-4 x+2 x^2-x^3) \log (4))+(-36+24 x-4 x^2 \log (4)+e^x (12+12 x-4 x^2 \log (4))) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx\)

Optimal. Leaf size=34 \[ \frac {4+\frac {1}{2} \left (3-e^x-x\right ) x \left (e^x+4 \log (x)\right )}{-3+x \log (4)} \]

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Rubi [C]  time = 3.82, antiderivative size = 767, normalized size of antiderivative = 22.56, number of steps used = 55, number of rules used = 18, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {27, 12, 6742, 43, 77, 2199, 2177, 2178, 2194, 2314, 31, 2351, 2316, 2315, 2295, 6688, 2176, 2554} \begin {gather*} \frac {9 e^{\frac {3}{\log (4)}} (3+\log (16)) \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^4(4)}-\frac {27 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^4(4)}-\frac {e^{\frac {6}{\log (4)}} \log (4096) \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{2 \log ^3(4)}-\frac {3 e^{\frac {3}{\log (4)}} (3+\log (256)) \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^3(4)}+\frac {3 e^{\frac {3}{\log (4)}} (3+\log (16)) \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^3(4)}-\frac {27 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^3(4)}-\frac {2^{\frac {6}{\log ^2(4)}-1} (3+\log (256)) \text {Ei}\left (\frac {x \log ^2(4)-\log (64)}{\log ^2(4)}\right )}{\log ^2(4)}+\frac {3 e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{\log ^2(4)}+\frac {3 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^2(4)}+\frac {2 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log (4)}+\frac {12 \text {Li}_2\left (1-\frac {1}{3} x \log (4)\right )}{\log ^2(4)}-\frac {12 \text {Li}_2\left (1-\frac {x \log ^2(4)}{\log (64)}\right )}{\log ^2(4)}-\frac {27 e^x}{2 \log ^3(4) (3-x \log (4))}+\frac {9 e^x (3+\log (16))}{2 \log ^3(4) (3-x \log (4))}-\frac {6 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}+\frac {6 \log (3-x \log (4))}{\log ^2(4)}-\frac {12 \log \left (\frac {3}{\log (4)}\right ) \log (x \log (4)-3)}{\log ^2(4)}+\frac {6 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {18}{\log ^2(4) (3-x \log (4))}+\frac {12 \log \left (\frac {\log (64)}{\log ^2(4)}\right ) \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}-\frac {3 e^x (3+\log (256))}{2 \log ^2(4) (3-x \log (4))}+\frac {e^x (3+\log (16))}{2 \log ^2(4)}-\frac {e^{2 x} \log (16)}{4 \log ^2(4)}-\frac {2 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {3 e^x}{\log ^2(4)}+\frac {6 x \log (x)}{\log (4) (3-x \log (4))}-\frac {6 x \log (x)}{3-x \log (4)}-\frac {2 x \log (x)}{\log (4)}-\frac {e^x x}{2 \log (4)}+\frac {6 e^x \log (x)}{\log (4) (3-x \log (4))}-\frac {2 e^x \log (x)}{\log (4)}-\frac {6 \log (3-x \log (4))}{\log (4)}+\frac {3 e^x}{2 \log (4) (3-x \log (4))}+\frac {3 e^{2 x}}{2 \log (4) (3-x \log (4))}-\frac {18}{\log (4) (3-x \log (4))}+\frac {e^x}{2 \log (4)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-36 + 12*x + (-8 + 12*x - 4*x^2)*Log[4] + E^(2*x)*(3 + 6*x - 2*x^2*Log[4]) + E^x*(3 - 3*x + 3*x^2 + (-4*x
+ 2*x^2 - x^3)*Log[4]) + (-36 + 24*x - 4*x^2*Log[4] + E^x*(12 + 12*x - 4*x^2*Log[4]))*Log[x])/(18 - 12*x*Log[
4] + 2*x^2*Log[4]^2),x]

[Out]

