3.77.15 \(\int ((-512 x-512 x \log (2)) \log (x)+(768 x+768 x \log (2)) \log ^2(x)+(-384 x-384 x \log (2)) \log ^3(x)+(64 x+64 x \log (2)) \log ^4(x)) \, dx\)

Optimal. Leaf size=17 \[ 2 x^2 (1+\log (2)) (-4+2 \log (x))^4 \]

________________________________________________________________________________________

Rubi [B]  time = 0.10, antiderivative size = 60, normalized size of antiderivative = 3.53, number of steps used = 19, number of rules used = 4, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6, 12, 2304, 2305} \begin {gather*} 32 x^2 (1+\log (2)) \log ^4(x)-256 x^2 (1+\log (2)) \log ^3(x)+768 x^2 (1+\log (2)) \log ^2(x)-1024 x^2 (1+\log (2)) \log (x)+512 x^2 (1+\log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-512*x - 512*x*Log[2])*Log[x] + (768*x + 768*x*Log[2])*Log[x]^2 + (-384*x - 384*x*Log[2])*Log[x]^3 + (64*
x + 64*x*Log[2])*Log[x]^4,x]

[Out]

512*x^2*(1 + Log[2]) - 1024*x^2*(1 + Log[2])*Log[x] + 768*x^2*(1 + Log[2])*Log[x]^2 - 256*x^2*(1 + Log[2])*Log
[x]^3 + 32*x^2*(1 + Log[2])*Log[x]^4

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 2304

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 2305

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int (-512 x-512 x \log (2)) \log (x) \, dx+\int (768 x+768 x \log (2)) \log ^2(x) \, dx+\int (-384 x-384 x \log (2)) \log ^3(x) \, dx+\int (64 x+64 x \log (2)) \log ^4(x) \, dx\\ &=\int x (-512-512 \log (2)) \log (x) \, dx+\int x (768+768 \log (2)) \log ^2(x) \, dx+\int x (-384-384 \log (2)) \log ^3(x) \, dx+\int x (64+64 \log (2)) \log ^4(x) \, dx\\ &=(64 (1+\log (2))) \int x \log ^4(x) \, dx-(384 (1+\log (2))) \int x \log ^3(x) \, dx-(512 (1+\log (2))) \int x \log (x) \, dx+(768 (1+\log (2))) \int x \log ^2(x) \, dx\\ &=128 x^2 (1+\log (2))-256 x^2 (1+\log (2)) \log (x)+384 x^2 (1+\log (2)) \log ^2(x)-192 x^2 (1+\log (2)) \log ^3(x)+32 x^2 (1+\log (2)) \log ^4(x)-(128 (1+\log (2))) \int x \log ^3(x) \, dx+(576 (1+\log (2))) \int x \log ^2(x) \, dx-(768 (1+\log (2))) \int x \log (x) \, dx\\ &=320 x^2 (1+\log (2))-640 x^2 (1+\log (2)) \log (x)+672 x^2 (1+\log (2)) \log ^2(x)-256 x^2 (1+\log (2)) \log ^3(x)+32 x^2 (1+\log (2)) \log ^4(x)+(192 (1+\log (2))) \int x \log ^2(x) \, dx-(576 (1+\log (2))) \int x \log (x) \, dx\\ &=464 x^2 (1+\log (2))-928 x^2 (1+\log (2)) \log (x)+768 x^2 (1+\log (2)) \log ^2(x)-256 x^2 (1+\log (2)) \log ^3(x)+32 x^2 (1+\log (2)) \log ^4(x)-(192 (1+\log (2))) \int x \log (x) \, dx\\ &=512 x^2 (1+\log (2))-1024 x^2 (1+\log (2)) \log (x)+768 x^2 (1+\log (2)) \log ^2(x)-256 x^2 (1+\log (2)) \log ^3(x)+32 x^2 (1+\log (2)) \log ^4(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.02, size = 48, normalized size = 2.82 \begin {gather*} 64 (1+\log (2)) \left (8 x^2-16 x^2 \log (x)+12 x^2 \log ^2(x)-4 x^2 \log ^3(x)+\frac {1}{2} x^2 \log ^4(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-512*x - 512*x*Log[2])*Log[x] + (768*x + 768*x*Log[2])*Log[x]^2 + (-384*x - 384*x*Log[2])*Log[x]^3
+ (64*x + 64*x*Log[2])*Log[x]^4,x]

