3.46.67 \(\int \frac {-2 x^3 \log ^2(25) \log ^2(x)+(x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx\)

Optimal. Leaf size=21 \[ \left (4-\frac {x^2 \log (25) \log (x)}{16 \log ^2(4 x)}\right )^2 \]

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Rubi [A]  time = 0.27, antiderivative size = 39, normalized size of antiderivative = 1.86, number of steps used = 5, number of rules used = 3, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {12, 6688, 6712} \begin {gather*} \frac {x^4 \log ^2(25) \log ^2(x)}{256 \log ^4(4 x)}-\frac {x^2 \log (25) \log (x)}{2 \log ^2(4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x^3*Log[25]^2*Log[x]^2 + (x^3*Log[25]^2*Log[x] + 2*x^3*Log[25]^2*Log[x]^2)*Log[4*x] + 128*x*Log[25]*Lo
g[x]*Log[4*x]^2 + (-64*x*Log[25] - 128*x*Log[25]*Log[x])*Log[4*x]^3)/(128*Log[4*x]^5),x]

[Out]

(x^4*Log[25]^2*Log[x]^2)/(256*Log[4*x]^4) - (x^2*Log[25]*Log[x])/(2*Log[4*x]^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 6688

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

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{128} \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{\log ^5(4 x)} \, dx\\ &=\frac {1}{128} \int \frac {x \log (25) (2 \log (x) (-1+\log (4 x))+\log (4 x)) \left (x^2 \log (25) \log (x)-64 \log ^2(4 x)\right )}{\log ^5(4 x)} \, dx\\ &=\frac {1}{128} \log (25) \int \frac {x (2 \log (x) (-1+\log (4 x))+\log (4 x)) \left (x^2 \log (25) \log (x)-64 \log ^2(4 x)\right )}{\log ^5(4 x)} \, dx\\ &=\frac {1}{128} \operatorname {Subst}\left (\int (-64+x) \, dx,x,\frac {x^2 \log (25) \log (x)}{\log ^2(4 x)}\right )\\ &=\frac {x^4 \log ^2(25) \log ^2(x)}{256 \log ^4(4 x)}-\frac {x^2 \log (25) \log (x)}{2 \log ^2(4 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 34, normalized size = 1.62 \begin {gather*} \frac {x^2 \log (25) \log (x) \left (x^2 \log (25) \log (x)-128 \log ^2(4 x)\right )}{256 \log ^4(4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^3*Log[25]^2*Log[x]^2 + (x^3*Log[25]^2*Log[x] + 2*x^3*Log[25]^2*Log[x]^2)*Log[4*x] + 128*x*Log[
25]*Log[x]*Log[4*x]^2 + (-64*x*Log[25] - 128*x*Log[25]*Log[x])*Log[4*x]^3)/(128*Log[4*x]^5),x]

[Out]

(x^2*Log[25]*Log[x]*(x^2*Log[25]*Log[x] - 128*Log[4*x]^2))/(256*Log[4*x]^4)

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fricas [B]  time = 0.73, size = 90, normalized size = 4.29 \begin {gather*} -\frac {256 \, x^{2} \log \relax (5) \log \relax (2)^{2} \log \relax (x) + 64 \, x^{2} \log \relax (5) \log \relax (x)^{3} - {\left (x^{4} \log \relax (5)^{2} - 256 \, x^{2} \log \relax (5) \log \relax (2)\right )} \log \relax (x)^{2}}{64 \, {\left (16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log(5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2
*log(x)^2+4*x^3*log(5)^2*log(x))*log(4*x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm="fricas")

[Out]

-1/64*(256*x^2*log(5)*log(2)^2*log(x) + 64*x^2*log(5)*log(x)^3 - (x^4*log(5)^2 - 256*x^2*log(5)*log(2))*log(x)
^2)/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4)

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giac [B]  time = 0.15, size = 207, normalized size = 9.86 \begin {gather*} \frac {x^{4} \log \relax (5)^{2} \log \relax (x)^{2}}{64 \, {\left (16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}\right )}} - \frac {4 \, x^{2} \log \relax (5) \log \relax (2)^{2} \log \relax (x)}{16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}} - \frac {4 \, x^{2} \log \relax (5) \log \relax (2) \log \relax (x)^{2}}{16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}} - \frac {x^{2} \log \relax (5) \log \relax (x)^{3}}{16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log(5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2
*log(x)^2+4*x^3*log(5)^2*log(x))*log(4*x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm="giac")

[Out]

1/64*x^4*log(5)^2*log(x)^2/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(
x)^4) - 4*x^2*log(5)*log(2)^2*log(x)/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x
)^3 + log(x)^4) - 4*x^2*log(5)*log(2)*log(x)^2/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*lo
g(2)*log(x)^3 + log(x)^4) - x^2*log(5)*log(x)^3/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*l
og(2)*log(x)^3 + log(x)^4)