(-27*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/(2*Log[4]^4) - (27*E^(3/Log[4])*ExpIntegralEi[-((3
- x*Log[4])/Log[4])])/(2*Log[4]^3) - (3*E^x)/Log[4]^2 + (3*E^(6/Log[4])*ExpIntegralEi[(-2*(3 - x*Log[4]))/Log[
4]])/Log[4]^2 + (3*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/(2*Log[4]^2) + E^x/(2*Log[4]) - (E^x*
x)/(2*Log[4]) + (2*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/Log[4] - (27*E^x)/(2*Log[4]^3*(3 - x*
Log[4])) + 18/(Log[4]^2*(3 - x*Log[4])) - 18/(Log[4]*(3 - x*Log[4])) + (3*E^x)/(2*Log[4]*(3 - x*Log[4])) + (3*
E^(2*x))/(2*Log[4]*(3 - x*Log[4])) - (2*(3 - Log[4])*(3 - Log[16]))/(Log[4]^2*(3 - x*Log[4])) - (E^(2*x)*Log[1
6])/(4*Log[4]^2) + (9*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*(3 + Log[16]))/(2*Log[4]^4) + (3*E^
(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*(3 + Log[16]))/Log[4]^3 + (E^x*(3 + Log[16]))/(2*Log[4]^2)
+ (9*E^x*(3 + Log[16]))/(2*Log[4]^3*(3 - x*Log[4])) - (3*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*
(3 + Log[256]))/(2*Log[4]^3) - (2^(-1 + 6/Log[4]^2)*ExpIntegralEi[(x*Log[4]^2 - Log[64])/Log[4]^2]*(3 + Log[25
6]))/Log[4]^2 - (3*E^x*(3 + Log[256]))/(2*Log[4]^2*(3 - x*Log[4])) - (E^(6/Log[4])*ExpIntegralEi[(-2*(3 - x*Lo
g[4]))/Log[4]]*Log[4096])/(2*Log[4]^3) - (2*E^x*Log[x])/Log[4] - (2*x*Log[x])/Log[4] - (6*x*Log[x])/(3 - x*Log
[4]) + (6*E^x*Log[x])/(Log[4]*(3 - x*Log[4])) + (6*x*Log[x])/(Log[4]*(3 - x*Log[4])) + (6*Log[3 - x*Log[4]])/L
og[4]^2 - (6*(2 - Log[4])*Log[3 - x*Log[4]])/Log[4]^2 - (6*Log[3 - x*Log[4]])/Log[4] - (12*Log[3/Log[4]]*Log[-
3 + x*Log[4]])/Log[4]^2 + (6*Log[x*Log[4]^2 - Log[64]])/Log[4]^2 + (12*Log[x*Log[4]^2 - Log[64]]*Log[Log[64]/L
og[4]^2])/Log[4]^2 + (12*PolyLog[2, 1 - (x*Log[4])/3])/Log[4]^2 - (12*PolyLog[2, 1 - (x*Log[4]^2)/Log[64]])/Lo
g[4]^2

