3.59.12 \(\int \frac {(-12 x^7+4 x^9+(12 x^6-8 x^8) \log (4)) \log ^3(\frac {5 x-5 \log (4)}{x^2})+(-8 x^9+8 x^8 \log (4)) \log ^4(\frac {5 x-5 \log (4)}{x^2})+\log ^3(x) (-324 x+108 x^3+(324-216 x^2) \log (4)+(-216 x^3+216 x^2 \log (4)) \log (\frac {5 x-5 \log (4)}{x^2}))+\log ^2(x) ((-324 x^3+108 x^5+(324 x^2-216 x^4) \log (4)) \log (\frac {5 x-5 \log (4)}{x^2})+(-216 x^5+216 x^4 \log (4)) \log ^2(\frac {5 x-5 \log (4)}{x^2}))+\log (x) ((-108 x^5+36 x^7+(108 x^4-72 x^6) \log (4)) \log ^2(\frac {5 x-5 \log (4)}{x^2})+(-72 x^7+72 x^6 \log (4)) \log ^3(\frac {5 x-5 \log (4)}{x^2}))}{-81 x^2+81 x \log (4)} \, dx\)

Optimal. Leaf size=26 \[ \left (\log (x)+\frac {1}{3} x^2 \log \left (\frac {5}{x}-\frac {5 \log (4)}{x^2}\right )\right )^4 \]

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Rubi [A]  time = 0.63, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 284, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {1593, 6688, 12, 6686} \begin {gather*} \frac {1}{81} \left (x^2 \log \left (\frac {5 (x-\log (4))}{x^2}\right )+3 \log (x)\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-12*x^7 + 4*x^9 + (12*x^6 - 8*x^8)*Log[4])*Log[(5*x - 5*Log[4])/x^2]^3 + (-8*x^9 + 8*x^8*Log[4])*Log[(5*
x - 5*Log[4])/x^2]^4 + Log[x]^3*(-324*x + 108*x^3 + (324 - 216*x^2)*Log[4] + (-216*x^3 + 216*x^2*Log[4])*Log[(
5*x - 5*Log[4])/x^2]) + Log[x]^2*((-324*x^3 + 108*x^5 + (324*x^2 - 216*x^4)*Log[4])*Log[(5*x - 5*Log[4])/x^2]
+ (-216*x^5 + 216*x^4*Log[4])*Log[(5*x - 5*Log[4])/x^2]^2) + Log[x]*((-108*x^5 + 36*x^7 + (108*x^4 - 72*x^6)*L
og[4])*Log[(5*x - 5*Log[4])/x^2]^2 + (-72*x^7 + 72*x^6*Log[4])*Log[(5*x - 5*Log[4])/x^2]^3))/(-81*x^2 + 81*x*L
og[4]),x]

[Out]