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maple [B]  time = 0.43, size = 42, normalized size = 2.00

method	result	size
default	\(-\frac {\ln \relax (5) x^{2} \ln \relax (x )}{\left (\ln \relax (x )+2 \ln \relax (2)\right )^{2}}+\frac {\ln \relax (5)^{2} x^{4} \ln \relax (x )^{2}}{64 \left (\ln \relax (x )+2 \ln \relax (2)\right )^{4}}\)	\(42\)
risch	\(-\frac {\ln \relax (5) x^{2} \ln \relax (x ) \left (-x^{2} \ln \relax (5) \ln \relax (x )+256 \ln \relax (2)^{2}+256 \ln \relax (2) \ln \relax (x )+64 \ln \relax (x )^{2}\right )}{4 \left (4 \ln \relax (2)+2 \ln \relax (x )\right )^{4}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/128*((-256*x*ln(5)*ln(x)-128*x*ln(5))*ln(4*x)^3+256*x*ln(5)*ln(x)*ln(4*x)^2+(8*x^3*ln(5)^2*ln(x)^2+4*x^3
*ln(5)^2*ln(x))*ln(4*x)-8*x^3*ln(5)^2*ln(x)^2)/ln(4*x)^5,x,method=_RETURNVERBOSE)

[Out]

-ln(5)*x^2*ln(x)/(ln(x)+2*ln(2))^2+1/64*ln(5)^2*x^4*ln(x)^2/(ln(x)+2*ln(2))^4

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maxima [B]  time = 0.50, size = 90, normalized size = 4.29 \begin {gather*} -\frac {256 \, x^{2} \log \relax (5) \log \relax (2)^{2} \log \relax (x) + 64 \, x^{2} \log \relax (5) \log \relax (x)^{3} - {\left (x^{4} \log \relax (5)^{2} - 256 \, x^{2} \log \relax (5) \log \relax (2)\right )} \log \relax (x)^{2}}{64 \, {\left (16 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{3} \log \relax (x) + 24 \, \log \relax (2)^{2} \log \relax (x)^{2} + 8 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log(5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2
*log(x)^2+4*x^3*log(5)^2*log(x))*log(4*x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm="maxima")

[Out]

-1/64*(256*x^2*log(5)*log(2)^2*log(x) + 64*x^2*log(5)*log(x)^3 - (x^4*log(5)^2 - 256*x^2*log(5)*log(2))*log(x)
^2)/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4)

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mupad [B]  time = 3.76, size = 1277, normalized size = 60.81 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(4*x)*(8*x^3*log(5)^2*log(x)^2 + 4*x^3*log(5)^2*log(x)))/128 - (log(4*x)^3*(128*x*log(5) + 256*x*log(
5)*log(x)))/128 - (x^3*log(5)^2*log(x)^2)/16 + 2*x*log(4*x)^2*log(5)*log(x))/log(4*x)^5,x)

[Out]