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

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 2314

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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 {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{2 (-3+x \log (4))^2} \, dx\\ &=\frac {1}{2} \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{(-3+x \log (4))^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {36}{(-3+x \log (4))^2}+\frac {12 x}{(-3+x \log (4))^2}-\frac {4 (-2+x) (-1+x) \log (4)}{(-3+x \log (4))^2}-\frac {e^{2 x} \left (-3-6 x+x^2 \log (16)\right )}{(-3+x \log (4))^2}-\frac {36 \log (x)}{(-3+x \log (4))^2}+\frac {24 x \log (x)}{(-3+x \log (4))^2}-\frac {4 x^2 \log (4) \log (x)}{(-3+x \log (4))^2}+\frac {e^x \left (3-x^3 \log (4)+3 x^2 \left (1+\frac {\log (16)}{3}\right )-3 x \left (1+\log \left (4\ 2^{2/3}\right )\right )+12 \log (x)+12 x \log (x)-4 x^2 \log (4) \log (x)\right )}{(3-x \log (4))^2}\right ) \, dx\\ &=-\frac {18}{\log (4) (3-x \log (4))}-\frac {1}{2} \int \frac {e^{2 x} \left (-3-6 x+x^2 \log (16)\right )}{(-3+x \log (4))^2} \, dx+\frac {1}{2} \int \frac {e^x \left (3-x^3 \log (4)+3 x^2 \left (1+\frac {\log (16)}{3}\right )-3 x \left (1+\log \left (4\ 2^{2/3}\right )\right )+12 \log (x)+12 x \log (x)-4 x^2 \log (4) \log (x)\right )}{(3-x \log (4))^2} \, dx+6 \int \frac {x}{(-3+x \log (4))^2} \, dx+12 \int \frac {x \log (x)}{(-3+x \log (4))^2} \, dx-18 \int \frac {\log (x)}{(-3+x \log (4))^2} \, dx-(2 \log (4)) \int \frac {(-2+x) (-1+x)}{(-3+x \log (4))^2} \, dx-(2 \log (4)) \int \frac {x^2 \log (x)}{(-3+x \log (4))^2} \, dx\\ &=-\frac {18}{\log (4) (3-x \log (4))}-\frac {6 x \log (x)}{3-x \log (4)}-\frac {1}{2} \int \left (-\frac {3 e^{2 x}}{(-3+x \log (4))^2}+\frac {e^{2 x} \log (16)}{\log ^2(4)}+\frac {e^{2 x} \log (4096)}{\log ^2(4) (-3+x \log (4))}\right ) \, dx+\frac {1}{2} \int \frac {e^x \left (3-x^3 \log (4)+x^2 (3+\log (16))-x (3+\log (256))-4 \left (-3-3 x+x^2 \log (4)\right ) \log (x)\right )}{(3-x \log (4))^2} \, dx-6 \int \frac {1}{-3+x \log (4)} \, dx+6 \int \left (\frac {3}{\log (4) (-3+x \log (4))^2}+\frac {1}{x \log ^2(4)-\log (64)}\right ) \, dx+12 \int \left (\frac {3 \log (x)}{\log (4) (-3+x \log (4))^2}+\frac {\log (x)}{x \log ^2(4)-\log (64)}\right ) \, dx-(2 \log (4)) \int \left (\frac {1}{\log ^2(4)}+\frac {(3-2 \log (4)) (3-\log (4))}{\log ^2(4) (3-x \log (4))^2}-\frac {3 (-2+\log (4))}{\log ^2(4) (-3+x \log (4))}\right ) \, dx-(2 \log (4)) \int \left (\frac {\log (x)}{\log ^2(4)}+\frac {9 \log (x)}{\log ^2(4) (-3+x \log (4))^2}+\frac {6 \log (x)}{\log ^2(4) (-3+x \log (4))}\right ) \, dx\\ &=-\frac {2 x}{\log (4)}+\frac {18}{\log ^2(4) (3-x \log (4))}-\frac {18}{\log (4) (3-x \log (4))}-\frac {2 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {6 x \log (x)}{3-x \log (4)}-\frac {6 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}-\frac {6 \log (3-x \log (4))}{\log (4)}+\frac {6 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {1}{2} \int \left (\frac {3 e^x}{(-3+x \log (4))^2}-\frac {e^x x^3 \log (4)}{(-3+x \log (4))^2}+\frac {e^x x^2 (3+\log (16))}{(-3+x \log (4))^2}-\frac {e^x x (3+\log (256))}{(-3+x \log (4))^2}-\frac {4 e^x \left (-3-3 x+x^2 \log (4)\right ) \log (x)}{(-3+x \log (4))^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{2 x}}{(-3+x \log (4))^2} \, dx+12 \int \frac {\log (x)}{x \log ^2(4)-\log (64)} \, dx-\frac {2 \int \log (x) \, dx}{\log (4)}-\frac {12 \int \frac {\log (x)}{-3+x \log (4)} \, dx}{\log (4)}-\frac {18 \int \frac {\log (x)}{(-3+x \log (4))^2} \, dx}{\log (4)}+\frac {36 \int \frac {\log (x)}{(-3+x \log (4))^2} \, dx}{\log (4)}-\frac {\log (16) \int e^{2 x} \, dx}{2 \log ^2(4)}-\frac {\log (4096) \int \frac {e^{2 x}}{-3+x \log (4)} \, dx}{2 \log ^2(4)}\\ &=\frac {18}{\log ^2(4) (3-x \log (4))}-\frac {18}{\log (4) (3-x \log (4))}+\frac {3 e^{2 x}}{2 \log (4) (3-x \log (4))}-\frac {2 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {e^{2 x} \log (16)}{4 \log ^2(4)}-\frac {e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right ) \log (4096)}{2 \log ^3(4)}-\frac {2 x \log (x)}{\log (4)}-\frac {6 x \log (x)}{3-x \log (4)}+\frac {6 x \log (x)}{\log (4) (3-x \log (4))}-\frac {6 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}-\frac {6 \log (3-x \log (4))}{\log (4)}-\frac {12 \log \left (\frac {3}{\log (4)}\right ) \log (-3+x \log (4))}{\log ^2(4)}+\frac {6 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {12 \log \left (x \log ^2(4)-\log (64)\right ) \log \left (\frac {\log (64)}{\log ^2(4)}\right )}{\log ^2(4)}+\frac {3}{2} \int \frac {e^x}{(-3+x \log (4))^2} \, dx-2 \int \frac {e^x \left (-3-3 x+x^2 \log (4)\right ) \log (x)}{(-3+x \log (4))^2} \, dx+12 \int \frac {\log \left (\frac {x \log ^2(4)}{\log (64)}\right )}{x \log ^2(4)-\log (64)} \, dx+\frac {3 \int \frac {e^{2 x}}{-3+x \log (4)} \, dx}{\log (4)}-\frac {6 \int \frac {1}{-3+x \log (4)} \, dx}{\log (4)}+\frac {12 \int \frac {1}{-3+x \log (4)} \, dx}{\log (4)}-\frac {12 \int \frac {\log \left (\frac {1}{3} x \log (4)\right )}{-3+x \log (4)} \, dx}{\log (4)}-\frac {1}{2} \log (4) \int \frac {e^x x^3}{(-3+x \log (4))^2} \, dx+\frac {1}{2} (3+\log (16)) \int \frac {e^x x^2}{(-3+x \log (4))^2} \, dx+\frac {1}{2} (-3-\log (256)) \int \frac {e^x x}{(-3+x \log (4))^2} \, dx\\ &=\frac {3 e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{\log ^2(4)}+\frac {18}{\log ^2(4) (3-x \log (4))}-\frac {18}{\log (4) (3-x \log (4))}+\frac {3 e^x}{2 \log (4) (3-x \log (4))}+\frac {3 e^{2 x}}{2 \log (4) (3-x \log (4))}-\frac {2 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {e^{2 x} \log (16)}{4 \log ^2(4)}-\frac {e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right ) \log (4096)}{2 \log ^3(4)}-\frac {2 e^x \log (x)}{\log (4)}-\frac {2 x \log (x)}{\log (4)}-\frac {6 x \log (x)}{3-x \log (4)}+\frac {6 e^x \log (x)}{\log (4) (3-x \log (4))}+\frac {6 x \log (x)}{\log (4) (3-x \log (4))}+\frac {6 \log (3-x \log (4))}{\log ^2(4)}-\frac {6 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}-\frac {6 \log (3-x \log (4))}{\log (4)}-\frac {12 \log \left (\frac {3}{\log (4)}\right ) \log (-3+x \log (4))}{\log ^2(4)}+\frac {6 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {12 \log \left (x \log ^2(4)-\log (64)\right ) \log \left (\frac {\log (64)}{\log ^2(4)}\right )}{\log ^2(4)}+\frac {12 \text {Li}_2\left (1-\frac {1}{3} x \log (4)\right )}{\log ^2(4)}-\frac {12 \text {Li}_2\left (1-\frac {x \log ^2(4)}{\log (64)}\right )}{\log ^2(4)}+2 \int \frac {e^x}{-3+x \log (4)} \, dx+\frac {3 \int \frac {e^x}{-3+x \log (4)} \, dx}{2 \log (4)}-\frac {1}{2} \log (4) \int \left (\frac {6 e^x}{\log ^3(4)}+\frac {e^x x}{\log ^2(4)}+\frac {27 e^x}{\log ^3(4) (-3+x \log (4))^2}+\frac {27 e^x}{\log ^3(4) (-3+x \log (4))}\right ) \, dx+\frac {1}{2} (3+\log (16)) \int \left (\frac {e^x}{\log ^2(4)}+\frac {9 e^x}{\log ^2(4) (-3+x \log (4))^2}+\frac {6 e^x}{\log ^2(4) (-3+x \log (4))}\right ) \, dx+\frac {1}{2} (-3-\log (256)) \int \left (\frac {3 e^x}{\log (4) (-3+x \log (4))^2}+\frac {e^x}{x \log ^2(4)-\log (64)}\right ) \, dx\\ &=\frac {3 e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{\log ^2(4)}+\frac {3 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{2 \log ^2(4)}+\frac {2 e^{\frac {3}{\log (4)}} \text {Ei}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log (4)}+\frac {18}{\log ^2(4) (3-x \log (4))}-\frac {18}{\log (4) (3-x \log (4))}+\frac {3 e^x}{2 \log (4) (3-x \log (4))}+\frac {3 e^{2 x}}{2 \log (4) (3-x \log (4))}-\frac {2 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {e^{2 x} \log (16)}{4 \log ^2(4)}-\frac {e^{\frac {6}{\log (4)}} \text {Ei}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right ) \log (4096)}{2 \log ^3(4)}-\frac {2 e^x \log (x)}{\log (4)}-\frac {2 x \log (x)}{\log (4)}-\frac {6 x \log (x)}{3-x \log (4)}+\frac {6 e^x \log (x)}{\log (4) (3-x \log (4))}+\frac {6 x \log (x)}{\log (4) (3-x \log (4))}+\frac {6 \log (3-x \log (4))}{\log ^2(4)}-\frac {6 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}-\frac {6 \log (3-x \log (4))}{\log (4)}-\frac {12 \log \left (\frac {3}{\log (4)}\right ) \log (-3+x \log (4))}{\log ^2(4)}+\frac {6 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {12 \log \left (x \log ^2(4)-\log (64)\right ) \log \left (\frac {\log (64)}{\log ^2(4)}\right )}{\log ^2(4)}+\frac {12 \text {Li}_2\left (1-\frac {1}{3} x \log (4)\right )}{\log ^2(4)}-\frac {12 \text {Li}_2\left (1-\frac {x \log ^2(4)}{\log (64)}\right )}{\log ^2(4)}-\frac {3 \int e^x \, dx}{\log ^2(4)}-\frac {27 \int \frac {e^x}{(-3+x \log (4))^2} \, dx}{2 \log ^2(4)}-\frac {27 \int \frac {e^x}{-3+x \log (4)} \, dx}{2 \log ^2(4)}-\frac {\int e^x x \, dx}{2 \log (4)}+\frac {(3+\log (16)) \int e^x \, dx}{2 \log ^2(4)}+\frac {(3 (3+\log (16))) \int \frac {e^x}{-3+x \log (4)} \, dx}{\log ^2(4)}+\frac {(9 (3+\log (16))) \int \frac {e^x}{(-3+x \log (4))^2} \, dx}{2 \log ^2(4)}+\frac {1}{2} (-3-\log (256)) \int \frac {e^x}{x \log ^2(4)-\log (64)} \, dx+\frac {(3 (-3-\log (256))) \int \frac {e^x}{(-3+x \log (4))^2} \, dx}{2 \log (4)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.42, size = 146, normalized size = 4.29 \begin {gather*} \frac {6 e^x \log ^3(4)+e^{2 x} x \log ^4(4)+e^x x^2 \log ^4(4)-4 \log ^3(4) \log (16)-e^x x \log ^3(4) \log (64)-e^x \log (4) \log (16) \log (64)-e^{\frac {3}{\log (4)}} \text {Ei}\left (x-\frac {3}{\log (4)}\right ) (-3+x \log (4)) \left (4 \log ^3(4)-\log ^2(4) \log (256)-\log (64) \log (256)+\log (16) \log (4096)\right )+4 x \left (-3+e^x+x\right ) \log ^4(4) \log (x)}{2 \log ^4(4) (3-x \log (4))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + 12*x + (-8 + 12*x - 4*x^2)*Log[4] + E^(2*x)*(3 + 6*x - 2*x^2*Log[4]) + E^x*(3 - 3*x + 3*x^2 +
(-4*x + 2*x^2 - x^3)*Log[4]) + (-36 + 24*x - 4*x^2*Log[4] + E^x*(12 + 12*x - 4*x^2*Log[4]))*Log[x])/(18 - 12*
x*Log[4] + 2*x^2*Log[4]^2),x]