(3*Log[x] + x^2*Log[(5*(x - Log[4]))/x^2])^4/81

Rule 12

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-12 x^7+4 x^9+\left (12 x^6-8 x^8\right ) \log (4)\right ) \log ^3\left (\frac {5 x-5 \log (4)}{x^2}\right )+\left (-8 x^9+8 x^8 \log (4)\right ) \log ^4\left (\frac {5 x-5 \log (4)}{x^2}\right )+\log ^3(x) \left (-324 x+108 x^3+\left (324-216 x^2\right ) \log (4)+\left (-216 x^3+216 x^2 \log (4)\right ) \log \left (\frac {5 x-5 \log (4)}{x^2}\right )\right )+\log ^2(x) \left (\left (-324 x^3+108 x^5+\left (324 x^2-216 x^4\right ) \log (4)\right ) \log \left (\frac {5 x-5 \log (4)}{x^2}\right )+\left (-216 x^5+216 x^4 \log (4)\right ) \log ^2\left (\frac {5 x-5 \log (4)}{x^2}\right )\right )+\log (x) \left (\left (-108 x^5+36 x^7+\left (108 x^4-72 x^6\right ) \log (4)\right ) \log ^2\left (\frac {5 x-5 \log (4)}{x^2}\right )+\left (-72 x^7+72 x^6 \log (4)\right ) \log ^3\left (\frac {5 x-5 \log (4)}{x^2}\right )\right )}{x (-81 x+81 \log (4))} \, dx\\ &=\int \frac {4 \left (3 \log (x)+x^2 \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )^3 \left (3 x-x^3-3 \log (4)+x^2 \log (16)+2 x^2 (x-\log (4)) \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )}{81 x (x-\log (4))} \, dx\\ &=\frac {4}{81} \int \frac {\left (3 \log (x)+x^2 \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )^3 \left (3 x-x^3-3 \log (4)+x^2 \log (16)+2 x^2 (x-\log (4)) \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )}{x (x-\log (4))} \, dx\\ &=\frac {1}{81} \left (3 \log (x)+x^2 \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{81} \left (3 \log (x)+x^2 \log \left (\frac {5 (x-\log (4))}{x^2}\right )\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12*x^7 + 4*x^9 + (12*x^6 - 8*x^8)*Log[4])*Log[(5*x - 5*Log[4])/x^2]^3 + (-8*x^9 + 8*x^8*Log[4])*L
og[(5*x - 5*Log[4])/x^2]^4 + Log[x]^3*(-324*x + 108*x^3 + (324 - 216*x^2)*Log[4] + (-216*x^3 + 216*x^2*Log[4])
*Log[(5*x - 5*Log[4])/x^2]) + Log[x]^2*((-324*x^3 + 108*x^5 + (324*x^2 - 216*x^4)*Log[4])*Log[(5*x - 5*Log[4])
/x^2] + (-216*x^5 + 216*x^4*Log[4])*Log[(5*x - 5*Log[4])/x^2]^2) + Log[x]*((-108*x^5 + 36*x^7 + (108*x^4 - 72*
x^6)*Log[4])*Log[(5*x - 5*Log[4])/x^2]^2 + (-72*x^7 + 72*x^6*Log[4])*Log[(5*x - 5*Log[4])/x^2]^3))/(-81*x^2 +
81*x*Log[4]),x]

[Out]

(3*Log[x] + x^2*Log[(5*(x - Log[4]))/x^2])^4/81

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fricas [B]  time = 0.65, size = 89, normalized size = 3.42 \begin {gather*} \frac {1}{81} \, x^{8} \log \left (\frac {5 \, {\left (x - 2 \, \log \relax (2)\right )}}{x^{2}}\right )^{4} + \frac {4}{27} \, x^{6} \log \relax (x) \log \left (\frac {5 \, {\left (x - 2 \, \log \relax (2)\right )}}{x^{2}}\right )^{3} + \frac {2}{3} \, x^{4} \log \relax (x)^{2} \log \left (\frac {5 \, {\left (x - 2 \, \log \relax (2)\right )}}{x^{2}}\right )^{2} + \frac {4}{3} \, x^{2} \log \relax (x)^{3} \log \left (\frac {5 \, {\left (x - 2 \, \log \relax (2)\right )}}{x^{2}}\right ) + \log \relax (x)^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((432*x^2*log(2)-216*x^3)*log((-10*log(2)+5*x)/x^2)+2*(-216*x^2+324)*log(2)+108*x^3-324*x)*log(x)^3
+((432*x^4*log(2)-216*x^5)*log((-10*log(2)+5*x)/x^2)^2+(2*(-216*x^4+324*x^2)*log(2)+108*x^5-324*x^3)*log((-10*
log(2)+5*x)/x^2))*log(x)^2+((144*x^6*log(2)-72*x^7)*log((-10*log(2)+5*x)/x^2)^3+(2*(-72*x^6+108*x^4)*log(2)+36
*x^7-108*x^5)*log((-10*log(2)+5*x)/x^2)^2)*log(x)+(16*x^8*log(2)-8*x^9)*log((-10*log(2)+5*x)/x^2)^4+(2*(-8*x^8
+12*x^6)*log(2)+4*x^9-12*x^7)*log((-10*log(2)+5*x)/x^2)^3)/(162*x*log(2)-81*x^2),x, algorithm="fricas")

[Out]

1/81*x^8*log(5*(x - 2*log(2))/x^2)^4 + 4/27*x^6*log(x)*log(5*(x - 2*log(2))/x^2)^3 + 2/3*x^4*log(x)^2*log(5*(x
- 2*log(2))/x^2)^2 + 4/3*x^2*log(x)^3*log(5*(x - 2*log(2))/x^2) + log(x)^4