(x^4*log(5)^2)/8 + log(x)^2*((x^4*log(5)^2)/6 - x^2*(4*log(5) + (4*log(5)*(log(4*x) - log(x)))/3)) - x^2*(2*lo
g(5) + (4*log(5)*(log(4*x) - log(x))^2)/3 + 4*log(5)*(log(4*x) - log(x))) + ((x*(x^3*log(5)^2 - 32*x*log(5)*(l
og(4*x) - log(x)) + 256*x*log(5)*(log(4*x) - log(x))^2 + 192*x*log(5)*(log(4*x) - log(x))^3 - 6*x^3*log(5)^2*(
log(4*x) - log(x))))/384 + (x*log(x)^2*(12*x*log(5) - x^3*log(5)^2 + 68*x*log(5)*(log(4*x) - log(x)) + 24*x*lo
g(5)*(log(4*x) - log(x))^2 - x^3*log(5)^2*(log(4*x) - log(x))))/24 + (x*log(x)*(x^3*log(5)^2 - 48*x*log(5) + 2
24*x*log(5)*(log(4*x) - log(x)) + 416*x*log(5)*(log(4*x) - log(x))^2 + 64*x*log(5)*(log(4*x) - log(x))^3 - 12*
x^3*log(5)^2*(log(4*x) - log(x))))/192 + (x^2*log(5)*log(x)^4)/3 + (x*log(x)^3*(28*x*log(5) - x^3*log(5)^2 + 2
4*x*log(5)*(log(4*x) - log(x))))/24)/(2*log(x)*(log(4*x) - log(x)) + log(x)^2 + (log(4*x) - log(x))^2) + ((x^2
*log(5)*log(x)^4)/2 + (x^2*log(5)*(log(4*x) - log(x))^3)/4 + (x^2*log(5)*log(x)^2*(32*log(x) - 32*log(4*x) + 1
92*(log(4*x) - log(x))^2 + x^2*log(5) - 2*x^2*log(5)*(log(4*x) - log(x))))/128 - (x^2*log(5)*log(x)^3*(96*log(
x) - 96*log(4*x) + x^2*log(5) + 16))/64 + (x^2*log(5)*log(x)*(log(4*x) - log(x))*(32*log(4*x) - 32*log(x) + 64
*(log(4*x) - log(x))^2 - x^2*log(5)))/128)/(log(x)^4 + (log(4*x) - log(x))^4 + 4*log(x)*(log(4*x) - log(x))^3
+ 4*log(x)^3*(log(4*x) - log(x)) + 6*log(x)^2*(log(4*x) - log(x))^2) + ((x*(3*x^3*log(5)^2 - 48*x*log(5) + 192
*x*log(5)*(log(4*x) - log(x)) + 672*x*log(5)*(log(4*x) - log(x))^2 + 256*x*log(5)*(log(4*x) - log(x))^3 - 24*x
^3*log(5)^2*(log(4*x) - log(x))))/192 + (x*log(x)^2*(108*x*log(5) - 7*x^3*log(5)^2 + 208*x*log(5)*(log(4*x) -
log(x)) + 48*x*log(5)*(log(4*x) - log(x))^2 - 4*x^3*log(5)^2*(log(4*x) - log(x))))/24 + (x*log(x)*(24*x*log(5)
- 3*x^3*log(5)^2 + 384*x*log(5)*(log(4*x) - log(x)) + 304*x*log(5)*(log(4*x) - log(x))^2 + 32*x*log(5)*(log(4
*x) - log(x))^3 - 16*x^3*log(5)^2*(log(4*x) - log(x))))/48 + (2*x^2*log(5)*log(x)^4)/3 + (x*log(x)^3*(22*x*log
(5) - x^3*log(5)^2 + 12*x*log(5)*(log(4*x) - log(x))))/6)/log(4*x) + log(x)*((x^4*log(5)^2)/3 - x^2*(6*log(5)
+ (2*log(5)*(log(4*x) - log(x))^2)/3 + (16*log(5)*(log(4*x) - log(x)))/3)) + ((x*(32*x*log(5)*(log(4*x) - log(
x))^2 + 128*x*log(5)*(log(4*x) - log(x))^3 - x^3*log(5)^2*(log(4*x) - log(x))))/384 + (x*log(x)*(x^3*log(5)^2
- 32*x*log(5)*(log(4*x) - log(x)) + 224*x*log(5)*(log(4*x) - log(x))^2 + 64*x*log(5)*(log(4*x) - log(x))^3 - 4
*x^3*log(5)^2*(log(4*x) - log(x))))/192 - (x*log(x)^2*(x^3*log(5)^2 + 48*x*log(5) - 256*x*log(5)*(log(4*x) - l
og(x)) - 192*x*log(5)*(log(4*x) - log(x))^2 + 4*x^3*log(5)^2*(log(4*x) - log(x))))/192 + (x^2*log(5)*log(x)^4)
/3 + (x*log(x)^3*(24*x*log(5) - x^3*log(5)^2 + 48*x*log(5)*(log(4*x) - log(x))))/48)/(log(x)^3 + (log(4*x) - l
og(x))^3 + 3*log(x)*(log(4*x) - log(x))^2 + 3*log(x)^2*(log(4*x) - log(x))) - (2*x^2*log(5)*log(x)^3)/3

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sympy [B]  time = 0.37, size = 100, normalized size = 4.76 \begin {gather*} \frac {- 64 x^{2} \log {\relax (5 )} \log {\relax (x )}^{3} - 256 x^{2} \log {\relax (2 )}^{2} \log {\relax (5 )} \log {\relax (x )} + \left (x^{4} \log {\relax (5 )}^{2} - 256 x^{2} \log {\relax (2 )} \log {\relax (5 )}\right ) \log {\relax (x )}^{2}}{64 \log {\relax (x )}^{4} + 512 \log {\relax (2 )} \log {\relax (x )}^{3} + 1536 \log {\relax (2 )}^{2} \log {\relax (x )}^{2} + 2048 \log {\relax (2 )}^{3} \log {\relax (x )} + 1024 \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/128*((-256*x*ln(5)*ln(x)-128*x*ln(5))*ln(4*x)**3+256*x*ln(5)*ln(x)*ln(4*x)**2+(8*x**3*ln(5)**2*ln(
x)**2+4*x**3*ln(5)**2*ln(x))*ln(4*x)-8*x**3*ln(5)**2*ln(x)**2)/ln(4*x)**5,x)

[Out]

(-64*x**2*log(5)*log(x)**3 - 256*x**2*log(2)**2*log(5)*log(x) + (x**4*log(5)**2 - 256*x**2*log(2)*log(5))*log(
x)**2)/(64*log(x)**4 + 512*log(2)*log(x)**3 + 1536*log(2)**2*log(x)**2 + 2048*log(2)**3*log(x) + 1024*log(2)**
4)

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