[Out]

(6*E^x*Log[4]^3 + E^(2*x)*x*Log[4]^4 + E^x*x^2*Log[4]^4 - 4*Log[4]^3*Log[16] - E^x*x*Log[4]^3*Log[64] - E^x*Lo
g[4]*Log[16]*Log[64] - E^(3/Log[4])*ExpIntegralEi[x - 3/Log[4]]*(-3 + x*Log[4])*(4*Log[4]^3 - Log[4]^2*Log[256
] - Log[64]*Log[256] + Log[16]*Log[4096]) + 4*x*(-3 + E^x + x)*Log[4]^4*Log[x])/(2*Log[4]^4*(3 - x*Log[4]))

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fricas [A]  time = 0.70, size = 44, normalized size = 1.29 \begin {gather*} -\frac {x e^{\left (2 \, x\right )} + {\left (x^{2} - 3 \, x\right )} e^{x} + 4 \, {\left (x^{2} + x e^{x} - 3 \, x\right )} \log \relax (x) - 8}{2 \, {\left (2 \, x \log \relax (2) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(-4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^
3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x)+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, a
lgorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 49, normalized size = 1.44 \begin {gather*} -\frac {x^{2} e^{x} + 4 \, x^{2} \log \relax (x) + 4 \, x e^{x} \log \relax (x) + x e^{\left (2 \, x\right )} - 3 \, x e^{x} - 12 \, x \log \relax (x) - 8}{2 \, {\left (2 \, x \log \relax (2) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(-4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^
3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x)+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, a
lgorithm="giac")

[Out]

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

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maple [B]  time = 0.31, size = 115, normalized size = 3.38

method	result	size
default	\(\frac {3 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}-4 x \,{\mathrm e}^{x} \ln \relax (x )}{4 x \ln \relax (2)-6}+\frac {4}{2 x \ln \relax (2)-3}-\frac {\ln \relax (x ) x}{\ln \relax (2)}-\frac {3 \ln \relax (x ) x}{\ln \relax (2) \left (2 x \ln \relax (2)-3\right )}+\frac {6 \ln \relax (x ) x}{2 x \ln \relax (2)-3}-\frac {3 \,{\mathrm e}^{2 x}}{8 \ln \relax (2)^{2} \left (x -\frac {3}{2 \ln \relax (2)}\right )}-\frac {{\mathrm e}^{2 x}}{4 \ln \relax (2)}\)	\(115\)
risch	\(-\frac {\left (4 x^{2} \ln \relax (2)^{2}+4 x \ln \relax (2)^{2} {\mathrm e}^{x}-6 x \ln \relax (2)-18 \ln \relax (2)+9\right ) \ln \relax (x )}{2 \ln \relax (2)^{2} \left (2 x \ln \relax (2)-3\right )}+\frac {-{\mathrm e}^{x} \ln \relax (2)^{2} x^{2}-x \ln \relax (2)^{2} {\mathrm e}^{2 x}+12 \ln \relax (2)^{2} \ln \left (-x \right ) x +3 x \ln \relax (2)^{2} {\mathrm e}^{x}-6 \ln \relax (2) \ln \left (-x \right ) x +8 \ln \relax (2)^{2}-18 \ln \relax (2) \ln \left (-x \right )+9 \ln \left (-x \right )}{2 \ln \relax (2)^{2} \left (2 x \ln \relax (2)-3\right )}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-8*x^2*ln(2)+12*x+12)*exp(x)-8*x^2*ln(2)+24*x-36)*ln(x)+(-4*x^2*ln(2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*
x)*ln(2)+3*x^2-3*x+3)*exp(x)+2*(-4*x^2+12*x-8)*ln(2)+12*x-36)/(8*x^2*ln(2)^2-24*x*ln(2)+18),x,method=_RETURNVE
RBOSE)

[Out]