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giac [B]  time = 2.56, size = 140, normalized size = 5.38 \begin {gather*} \frac {1}{81} \, x^{8} \log \left (5 \, x - 10 \, \log \relax (2)\right )^{4} - \frac {4}{81} \, {\left (2 \, x^{8} - 3 \, x^{6}\right )} \log \left (5 \, x - 10 \, \log \relax (2)\right )^{3} \log \relax (x) + \frac {2}{27} \, {\left (4 \, x^{8} - 12 \, x^{6} + 9 \, x^{4}\right )} \log \left (5 \, x - 10 \, \log \relax (2)\right )^{2} \log \relax (x)^{2} - \frac {4}{81} \, {\left (8 \, x^{8} - 36 \, x^{6} + 54 \, x^{4} - 27 \, x^{2}\right )} \log \left (5 \, x - 10 \, \log \relax (2)\right ) \log \relax (x)^{3} + \frac {1}{81} \, {\left (16 \, x^{8} - 96 \, x^{6} + 216 \, x^{4} - 216 \, x^{2} + 81\right )} \log \relax (x)^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((432*x^2*log(2)-216*x^3)*log((-10*log(2)+5*x)/x^2)+2*(-216*x^2+324)*log(2)+108*x^3-324*x)*log(x)^3
+((432*x^4*log(2)-216*x^5)*log((-10*log(2)+5*x)/x^2)^2+(2*(-216*x^4+324*x^2)*log(2)+108*x^5-324*x^3)*log((-10*
log(2)+5*x)/x^2))*log(x)^2+((144*x^6*log(2)-72*x^7)*log((-10*log(2)+5*x)/x^2)^3+(2*(-72*x^6+108*x^4)*log(2)+36
*x^7-108*x^5)*log((-10*log(2)+5*x)/x^2)^2)*log(x)+(16*x^8*log(2)-8*x^9)*log((-10*log(2)+5*x)/x^2)^4+(2*(-8*x^8
+12*x^6)*log(2)+4*x^9-12*x^7)*log((-10*log(2)+5*x)/x^2)^3)/(162*x*log(2)-81*x^2),x, algorithm="giac")

[Out]

1/81*x^8*log(5*x - 10*log(2))^4 - 4/81*(2*x^8 - 3*x^6)*log(5*x - 10*log(2))^3*log(x) + 2/27*(4*x^8 - 12*x^6 +
9*x^4)*log(5*x - 10*log(2))^2*log(x)^2 - 4/81*(8*x^8 - 36*x^6 + 54*x^4 - 27*x^2)*log(5*x - 10*log(2))*log(x)^3
+ 1/81*(16*x^8 - 96*x^6 + 216*x^4 - 216*x^2 + 81)*log(x)^4

