\(\int \frac {1}{80} (240+8 x+4 x^2+(-8 x^2-4 x^3) \log (2)+(2 x^3+x^4) \log ^2(2)+(16 x+12 x^2+(-24 x^2-16 x^3) \log (2)+(8 x^3+5 x^4) \log ^2(2)) \log (x)) \, dx\) [7986]

Optimal result

Integrand size = 84, antiderivative size = 25 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=x+(2+x) \left (2+\frac {1}{80} x^2 (-2+x \log (2))^2 \log (x)\right ) \]

[Out]

(2+x)*(1/80*x^2*(x*ln(2)-2)^2*ln(x)+2)+x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(25)=50\).

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.88, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12, 2403, 2341} \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=\frac {1}{80} x^5 \log ^2(2) \log (x)+\frac {1}{160} x^4 \log ^2(2)-\frac {1}{40} x^4 (2-\log (2)) \log (2) \log (x)+\frac {1}{160} x^4 (2-\log (2)) \log (2)-\frac {1}{80} x^4 \log (2)+\frac {x^3}{60}+\frac {1}{20} x^3 (1-\log (4)) \log (x)-\frac {1}{60} x^3 (1-\log (4))-\frac {1}{30} x^3 \log (2)+\frac {1}{10} x^2 \log (x)+3 x \]

[In]

Int[(240 + 8*x + 4*x^2 + (-8*x^2 - 4*x^3)*Log[2] + (2*x^3 + x^4)*Log[2]^2 + (16*x + 12*x^2 + (-24*x^2 - 16*x^3
)*Log[2] + (8*x^3 + 5*x^4)*Log[2]^2)*Log[x])/80,x]

[Out]

3*x + x^3/60 - (x^3*Log[2])/30 - (x^4*Log[2])/80 + (x^4*(2 - Log[2])*Log[2])/160 + (x^4*Log[2]^2)/160 - (x^3*(
1 - Log[4]))/60 + (x^2*Log[x])/10 - (x^4*(2 - Log[2])*Log[2]*Log[x])/40 + (x^5*Log[2]^2*Log[x])/80 + (x^3*(1 -
Log[4])*Log[x])/20

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 2403

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{80} \int \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx \\ & = 3 x+\frac {x^2}{20}+\frac {x^3}{60}+\frac {1}{80} \int \left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x) \, dx+\frac {1}{80} \log (2) \int \left (-8 x^2-4 x^3\right ) \, dx+\frac {1}{80} \log ^2(2) \int \left (2 x^3+x^4\right ) \, dx \\ & = 3 x+\frac {x^2}{20}+\frac {x^3}{60}-\frac {1}{30} x^3 \log (2)-\frac {1}{80} x^4 \log (2)+\frac {1}{160} x^4 \log ^2(2)+\frac {1}{400} x^5 \log ^2(2)+\frac {1}{80} \int \left (16 x \log (x)+8 x^3 (-2+\log (2)) \log (2) \log (x)+5 x^4 \log ^2(2) \log (x)-12 x^2 (-1+\log (4)) \log (x)\right ) \, dx \\ & = 3 x+\frac {x^2}{20}+\frac {x^3}{60}-\frac {1}{30} x^3 \log (2)-\frac {1}{80} x^4 \log (2)+\frac {1}{160} x^4 \log ^2(2)+\frac {1}{400} x^5 \log ^2(2)+\frac {1}{5} \int x \log (x) \, dx-\frac {1}{10} ((2-\log (2)) \log (2)) \int x^3 \log (x) \, dx+\frac {1}{16} \log ^2(2) \int x^4 \log (x) \, dx+\frac {1}{20} (3 (1-\log (4))) \int x^2 \log (x) \, dx \\ & = 3 x+\frac {x^3}{60}-\frac {1}{30} x^3 \log (2)-\frac {1}{80} x^4 \log (2)+\frac {1}{160} x^4 (2-\log (2)) \log (2)+\frac {1}{160} x^4 \log ^2(2)-\frac {1}{60} x^3 (1-\log (4))+\frac {1}{10} x^2 \log (x)-\frac {1}{40} x^4 (2-\log (2)) \log (2) \log (x)+\frac {1}{80} x^5 \log ^2(2) \log (x)+\frac {1}{20} x^3 (1-\log (4)) \log (x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(25)=50\).

