\(\int \frac {1}{8} (-16 e^4+(16 e^4-e^8 x) \log (3)+e^8 x \log ^2(3)+(16 e^4-e^8 x+2 e^8 x \log (3)) \log (\frac {3}{x})+e^8 x \log ^2(\frac {3}{x})) \, dx\) [4317]

Optimal result

Integrand size = 74, antiderivative size = 29 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {\left (x+x \left (3+\frac {1}{4} e^4 x \left (\log (3)+\log \left (\frac {3}{x}\right )\right )\right )\right )^2}{x^2} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(29)=58\).

Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 6, 2350, 9, 2342, 2341} \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {e^8 x^2}{32}+\frac {1}{16} e^8 x^2 \log ^2\left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log ^2(3)-\frac {1}{16} e^8 x^2 (1-\log (9)) \log \left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log \left (\frac {3}{x}\right )-2 e^4 x+2 e^4 x \log \left (\frac {3}{x}\right )-\frac {\left (32-e^4 x (1-\log (9))\right )^2}{32 (1-\log (9))}-\frac {1}{16} \left (16-e^4 x\right )^2 \log (3) \]

[In]

Int[(-16*E^4 + (16*E^4 - E^8*x)*Log[3] + E^8*x*Log[3]^2 + (16*E^4 - E^8*x + 2*E^8*x*Log[3])*Log[3/x] + E^8*x*L
og[3/x]^2)/8,x]

[Out]

-2*E^4*x + (E^8*x^2)/32 - ((16 - E^4*x)^2*Log[3])/16 + (E^8*x^2*Log[3]^2)/16 - (32 - E^4*x*(1 - Log[9]))^2/(32
*(1 - Log[9])) + 2*E^4*x*Log[3/x] + (E^8*x^2*Log[3/x])/16 - (E^8*x^2*(1 - Log[9])*Log[3/x])/16 + (E^8*x^2*Log[
3/x]^2)/16

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 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[a*((b + c*x)^2/(2*c)), x] /; FreeQ[{a, b, c}, 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 2341

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 2342

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 2350

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx \\ & = -2 e^4 x-\frac {1}{16} \left (16-e^4 x\right )^2 \log (3)+\frac {1}{16} e^8 x^2 \log ^2(3)+\frac {1}{8} \int \left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right ) \, dx+\frac {1}{8} e^8 \int x \log ^2\left (\frac {3}{x}\right ) \, dx \\ & = -2 e^4 x-\frac {1}{16} \left (16-e^4 x\right )^2 \log (3)+\frac {1}{16} e^8 x^2 \log ^2(3)+\frac {1}{16} e^8 x^2 \log ^2\left (\frac {3}{x}\right )+\frac {1}{8} \int \left (16 e^4+e^8 x (-1+2 \log (3))\right ) \log \left (\frac {3}{x}\right ) \, dx+\frac {1}{8} e^8 \int x \log \left (\frac {3}{x}\right ) \, dx \\ & = -2 e^4 x+\frac {e^8 x^2}{32}-\frac {1}{16} \left (16-e^4 x\right )^2 \log (3)+\frac {1}{16} e^8 x^2 \log ^2(3)+2 e^4 x \log \left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log \left (\frac {3}{x}\right )-\frac {1}{16} e^8 x^2 (1-\log (9)) \log \left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log ^2\left (\frac {3}{x}\right )+\frac {1}{8} \int \frac {1}{2} e^4 \left (32-e^4 x (1-\log (9))\right ) \, dx \\ & = -2 e^4 x+\frac {e^8 x^2}{32}-\frac {1}{16} \left (16-e^4 x\right )^2 \log (3)+\frac {1}{16} e^8 x^2 \log ^2(3)-\frac {\left (32-e^4 x (1-\log (9))\right )^2}{32 (1-\log (9))}+2 e^4 x \log \left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log \left (\frac {3}{x}\right )-\frac {1}{16} e^8 x^2 (1-\log (9)) \log \left (\frac {3}{x}\right )+\frac {1}{16} e^8 x^2 \log ^2\left (\frac {3}{x}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(29)=58\).

Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {1}{32} e^4 x \left (64 \log (3)+e^4 x \log (3)-e^4 x \log (9)+e^4 x \log ^2(9)+\left (64+e^4 x (1+\log (81))\right ) \log \left (\frac {3}{x}\right )+e^4 x \log \left (\frac {1}{x}\right ) \left (-1+\log (9)+2 \log \left (\frac {3}{x}\right )\right )\right ) \]

[In]

Integrate[(-16*E^4 + (16*E^4 - E^8*x)*Log[3] + E^8*x*Log[3]^2 + (16*E^4 - E^8*x + 2*E^8*x*Log[3])*Log[3/x] + E
^8*x*Log[3/x]^2)/8,x]

[Out]

(E^4*x*(64*Log[3] + E^4*x*Log[3] - E^4*x*Log[9] + E^4*x*Log[9]^2 + (64 + E^4*x*(1 + Log[81]))*Log[3/x] + E^4*x
*Log[x^(-1)]*(-1 + Log[9] + 2*Log[3/x])))/32

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(29)=58\).

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10

method	result	size
risch	\(\frac {x^{2} {\mathrm e}^{8} \ln \left (\frac {3}{x}\right )^{2}}{16}+2 \,{\mathrm e}^{4} x \ln \left (3\right )+2 \,{\mathrm e}^{4} \ln \left (\frac {3}{x}\right ) x +\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2} \ln \left (\frac {3}{x}\right )}{8}\)	\(61\)
norman	\(\frac {x^{2} {\mathrm e}^{8} \ln \left (\frac {3}{x}\right )^{2}}{16}+2 \,{\mathrm e}^{4} x \ln \left (3\right )+2 \,{\mathrm e}^{4} \ln \left (\frac {3}{x}\right ) x +\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2} \ln \left (\frac {3}{x}\right )}{8}\)	\(67\)
parallelrisch	\(\frac {x^{2} {\mathrm e}^{8} \ln \left (\frac {3}{x}\right )^{2}}{16}+2 \,{\mathrm e}^{4} x \ln \left (3\right )+2 \,{\mathrm e}^{4} \ln \left (\frac {3}{x}\right ) x +\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2} \ln \left (\frac {3}{x}\right )}{8}\)	\(67\)
default	\(-\frac {\ln \left (3\right ) {\mathrm e}^{4} \left (\frac {x^{2} {\mathrm e}^{4}}{2}-16 x \right )}{8}-\frac {9 \,{\mathrm e}^{8} \ln \left (3\right ) \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{4}+\frac {9 \,{\mathrm e}^{8} \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{8}-6 \,{\mathrm e}^{4} \left (-\frac {x \ln \left (\frac {3}{x}\right )}{3}-\frac {x}{3}\right )+\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {{\mathrm e}^{8} \left (\frac {x^{2} \ln \left (\frac {3}{x}\right )^{2}}{2}+\frac {x^{2} \ln \left (\frac {3}{x}\right )}{2}+\frac {x^{2}}{4}\right )}{8}-2 x \,{\mathrm e}^{4}\)	\(138\)
derivativedivides	\(-\frac {9 \,{\mathrm e}^{8} \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )^{2}}{18}-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{8}-\frac {9 \,{\mathrm e}^{8} \ln \left (3\right ) \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{4}+\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {9 \,{\mathrm e}^{8} \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{8}-6 \,{\mathrm e}^{4} \left (-\frac {x \ln \left (\frac {3}{x}\right )}{3}-\frac {x}{3}\right )-\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2}}{16}+2 \,{\mathrm e}^{4} x \ln \left (3\right )-2 x \,{\mathrm e}^{4}\)	\(139\)
parts	\(-\frac {9 \,{\mathrm e}^{8} \ln \left (3\right ) \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{4}+\frac {9 \,{\mathrm e}^{8} \left (-\frac {x^{2} \ln \left (\frac {3}{x}\right )}{18}-\frac {x^{2}}{36}\right )}{8}-6 \,{\mathrm e}^{4} \left (-\frac {x \ln \left (\frac {3}{x}\right )}{3}-\frac {x}{3}\right )+2 \,{\mathrm e}^{4} x \ln \left (3\right )-\frac {{\mathrm e}^{8} \ln \left (3\right ) x^{2}}{16}+\frac {{\mathrm e}^{8} \ln \left (3\right )^{2} x^{2}}{16}+\frac {{\mathrm e}^{8} \left (\frac {x^{2} \ln \left (\frac {3}{x}\right )^{2}}{2}+\frac {x^{2} \ln \left (\frac {3}{x}\right )}{2}+\frac {x^{2}}{4}\right )}{8}-2 x \,{\mathrm e}^{4}\)	\(139\)