[Out]

64*(1 + Log[2])*(8*x^2 - 16*x^2*Log[x] + 12*x^2*Log[x]^2 - 4*x^2*Log[x]^3 + (x^2*Log[x]^4)/2)

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 75, normalized size = 4.41 \begin {gather*} 32 \, {\left (x^{2} \log \relax (2) + x^{2}\right )} \log \relax (x)^{4} - 256 \, {\left (x^{2} \log \relax (2) + x^{2}\right )} \log \relax (x)^{3} + 512 \, x^{2} \log \relax (2) + 768 \, {\left (x^{2} \log \relax (2) + x^{2}\right )} \log \relax (x)^{2} + 512 \, x^{2} - 1024 \, {\left (x^{2} \log \relax (2) + x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((64*x*log(2)+64*x)*log(x)^4+(-384*x*log(2)-384*x)*log(x)^3+(768*x*log(2)+768*x)*log(x)^2+(-512*x*log
(2)-512*x)*log(x),x, algorithm="fricas")

[Out]

32*(x^2*log(2) + x^2)*log(x)^4 - 256*(x^2*log(2) + x^2)*log(x)^3 + 512*x^2*log(2) + 768*(x^2*log(2) + x^2)*log
(x)^2 + 512*x^2 - 1024*(x^2*log(2) + x^2)*log(x)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 89, normalized size = 5.24 \begin {gather*} 32 \, x^{2} \log \relax (2) \log \relax (x)^{4} - 256 \, x^{2} \log \relax (2) \log \relax (x)^{3} + 32 \, x^{2} \log \relax (x)^{4} + 768 \, x^{2} \log \relax (2) \log \relax (x)^{2} - 256 \, x^{2} \log \relax (x)^{3} - 1024 \, x^{2} \log \relax (2) \log \relax (x) + 768 \, x^{2} \log \relax (x)^{2} + 512 \, x^{2} \log \relax (2) - 1024 \, x^{2} \log \relax (x) + 512 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((64*x*log(2)+64*x)*log(x)^4+(-384*x*log(2)-384*x)*log(x)^3+(768*x*log(2)+768*x)*log(x)^2+(-512*x*log
(2)-512*x)*log(x),x, algorithm="giac")

[Out]

32*x^2*log(2)*log(x)^4 - 256*x^2*log(2)*log(x)^3 + 32*x^2*log(x)^4 + 768*x^2*log(2)*log(x)^2 - 256*x^2*log(x)^
3 - 1024*x^2*log(2)*log(x) + 768*x^2*log(x)^2 + 512*x^2*log(2) - 1024*x^2*log(x) + 512*x^2

________________________________________________________________________________________

maple [B]  time = 0.04, size = 64, normalized size = 3.76

method	result	size
risch	\(32 x^{2} \left (1+\ln \relax (2)\right ) \ln \relax (x )^{4}-256 x^{2} \left (1+\ln \relax (2)\right ) \ln \relax (x )^{3}+768 x^{2} \left (1+\ln \relax (2)\right ) \ln \relax (x )^{2}-1024 x^{2} \left (1+\ln \relax (2)\right ) \ln \relax (x )+512 x^{2} \ln \relax (2)+512 x^{2}\)	\(64\)
norman	\(\left (512 \ln \relax (2)+512\right ) x^{2}+\left (-1024-1024 \ln \relax (2)\right ) x^{2} \ln \relax (x )+\left (-256 \ln \relax (2)-256\right ) x^{2} \ln \relax (x )^{3}+\left (32 \ln \relax (2)+32\right ) x^{2} \ln \relax (x )^{4}+\left (768 \ln \relax (2)+768\right ) x^{2} \ln \relax (x )^{2}\)	\(66\)
default	\(-1024 x^{2} \ln \relax (x )+512 x^{2}-1024 x^{2} \ln \relax (2) \ln \relax (x )+512 x^{2} \ln \relax (2)-256 x^{2} \ln \relax (x )^{3}+768 x^{2} \ln \relax (x )^{2}-256 \ln \relax (2) x^{2} \ln \relax (x )^{3}+768 x^{2} \ln \relax (2) \ln \relax (x )^{2}+32 x^{2} \ln \relax (x )^{4}+32 \ln \relax (2) x^{2} \ln \relax (x )^{4}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x*ln(2)+64*x)*ln(x)^4+(-384*x*ln(2)-384*x)*ln(x)^3+(768*x*ln(2)+768*x)*ln(x)^2+(-512*x*ln(2)-512*x)*ln
(x),x,method=_RETURNVERBOSE)