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=3 x+\frac {1}{10} x^2 \log (x)+\frac {1}{20} x^3 \log (x)-\frac {1}{10} x^3 \log (2) \log (x)-\frac {1}{20} x^4 \log (2) \log (x)+\frac {1}{40} x^4 \log ^2(2) \log (x)+\frac {1}{80} x^5 \log ^2(2) \log (x) \]

[In]

Integrate[(240 + 8*x + 4*x^2 + (-8*x^2 - 4*x^3)*Log[2] + (2*x^3 + x^4)*Log[2]^2 + (16*x + 12*x^2 + (-24*x^2 -
16*x^3)*Log[2] + (8*x^3 + 5*x^4)*Log[2]^2)*Log[x])/80,x]

[Out]

3*x + (x^2*Log[x])/10 + (x^3*Log[x])/20 - (x^3*Log[2]*Log[x])/10 - (x^4*Log[2]*Log[x])/20 + (x^4*Log[2]^2*Log[
x])/40 + (x^5*Log[2]^2*Log[x])/80

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(23)=46\).

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {\left (x^{5} \ln \left (2\right )^{2}+2 x^{4} \ln \left (2\right )^{2}-4 x^{4} \ln \left (2\right )-8 x^{3} \ln \left (2\right )+4 x^{3}+8 x^{2}\right ) \ln \left (x \right )}{80}+3 x\)	\(51\)
norman	\(\left (\frac {\ln \left (2\right )^{2}}{40}-\frac {\ln \left (2\right )}{20}\right ) x^{4} \ln \left (x \right )+\left (-\frac {\ln \left (2\right )}{10}+\frac {1}{20}\right ) x^{3} \ln \left (x \right )+3 x +\frac {x^{2} \ln \left (x \right )}{10}+\frac {\ln \left (2\right )^{2} \ln \left (x \right ) x^{5}}{80}\)	\(52\)
parallelrisch	\(\frac {\ln \left (2\right )^{2} \ln \left (x \right ) x^{5}}{80}+\frac {x^{4} \ln \left (2\right )^{2} \ln \left (x \right )}{40}-\frac {\ln \left (2\right ) x^{4} \ln \left (x \right )}{20}-\frac {\ln \left (x \right ) \ln \left (2\right ) x^{3}}{10}+\frac {x^{3} \ln \left (x \right )}{20}+\frac {x^{2} \ln \left (x \right )}{10}+3 x\)	\(59\)
parts	\(\frac {\ln \left (2\right )^{2} \ln \left (x \right ) x^{5}}{80}+\frac {x^{4} \ln \left (2\right )^{2} \ln \left (x \right )}{40}-\frac {\ln \left (2\right ) x^{4} \ln \left (x \right )}{20}-\frac {\ln \left (x \right ) \ln \left (2\right ) x^{3}}{10}+\frac {x^{3} \ln \left (x \right )}{20}+\frac {x^{2} \ln \left (x \right )}{10}+3 x\)	\(59\)
default	\(3 x +\frac {\ln \left (2\right )^{2} \ln \left (x \right ) x^{5}}{80}-\frac {x^{5} \ln \left (2\right )^{2}}{400}+\frac {x^{4} \ln \left (2\right )^{2} \ln \left (x \right )}{40}-\frac {x^{4} \ln \left (2\right )^{2}}{160}-\frac {\ln \left (2\right ) x^{4} \ln \left (x \right )}{20}-\frac {\ln \left (x \right ) \ln \left (2\right ) x^{3}}{10}+\frac {x^{3} \ln \left (x \right )}{20}+\frac {x^{2} \ln \left (x \right )}{10}+\frac {\ln \left (2\right )^{2} \left (\frac {1}{5} x^{5}+\frac {1}{2} x^{4}\right )}{80}\)	\(94\)