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maple [C]  time = 10.66, size = 77075, normalized size = 2964.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(77075\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((432*x^2*ln(2)-216*x^3)*ln((-10*ln(2)+5*x)/x^2)+2*(-216*x^2+324)*ln(2)+108*x^3-324*x)*ln(x)^3+((432*x^4*
ln(2)-216*x^5)*ln((-10*ln(2)+5*x)/x^2)^2+(2*(-216*x^4+324*x^2)*ln(2)+108*x^5-324*x^3)*ln((-10*ln(2)+5*x)/x^2))
*ln(x)^2+((144*x^6*ln(2)-72*x^7)*ln((-10*ln(2)+5*x)/x^2)^3+(2*(-72*x^6+108*x^4)*ln(2)+36*x^7-108*x^5)*ln((-10*
ln(2)+5*x)/x^2)^2)*ln(x)+(16*x^8*ln(2)-8*x^9)*ln((-10*ln(2)+5*x)/x^2)^4+(2*(-8*x^8+12*x^6)*ln(2)+4*x^9-12*x^7)
*ln((-10*ln(2)+5*x)/x^2)^3)/(162*x*ln(2)-81*x^2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [B]  time = 0.52, size = 333, normalized size = 12.81 \begin {gather*} \frac {1}{81} \, x^{8} \log \relax (5)^{4} + \frac {1}{81} \, x^{8} \log \left (x - 2 \, \log \relax (2)\right )^{4} + \frac {1}{81} \, {\left (16 \, x^{8} - 96 \, x^{6} + 216 \, x^{4} - 216 \, x^{2} + 81\right )} \log \relax (x)^{4} + \frac {4}{81} \, {\left (x^{8} \log \relax (5) - {\left (2 \, x^{8} - 3 \, x^{6}\right )} \log \relax (x)\right )} \log \left (x - 2 \, \log \relax (2)\right )^{3} - \frac {4}{81} \, {\left (8 \, x^{8} \log \relax (5) - 36 \, x^{6} \log \relax (5) + 54 \, x^{4} \log \relax (5) - 27 \, x^{2} \log \relax (5)\right )} \log \relax (x)^{3} + \frac {2}{27} \, {\left (x^{8} \log \relax (5)^{2} + {\left (4 \, x^{8} - 12 \, x^{6} + 9 \, x^{4}\right )} \log \relax (x)^{2} - 2 \, {\left (2 \, x^{8} \log \relax (5) - 3 \, x^{6} \log \relax (5)\right )} \log \relax (x)\right )} \log \left (x - 2 \, \log \relax (2)\right )^{2} + \frac {2}{27} \, {\left (4 \, x^{8} \log \relax (5)^{2} - 12 \, x^{6} \log \relax (5)^{2} + 9 \, x^{4} \log \relax (5)^{2}\right )} \log \relax (x)^{2} + \frac {4}{81} \, {\left (x^{8} \log \relax (5)^{3} - {\left (8 \, x^{8} - 36 \, x^{6} + 54 \, x^{4} - 27 \, x^{2}\right )} \log \relax (x)^{3} + 3 \, {\left (4 \, x^{8} \log \relax (5) - 12 \, x^{6} \log \relax (5) + 9 \, x^{4} \log \relax (5)\right )} \log \relax (x)^{2} - 3 \, {\left (2 \, x^{8} \log \relax (5)^{2} - 3 \, x^{6} \log \relax (5)^{2}\right )} \log \relax (x)\right )} \log \left (x - 2 \, \log \relax (2)\right ) - \frac {4}{81} \, {\left (2 \, x^{8} \log \relax (5)^{3} - 3 \, x^{6} \log \relax (5)^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((432*x^2*log(2)-216*x^3)*log((-10*log(2)+5*x)/x^2)+2*(-216*x^2+324)*log(2)+108*x^3-324*x)*log(x)^3
+((432*x^4*log(2)-216*x^5)*log((-10*log(2)+5*x)/x^2)^2+(2*(-216*x^4+324*x^2)*log(2)+108*x^5-324*x^3)*log((-10*
log(2)+5*x)/x^2))*log(x)^2+((144*x^6*log(2)-72*x^7)*log((-10*log(2)+5*x)/x^2)^3+(2*(-72*x^6+108*x^4)*log(2)+36
*x^7-108*x^5)*log((-10*log(2)+5*x)/x^2)^2)*log(x)+(16*x^8*log(2)-8*x^9)*log((-10*log(2)+5*x)/x^2)^4+(2*(-8*x^8
+12*x^6)*log(2)+4*x^9-12*x^7)*log((-10*log(2)+5*x)/x^2)^3)/(162*x*log(2)-81*x^2),x, algorithm="maxima")

[Out]