[Out]

32*x^2*(1+ln(2))*ln(x)^4-256*x^2*(1+ln(2))*ln(x)^3+768*x^2*(1+ln(2))*ln(x)^2-1024*x^2*(1+ln(2))*ln(x)+512*x^2*
ln(2)+512*x^2

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 139, normalized size = 8.18 \begin {gather*} 16 \, {\left (2 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{4} - 4 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{3} + 6 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{2} - 6 \, {\left (\log \relax (2) + 1\right )} \log \relax (x) + 3 \, \log \relax (2) + 3\right )} x^{2} - 48 \, {\left (4 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{3} - 6 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{2} + 6 \, {\left (\log \relax (2) + 1\right )} \log \relax (x) - 3 \, \log \relax (2) - 3\right )} x^{2} + 192 \, {\left (2 \, {\left (\log \relax (2) + 1\right )} \log \relax (x)^{2} - 2 \, {\left (\log \relax (2) + 1\right )} \log \relax (x) + \log \relax (2) + 1\right )} x^{2} + 128 \, x^{2} {\left (\log \relax (2) + 1\right )} - 256 \, {\left (x^{2} \log \relax (2) + x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((64*x*log(2)+64*x)*log(x)^4+(-384*x*log(2)-384*x)*log(x)^3+(768*x*log(2)+768*x)*log(x)^2+(-512*x*log
(2)-512*x)*log(x),x, algorithm="maxima")

[Out]

16*(2*(log(2) + 1)*log(x)^4 - 4*(log(2) + 1)*log(x)^3 + 6*(log(2) + 1)*log(x)^2 - 6*(log(2) + 1)*log(x) + 3*lo
g(2) + 3)*x^2 - 48*(4*(log(2) + 1)*log(x)^3 - 6*(log(2) + 1)*log(x)^2 + 6*(log(2) + 1)*log(x) - 3*log(2) - 3)*
x^2 + 192*(2*(log(2) + 1)*log(x)^2 - 2*(log(2) + 1)*log(x) + log(2) + 1)*x^2 + 128*x^2*(log(2) + 1) - 256*(x^2
*log(2) + x^2)*log(x)

________________________________________________________________________________________

mupad [B]  time = 5.24, size = 15, normalized size = 0.88 \begin {gather*} 32\,x^2\,{\left (\ln \relax (x)-2\right )}^4\,\left (\ln \relax (2)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)^4*(64*x + 64*x*log(2)) - log(x)^3*(384*x + 384*x*log(2)) + log(x)^2*(768*x + 768*x*log(2)) - log(x)
*(512*x + 512*x*log(2)),x)

[Out]

32*x^2*(log(x) - 2)^4*(log(2) + 1)

________________________________________________________________________________________

sympy [B]  time = 0.29, size = 85, normalized size = 5.00 \begin {gather*} x^{2} \left (512 \log {\relax (2 )} + 512\right ) + \left (- 1024 x^{2} - 1024 x^{2} \log {\relax (2 )}\right ) \log {\relax (x )} + \left (- 256 x^{2} - 256 x^{2} \log {\relax (2 )}\right ) \log {\relax (x )}^{3} + \left (32 x^{2} \log {\relax (2 )} + 32 x^{2}\right ) \log {\relax (x )}^{4} + \left (768 x^{2} \log {\relax (2 )} + 768 x^{2}\right ) \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((64*x*ln(2)+64*x)*ln(x)**4+(-384*x*ln(2)-384*x)*ln(x)**3+(768*x*ln(2)+768*x)*ln(x)**2+(-512*x*ln(2)-
512*x)*ln(x),x)

[Out]

x**2*(512*log(2) + 512) + (-1024*x**2 - 1024*x**2*log(2))*log(x) + (-256*x**2 - 256*x**2*log(2))*log(x)**3 + (
32*x**2*log(2) + 32*x**2)*log(x)**4 + (768*x**2*log(2) + 768*x**2)*log(x)**2

________________________________________________________________________________________