[In]

int(1/8*x*exp(4)^2*ln(3/x)^2+1/8*(2*x*exp(4)^2*ln(3)-x*exp(4)^2+16*exp(4))*ln(3/x)+1/8*x*exp(4)^2*ln(3)^2+1/8*
(-x*exp(4)^2+16*exp(4))*ln(3)-2*exp(4),x,method=_RETURNVERBOSE)

[Out]

1/16*x^2*exp(8)*ln(3/x)^2+2*exp(4)*x*ln(3)+2*exp(4)*ln(3/x)*x+1/16*exp(8)*ln(3)^2*x^2+1/8*exp(8)*ln(3)*x^2*ln(
3/x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {1}{16} \, x^{2} e^{8} \log \left (3\right )^{2} + \frac {1}{16} \, x^{2} e^{8} \log \left (\frac {3}{x}\right )^{2} + 2 \, x e^{4} \log \left (3\right ) + \frac {1}{8} \, {\left (x^{2} e^{8} \log \left (3\right ) + 16 \, x e^{4}\right )} \log \left (\frac {3}{x}\right ) \]

[In]

integrate(1/8*x*exp(4)^2*log(3/x)^2+1/8*(2*x*exp(4)^2*log(3)-x*exp(4)^2+16*exp(4))*log(3/x)+1/8*x*exp(4)^2*log
(3)^2+1/8*(-x*exp(4)^2+16*exp(4))*log(3)-2*exp(4),x, algorithm="fricas")

[Out]

1/16*x^2*e^8*log(3)^2 + 1/16*x^2*e^8*log(3/x)^2 + 2*x*e^4*log(3) + 1/8*(x^2*e^8*log(3) + 16*x*e^4)*log(3/x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {x^{2} e^{8} \log {\left (\frac {3}{x} \right )}^{2}}{16} + \frac {x^{2} e^{8} \log {\left (3 \right )}^{2}}{16} + 2 x e^{4} \log {\left (3 \right )} + \left (\frac {x^{2} e^{8} \log {\left (3 \right )}}{8} + 2 x e^{4}\right ) \log {\left (\frac {3}{x} \right )} \]

[In]

integrate(1/8*x*exp(4)**2*ln(3/x)**2+1/8*(2*x*exp(4)**2*ln(3)-x*exp(4)**2+16*exp(4))*ln(3/x)+1/8*x*exp(4)**2*l
n(3)**2+1/8*(-x*exp(4)**2+16*exp(4))*ln(3)-2*exp(4),x)

[Out]

x**2*exp(8)*log(3/x)**2/16 + x**2*exp(8)*log(3)**2/16 + 2*x*exp(4)*log(3) + (x**2*exp(8)*log(3)/8 + 2*x*exp(4)
)*log(3/x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (28) = 56\).