1/2*(3*exp(x)*x-exp(x)*x^2-4*x*exp(x)*ln(x))/(2*x*ln(2)-3)+4/(2*x*ln(2)-3)-1/ln(2)*ln(x)*x-3/ln(2)*ln(x)*x/(2*
x*ln(2)-3)+6*ln(x)*x/(2*x*ln(2)-3)-3/8/ln(2)^2*exp(x)^2/(x-3/2/ln(2))-1/4/ln(2)*exp(x)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{2} \, {\left (\frac {9}{2 \, x \log \relax (2)^{4} - 3 \, \log \relax (2)^{3}} - \frac {2 \, x}{\log \relax (2)^{2}} - \frac {6 \, \log \left (2 \, x \log \relax (2) - 3\right )}{\log \relax (2)^{3}}\right )} \log \relax (2) - 3 \, {\left (\frac {3}{2 \, x \log \relax (2)^{3} - 3 \, \log \relax (2)^{2}} - \frac {\log \left (2 \, x \log \relax (2) - 3\right )}{\log \relax (2)^{2}}\right )} \log \relax (2) - \frac {4 \, x e^{x} \log \relax (2)^{2} \log \relax (x) - 4 \, x^{2} \log \relax (2)^{2} + x e^{\left (2 \, x\right )} \log \relax (2)^{2} + 6 \, x \log \relax (2) + {\left (4 \, x^{2} \log \relax (2)^{2} - 6 \, x \log \relax (2) - 18 \, \log \relax (2) + 9\right )} \log \relax (x)}{2 \, {\left (2 \, x \log \relax (2)^{3} - 3 \, \log \relax (2)^{2}\right )}} - \frac {3 \, e^{\left (\frac {3}{2 \, \log \relax (2)}\right )} E_{2}\left (-\frac {2 \, x \log \relax (2) - 3}{2 \, \log \relax (2)}\right )}{4 \, {\left (2 \, x \log \relax (2) - 3\right )} \log \relax (2)} + \frac {4 \, \log \relax (2)}{2 \, x \log \relax (2)^{2} - 3 \, \log \relax (2)} - \frac {3 \, {\left (2 \, \log \relax (2) - 1\right )} \log \left (2 \, x \log \relax (2) - 3\right )}{2 \, \log \relax (2)^{2}} + \frac {3 \, {\left (2 \, \log \relax (2) - 1\right )} \log \relax (x)}{2 \, \log \relax (2)^{2}} - \frac {9}{2 \, {\left (2 \, x \log \relax (2)^{3} - 3 \, \log \relax (2)^{2}\right )}} + \frac {9}{2 \, x \log \relax (2)^{2} - 3 \, \log \relax (2)} + \frac {3 \, \log \left (2 \, x \log \relax (2) - 3\right )}{2 \, \log \relax (2)^{2}} - \frac {1}{2} \, \int \frac {{\left (2 \, x^{3} \log \relax (2) - x^{2} {\left (4 \, \log \relax (2) + 3\right )} + 3 \, x + 12\right )} e^{x}}{4 \, x^{2} \log \relax (2)^{2} - 12 \, x \log \relax (2) + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(-4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^
3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x)+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, a
lgorithm="maxima")

[Out]