[In]

int(1/80*((5*x^4+8*x^3)*ln(2)^2+(-16*x^3-24*x^2)*ln(2)+12*x^2+16*x)*ln(x)+1/80*(x^4+2*x^3)*ln(2)^2+1/80*(-4*x^
3-8*x^2)*ln(2)+1/20*x^2+1/10*x+3,x,method=_RETURNVERBOSE)

[Out]

1/80*(x^5*ln(2)^2+2*x^4*ln(2)^2-4*x^4*ln(2)-8*x^3*ln(2)+4*x^3+8*x^2)*ln(x)+3*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=\frac {1}{80} \, {\left (4 \, x^{3} + {\left (x^{5} + 2 \, x^{4}\right )} \log \left (2\right )^{2} + 8 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (x\right ) + 3 \, x \]

[In]

integrate(1/80*((5*x^4+8*x^3)*log(2)^2+(-16*x^3-24*x^2)*log(2)+12*x^2+16*x)*log(x)+1/80*(x^4+2*x^3)*log(2)^2+1
/80*(-4*x^3-8*x^2)*log(2)+1/20*x^2+1/10*x+3,x, algorithm="fricas")

[Out]

1/80*(4*x^3 + (x^5 + 2*x^4)*log(2)^2 + 8*x^2 - 4*(x^4 + 2*x^3)*log(2))*log(x) + 3*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=3 x + \left (\frac {x^{5} \log {\left (2 \right )}^{2}}{80} - \frac {x^{4} \log {\left (2 \right )}}{20} + \frac {x^{4} \log {\left (2 \right )}^{2}}{40} - \frac {x^{3} \log {\left (2 \right )}}{10} + \frac {x^{3}}{20} + \frac {x^{2}}{10}\right ) \log {\left (x \right )} \]

[In]

integrate(1/80*((5*x**4+8*x**3)*ln(2)**2+(-16*x**3-24*x**2)*ln(2)+12*x**2+16*x)*ln(x)+1/80*(x**4+2*x**3)*ln(2)
**2+1/80*(-4*x**3-8*x**2)*ln(2)+1/20*x**2+1/10*x+3,x)

[Out]

3*x + (x**5*log(2)**2/80 - x**4*log(2)/20 + x**4*log(2)**2/40 - x**3*log(2)/10 + x**3/20 + x**2/10)*log(x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (23) = 46\).

Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.68 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=-\frac {1}{400} \, x^{5} \log \left (2\right )^{2} - \frac {1}{160} \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right )\right )} x^{4} + \frac {1}{60} \, x^{3} {\left (2 \, \log \left (2\right ) - 1\right )} + \frac {1}{60} \, x^{3} + \frac {1}{800} \, {\left (2 \, x^{5} + 5 \, x^{4}\right )} \log \left (2\right )^{2} - \frac {1}{240} \, {\left (3 \, x^{4} + 8 \, x^{3}\right )} \log \left (2\right ) + \frac {1}{80} \, {\left (4 \, x^{3} + {\left (x^{5} + 2 \, x^{4}\right )} \log \left (2\right )^{2} + 8 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (x\right ) + 3 \, x \]

[In]

integrate(1/80*((5*x^4+8*x^3)*log(2)^2+(-16*x^3-24*x^2)*log(2)+12*x^2+16*x)*log(x)+1/80*(x^4+2*x^3)*log(2)^2+1
/80*(-4*x^3-8*x^2)*log(2)+1/20*x^2+1/10*x+3,x, algorithm="maxima")

[Out]

-1/400*x^5*log(2)^2 - 1/160*(log(2)^2 - 2*log(2))*x^4 + 1/60*x^3*(2*log(2) - 1) + 1/60*x^3 + 1/800*(2*x^5 + 5*
x^4)*log(2)^2 - 1/240*(3*x^4 + 8*x^3)*log(2) + 1/80*(4*x^3 + (x^5 + 2*x^4)*log(2)^2 + 8*x^2 - 4*(x^4 + 2*x^3)*
log(2))*log(x) + 3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (23) = 46\).