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.72 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {1}{16} \, x^{2} e^{8} \log \left (3\right )^{2} + \frac {1}{16} \, x^{2} e^{8} \log \left (\frac {3}{x}\right )^{2} + \frac {1}{32} \, {\left (2 \, e^{8} \log \left (3\right ) - e^{8}\right )} x^{2} + \frac {1}{32} \, {\left (2 \, x^{2} \log \left (\frac {3}{x}\right ) + x^{2}\right )} e^{8} - \frac {1}{16} \, {\left (x^{2} e^{8} - 32 \, x e^{4}\right )} \log \left (3\right ) + \frac {1}{16} \, {\left (2 \, x^{2} e^{8} \log \left (3\right ) - x^{2} e^{8} + 32 \, x e^{4}\right )} \log \left (\frac {3}{x}\right ) \]

[In]

integrate(1/8*x*exp(4)^2*log(3/x)^2+1/8*(2*x*exp(4)^2*log(3)-x*exp(4)^2+16*exp(4))*log(3/x)+1/8*x*exp(4)^2*log
(3)^2+1/8*(-x*exp(4)^2+16*exp(4))*log(3)-2*exp(4),x, algorithm="maxima")

[Out]

1/16*x^2*e^8*log(3)^2 + 1/16*x^2*e^8*log(3/x)^2 + 1/32*(2*e^8*log(3) - e^8)*x^2 + 1/32*(2*x^2*log(3/x) + x^2)*
e^8 - 1/16*(x^2*e^8 - 32*x*e^4)*log(3) + 1/16*(2*x^2*e^8*log(3) - x^2*e^8 + 32*x*e^4)*log(3/x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.07 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {1}{16} \, x^{2} e^{8} \log \left (3\right )^{2} + \frac {1}{32} \, {\left (2 \, e^{8} \log \left (3\right ) + \frac {64 \, e^{4}}{x} - e^{8}\right )} x^{2} + \frac {1}{32} \, {\left (2 \, x^{2} \log \left (\frac {3}{x}\right )^{2} + 2 \, x^{2} \log \left (\frac {3}{x}\right ) + x^{2}\right )} e^{8} - 2 \, x e^{4} - \frac {1}{16} \, {\left (x^{2} e^{8} - 32 \, x e^{4}\right )} \log \left (3\right ) + \frac {1}{16} \, {\left (2 \, x^{2} e^{8} \log \left (3\right ) - x^{2} e^{8} + 32 \, x e^{4}\right )} \log \left (\frac {3}{x}\right ) \]

[In]

integrate(1/8*x*exp(4)^2*log(3/x)^2+1/8*(2*x*exp(4)^2*log(3)-x*exp(4)^2+16*exp(4))*log(3/x)+1/8*x*exp(4)^2*log
(3)^2+1/8*(-x*exp(4)^2+16*exp(4))*log(3)-2*exp(4),x, algorithm="giac")

[Out]

1/16*x^2*e^8*log(3)^2 + 1/32*(2*e^8*log(3) + 64*e^4/x - e^8)*x^2 + 1/32*(2*x^2*log(3/x)^2 + 2*x^2*log(3/x) + x
^2)*e^8 - 2*x*e^4 - 1/16*(x^2*e^8 - 32*x*e^4)*log(3) + 1/16*(2*x^2*e^8*log(3) - x^2*e^8 + 32*x*e^4)*log(3/x)

Mupad [B] (verification not implemented)

Time = 10.65 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {1}{8} \left (-16 e^4+\left (16 e^4-e^8 x\right ) \log (3)+e^8 x \log ^2(3)+\left (16 e^4-e^8 x+2 e^8 x \log (3)\right ) \log \left (\frac {3}{x}\right )+e^8 x \log ^2\left (\frac {3}{x}\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^4\,\left (\ln \left (\frac {1}{x}\right )+2\,\ln \left (3\right )\right )\,\left (x\,\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^4+2\,x\,{\mathrm {e}}^4\,\ln \left (3\right )+32\right )}{16} \]

[In]

int((log(3/x)*(16*exp(4) - x*exp(8) + 2*x*exp(8)*log(3)))/8 - 2*exp(4) + (log(3)*(16*exp(4) - x*exp(8)))/8 + (
x*exp(8)*log(3)^2)/8 + (x*exp(8)*log(3/x)^2)/8,x)

[Out]

(x*exp(4)*(log(1/x) + 2*log(3))*(x*log(1/x)*exp(4) + 2*x*exp(4)*log(3) + 32))/16