1/2*(9/(2*x*log(2)^4 - 3*log(2)^3) - 2*x/log(2)^2 - 6*log(2*x*log(2) - 3)/log(2)^3)*log(2) - 3*(3/(2*x*log(2)^
3 - 3*log(2)^2) - log(2*x*log(2) - 3)/log(2)^2)*log(2) - 1/2*(4*x*e^x*log(2)^2*log(x) - 4*x^2*log(2)^2 + x*e^(
2*x)*log(2)^2 + 6*x*log(2) + (4*x^2*log(2)^2 - 6*x*log(2) - 18*log(2) + 9)*log(x))/(2*x*log(2)^3 - 3*log(2)^2)
- 3/4*e^(3/2/log(2))*exp_integral_e(2, -1/2*(2*x*log(2) - 3)/log(2))/((2*x*log(2) - 3)*log(2)) + 4*log(2)/(2*
x*log(2)^2 - 3*log(2)) - 3/2*(2*log(2) - 1)*log(2*x*log(2) - 3)/log(2)^2 + 3/2*(2*log(2) - 1)*log(x)/log(2)^2
- 9/2/(2*x*log(2)^3 - 3*log(2)^2) + 9/(2*x*log(2)^2 - 3*log(2)) + 3/2*log(2*x*log(2) - 3)/log(2)^2 - 1/2*integ
rate((2*x^3*log(2) - x^2*(4*log(2) + 3) + 3*x + 12)*e^x/(4*x^2*log(2)^2 - 12*x*log(2) + 9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {12\,x+\ln \relax (x)\,\left (24\,x+{\mathrm {e}}^x\,\left (-8\,\ln \relax (2)\,x^2+12\,x+12\right )-8\,x^2\,\ln \relax (2)-36\right )-2\,\ln \relax (2)\,\left (4\,x^2-12\,x+8\right )+{\mathrm {e}}^{2\,x}\,\left (-4\,\ln \relax (2)\,x^2+6\,x+3\right )-{\mathrm {e}}^x\,\left (3\,x+2\,\ln \relax (2)\,\left (x^3-2\,x^2+4\,x\right )-3\,x^2-3\right )-36}{8\,{\ln \relax (2)}^2\,x^2-24\,\ln \relax (2)\,x+18} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + log(x)*(24*x + exp(x)*(12*x - 8*x^2*log(2) + 12) - 8*x^2*log(2) - 36) - 2*log(2)*(4*x^2 - 12*x + 8
) + exp(2*x)*(6*x - 4*x^2*log(2) + 3) - exp(x)*(3*x + 2*log(2)*(4*x - 2*x^2 + x^3) - 3*x^2 - 3) - 36)/(8*x^2*l
og(2)^2 - 24*x*log(2) + 18),x)

[Out]

int((12*x + log(x)*(24*x + exp(x)*(12*x - 8*x^2*log(2) + 12) - 8*x^2*log(2) - 36) - 2*log(2)*(4*x^2 - 12*x + 8
) + exp(2*x)*(6*x - 4*x^2*log(2) + 3) - exp(x)*(3*x + 2*log(2)*(4*x - 2*x^2 + x^3) - 3*x^2 - 3) - 36)/(8*x^2*l
og(2)^2 - 24*x*log(2) + 18), x)

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sympy [B]  time = 0.98, size = 153, normalized size = 4.50 \begin {gather*} \frac {\left (- 4 x^{2} \log {\relax (2 )} + 6 x\right ) e^{2 x} + \left (- 4 x^{3} \log {\relax (2 )} - 16 x^{2} \log {\relax (2 )} \log {\relax (x )} + 6 x^{2} + 12 x^{2} \log {\relax (2 )} + 24 x \log {\relax (x )} - 18 x\right ) e^{x}}{16 x^{2} \log {\relax (2 )}^{2} - 48 x \log {\relax (2 )} + 36} + \frac {3 \left (-1 + 2 \log {\relax (2 )}\right ) \log {\relax (x )}}{2 \log {\relax (2 )}^{2}} + \frac {\left (- 4 x^{2} \log {\relax (2 )}^{2} + 6 x \log {\relax (2 )} - 9 + 18 \log {\relax (2 )}\right ) \log {\relax (x )}}{4 x \log {\relax (2 )}^{3} - 6 \log {\relax (2 )}^{2}} + \frac {4}{2 x \log {\relax (2 )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x**2*ln(2)+12*x+12)*exp(x)-8*x**2*ln(2)+24*x-36)*ln(x)+(-4*x**2*ln(2)+6*x+3)*exp(x)**2+(2*(-x*
*3+2*x**2-4*x)*ln(2)+3*x**2-3*x+3)*exp(x)+2*(-4*x**2+12*x-8)*ln(2)+12*x-36)/(8*x**2*ln(2)**2-24*x*ln(2)+18),x)

[Out]

((-4*x**2*log(2) + 6*x)*exp(2*x) + (-4*x**3*log(2) - 16*x**2*log(2)*log(x) + 6*x**2 + 12*x**2*log(2) + 24*x*lo
g(x) - 18*x)*exp(x))/(16*x**2*log(2)**2 - 48*x*log(2) + 36) + 3*(-1 + 2*log(2))*log(x)/(2*log(2)**2) + (-4*x**
2*log(2)**2 + 6*x*log(2) - 9 + 18*log(2))*log(x)/(4*x*log(2)**3 - 6*log(2)**2) + 4/(2*x*log(2) - 3)

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