Time = 0.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.88 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=\frac {1}{80} \, x^{5} \log \left (2\right )^{2} \log \left (x\right ) - \frac {1}{400} \, x^{5} \log \left (2\right )^{2} + \frac {1}{40} \, x^{4} \log \left (2\right )^{2} \log \left (x\right ) - \frac {1}{160} \, x^{4} \log \left (2\right )^{2} - \frac {1}{20} \, x^{4} \log \left (2\right ) \log \left (x\right ) + \frac {1}{80} \, x^{4} \log \left (2\right ) - \frac {1}{10} \, x^{3} \log \left (2\right ) \log \left (x\right ) + \frac {1}{30} \, x^{3} \log \left (2\right ) + \frac {1}{20} \, x^{3} \log \left (x\right ) + \frac {1}{800} \, {\left (2 \, x^{5} + 5 \, x^{4}\right )} \log \left (2\right )^{2} + \frac {1}{10} \, x^{2} \log \left (x\right ) - \frac {1}{240} \, {\left (3 \, x^{4} + 8 \, x^{3}\right )} \log \left (2\right ) + 3 \, x \]

[In]

integrate(1/80*((5*x^4+8*x^3)*log(2)^2+(-16*x^3-24*x^2)*log(2)+12*x^2+16*x)*log(x)+1/80*(x^4+2*x^3)*log(2)^2+1
/80*(-4*x^3-8*x^2)*log(2)+1/20*x^2+1/10*x+3,x, algorithm="giac")

[Out]

1/80*x^5*log(2)^2*log(x) - 1/400*x^5*log(2)^2 + 1/40*x^4*log(2)^2*log(x) - 1/160*x^4*log(2)^2 - 1/20*x^4*log(2
)*log(x) + 1/80*x^4*log(2) - 1/10*x^3*log(2)*log(x) + 1/30*x^3*log(2) + 1/20*x^3*log(x) + 1/800*(2*x^5 + 5*x^4
)*log(2)^2 + 1/10*x^2*log(x) - 1/240*(3*x^4 + 8*x^3)*log(2) + 3*x

Mupad [B] (verification not implemented)

Time = 12.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {1}{80} \left (240+8 x+4 x^2+\left (-8 x^2-4 x^3\right ) \log (2)+\left (2 x^3+x^4\right ) \log ^2(2)+\left (16 x+12 x^2+\left (-24 x^2-16 x^3\right ) \log (2)+\left (8 x^3+5 x^4\right ) \log ^2(2)\right ) \log (x)\right ) \, dx=3\,x+\frac {x^2\,\ln \left (x\right )}{10}-\frac {x^4\,\ln \left (x\right )\,\left (\ln \left (16\right )-2\,{\ln \left (2\right )}^2\right )}{80}-\frac {x^3\,\ln \left (x\right )\,\left (\ln \left (256\right )-4\right )}{80}+\frac {x^5\,{\ln \left (2\right )}^2\,\ln \left (x\right )}{80} \]

[In]

int(x/10 + (log(x)*(16*x - log(2)*(24*x^2 + 16*x^3) + 12*x^2 + log(2)^2*(8*x^3 + 5*x^4)))/80 - (log(2)*(8*x^2
+ 4*x^3))/80 + (log(2)^2*(2*x^3 + x^4))/80 + x^2/20 + 3,x)

[Out]

3*x + (x^2*log(x))/10 - (x^4*log(x)*(log(16) - 2*log(2)^2))/80 - (x^3*log(x)*(log(256) - 4))/80 + (x^5*log(2)^
2*log(x))/80