1/81*x^8*log(5)^4 + 1/81*x^8*log(x - 2*log(2))^4 + 1/81*(16*x^8 - 96*x^6 + 216*x^4 - 216*x^2 + 81)*log(x)^4 +
4/81*(x^8*log(5) - (2*x^8 - 3*x^6)*log(x))*log(x - 2*log(2))^3 - 4/81*(8*x^8*log(5) - 36*x^6*log(5) + 54*x^4*l
og(5) - 27*x^2*log(5))*log(x)^3 + 2/27*(x^8*log(5)^2 + (4*x^8 - 12*x^6 + 9*x^4)*log(x)^2 - 2*(2*x^8*log(5) - 3
*x^6*log(5))*log(x))*log(x - 2*log(2))^2 + 2/27*(4*x^8*log(5)^2 - 12*x^6*log(5)^2 + 9*x^4*log(5)^2)*log(x)^2 +
4/81*(x^8*log(5)^3 - (8*x^8 - 36*x^6 + 54*x^4 - 27*x^2)*log(x)^3 + 3*(4*x^8*log(5) - 12*x^6*log(5) + 9*x^4*lo
g(5))*log(x)^2 - 3*(2*x^8*log(5)^2 - 3*x^6*log(5)^2)*log(x))*log(x - 2*log(2)) - 4/81*(2*x^8*log(5)^3 - 3*x^6*
log(5)^3)*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \relax (x)\,\left (\left (144\,x^6\,\ln \relax (2)-72\,x^7\right )\,{\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )}^3+\left (2\,\ln \relax (2)\,\left (108\,x^4-72\,x^6\right )-108\,x^5+36\,x^7\right )\,{\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )}^2\right )+{\ln \relax (x)}^2\,\left (\left (432\,x^4\,\ln \relax (2)-216\,x^5\right )\,{\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )}^2+\left (2\,\ln \relax (2)\,\left (324\,x^2-216\,x^4\right )-324\,x^3+108\,x^5\right )\,\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )\right )+{\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )}^4\,\left (16\,x^8\,\ln \relax (2)-8\,x^9\right )-{\ln \relax (x)}^3\,\left (324\,x-\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )\,\left (432\,x^2\,\ln \relax (2)-216\,x^3\right )+2\,\ln \relax (2)\,\left (216\,x^2-324\right )-108\,x^3\right )+{\ln \left (\frac {5\,x-10\,\ln \relax (2)}{x^2}\right )}^3\,\left (2\,\ln \relax (2)\,\left (12\,x^6-8\,x^8\right )-12\,x^7+4\,x^9\right )}{162\,x\,\ln \relax (2)-81\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(log((5*x - 10*log(2))/x^2)^3*(144*x^6*log(2) - 72*x^7) + log((5*x - 10*log(2))/x^2)^2*(2*log(2)*(
108*x^4 - 72*x^6) - 108*x^5 + 36*x^7)) + log(x)^2*(log((5*x - 10*log(2))/x^2)*(2*log(2)*(324*x^2 - 216*x^4) -
324*x^3 + 108*x^5) + log((5*x - 10*log(2))/x^2)^2*(432*x^4*log(2) - 216*x^5)) + log((5*x - 10*log(2))/x^2)^4*(
16*x^8*log(2) - 8*x^9) - log(x)^3*(324*x - log((5*x - 10*log(2))/x^2)*(432*x^2*log(2) - 216*x^3) + 2*log(2)*(2
16*x^2 - 324) - 108*x^3) + log((5*x - 10*log(2))/x^2)^3*(2*log(2)*(12*x^6 - 8*x^8) - 12*x^7 + 4*x^9))/(162*x*l
og(2) - 81*x^2),x)

[Out]

int((log(x)*(log((5*x - 10*log(2))/x^2)^3*(144*x^6*log(2) - 72*x^7) + log((5*x - 10*log(2))/x^2)^2*(2*log(2)*(
108*x^4 - 72*x^6) - 108*x^5 + 36*x^7)) + log(x)^2*(log((5*x - 10*log(2))/x^2)*(2*log(2)*(324*x^2 - 216*x^4) -
324*x^3 + 108*x^5) + log((5*x - 10*log(2))/x^2)^2*(432*x^4*log(2) - 216*x^5)) + log((5*x - 10*log(2))/x^2)^4*(
16*x^8*log(2) - 8*x^9) - log(x)^3*(324*x - log((5*x - 10*log(2))/x^2)*(432*x^2*log(2) - 216*x^3) + 2*log(2)*(2
16*x^2 - 324) - 108*x^3) + log((5*x - 10*log(2))/x^2)^3*(2*log(2)*(12*x^6 - 8*x^8) - 12*x^7 + 4*x^9))/(162*x*l
og(2) - 81*x^2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((432*x**2*ln(2)-216*x**3)*ln((-10*ln(2)+5*x)/x**2)+2*(-216*x**2+324)*ln(2)+108*x**3-324*x)*ln(x)**
3+((432*x**4*ln(2)-216*x**5)*ln((-10*ln(2)+5*x)/x**2)**2+(2*(-216*x**4+324*x**2)*ln(2)+108*x**5-324*x**3)*ln((
-10*ln(2)+5*x)/x**2))*ln(x)**2+((144*x**6*ln(2)-72*x**7)*ln((-10*ln(2)+5*x)/x**2)**3+(2*(-72*x**6+108*x**4)*ln
(2)+36*x**7-108*x**5)*ln((-10*ln(2)+5*x)/x**2)**2)*ln(x)+(16*x**8*ln(2)-8*x**9)*ln((-10*ln(2)+5*x)/x**2)**4+(2
*(-8*x**8+12*x**6)*ln(2)+4*x**9-12*x**7)*ln((-10*ln(2)+5*x)/x**2)**3)/(162*x*ln(2)-81*x**2),x)

[Out]

Timed out

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