3.31.15 \(\int \frac {-562500-130000 \log (x)-10200 \log ^2(x)-336 \log ^3(x)-4 \log ^4(x)+(-337500-64500 \log (x)-3540 \log ^2(x)-60 \log ^3(x)) \log (4+\log (x))+(-67500-10200 \log (x)-300 \log ^2(x)) \log ^2(4+\log (x))+(-4500-500 \log (x)) \log ^3(4+\log (x))}{4 x+x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ -\frac {1}{4}-(\log (x)-5 (-5-\log (4+\log (x))))^4 \]

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Rubi [A]  time = 0.31, antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {12, 6688, 6686} \begin {gather*} -(\log (x)+5 \log (\log (x)+4)+25)^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-562500 - 130000*Log[x] - 10200*Log[x]^2 - 336*Log[x]^3 - 4*Log[x]^4 + (-337500 - 64500*Log[x] - 3540*Log
[x]^2 - 60*Log[x]^3)*Log[4 + Log[x]] + (-67500 - 10200*Log[x] - 300*Log[x]^2)*Log[4 + Log[x]]^2 + (-4500 - 500
*Log[x])*Log[4 + Log[x]]^3)/(4*x + x*Log[x]),x]

[Out]

-(25 + Log[x] + 5*Log[4 + Log[x]])^4

Rule 12

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

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} &=\operatorname {Subst}\left (\int -\frac {4 \left (140625+32500 x+2550 x^2+84 x^3+x^4+84375 \log (4+x)+16125 x \log (4+x)+885 x^2 \log (4+x)+15 x^3 \log (4+x)+16875 \log ^2(4+x)+2550 x \log ^2(4+x)+75 x^2 \log ^2(4+x)+1125 \log ^3(4+x)+125 x \log ^3(4+x)\right )}{4+x} \, dx,x,\log (x)\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {140625+32500 x+2550 x^2+84 x^3+x^4+84375 \log (4+x)+16125 x \log (4+x)+885 x^2 \log (4+x)+15 x^3 \log (4+x)+16875 \log ^2(4+x)+2550 x \log ^2(4+x)+75 x^2 \log ^2(4+x)+1125 \log ^3(4+x)+125 x \log ^3(4+x)}{4+x} \, dx,x,\log (x)\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {(9+x) (25+x+5 \log (4+x))^3}{4+x} \, dx,x,\log (x)\right )\right )\\ &=-(25+\log (x)+5 \log (4+\log (x)))^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.73 \begin {gather*} -(\log (x)+5 (5+\log (4+\log (x))))^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-562500 - 130000*Log[x] - 10200*Log[x]^2 - 336*Log[x]^3 - 4*Log[x]^4 + (-337500 - 64500*Log[x] - 35
40*Log[x]^2 - 60*Log[x]^3)*Log[4 + Log[x]] + (-67500 - 10200*Log[x] - 300*Log[x]^2)*Log[4 + Log[x]]^2 + (-4500
- 500*Log[x])*Log[4 + Log[x]]^3)/(4*x + x*Log[x]),x]

[Out]

-(Log[x] + 5*(5 + Log[4 + Log[x]]))^4

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fricas [B]  time = 1.04, size = 87, normalized size = 3.95 \begin {gather*} -\log \relax (x)^{4} - 500 \, {\left (\log \relax (x) + 25\right )} \log \left (\log \relax (x) + 4\right )^{3} - 625 \, \log \left (\log \relax (x) + 4\right )^{4} - 100 \, \log \relax (x)^{3} - 150 \, {\left (\log \relax (x)^{2} + 50 \, \log \relax (x) + 625\right )} \log \left (\log \relax (x) + 4\right )^{2} - 3750 \, \log \relax (x)^{2} - 20 \, {\left (\log \relax (x)^{3} + 75 \, \log \relax (x)^{2} + 1875 \, \log \relax (x) + 15625\right )} \log \left (\log \relax (x) + 4\right ) - 62500 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*log(x)-4500)*log(log(x)+4)^3+(-300*log(x)^2-10200*log(x)-67500)*log(log(x)+4)^2+(-60*log(x)^3
-3540*log(x)^2-64500*log(x)-337500)*log(log(x)+4)-4*log(x)^4-336*log(x)^3-10200*log(x)^2-130000*log(x)-562500)
/(x*log(x)+4*x),x, algorithm="fricas")

[Out]

-log(x)^4 - 500*(log(x) + 25)*log(log(x) + 4)^3 - 625*log(log(x) + 4)^4 - 100*log(x)^3 - 150*(log(x)^2 + 50*lo
g(x) + 625)*log(log(x) + 4)^2 - 3750*log(x)^2 - 20*(log(x)^3 + 75*log(x)^2 + 1875*log(x) + 15625)*log(log(x) +
4) - 62500*log(x)

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giac [B]  time = 0.24, size = 107, normalized size = 4.86 \begin {gather*} -{\left (\log \relax (x) + 4\right )}^{4} - 500 \, {\left (\log \relax (x) + 25\right )} \log \left (\log \relax (x) + 4\right )^{3} - 625 \, \log \left (\log \relax (x) + 4\right )^{4} - 84 \, {\left (\log \relax (x) + 4\right )}^{3} - 150 \, {\left ({\left (\log \relax (x) + 4\right )}^{2} + 42 \, \log \relax (x) + 609\right )} \log \left (\log \relax (x) + 4\right )^{2} - 2646 \, {\left (\log \relax (x) + 4\right )}^{2} - 20 \, {\left ({\left (\log \relax (x) + 4\right )}^{3} + 63 \, {\left (\log \relax (x) + 4\right )}^{2} + 1323 \, \log \relax (x) + 5292\right )} \log \left (\log \relax (x) + 4\right ) - 37044 \, \log \relax (x) - 185220 \, \log \left (\log \relax (x) + 4\right ) - 148176 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*log(x)-4500)*log(log(x)+4)^3+(-300*log(x)^2-10200*log(x)-67500)*log(log(x)+4)^2+(-60*log(x)^3
-3540*log(x)^2-64500*log(x)-337500)*log(log(x)+4)-4*log(x)^4-336*log(x)^3-10200*log(x)^2-130000*log(x)-562500)
/(x*log(x)+4*x),x, algorithm="giac")

[Out]

-(log(x) + 4)^4 - 500*(log(x) + 25)*log(log(x) + 4)^3 - 625*log(log(x) + 4)^4 - 84*(log(x) + 4)^3 - 150*((log(
x) + 4)^2 + 42*log(x) + 609)*log(log(x) + 4)^2 - 2646*(log(x) + 4)^2 - 20*((log(x) + 4)^3 + 63*(log(x) + 4)^2
+ 1323*log(x) + 5292)*log(log(x) + 4) - 37044*log(x) - 185220*log(log(x) + 4) - 148176

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maple [B]  time = 0.06, size = 97, normalized size = 4.41

method	result	size
risch	\(-625 \ln \left (\ln \relax (x )+4\right )^{4}+\left (-500 \ln \relax (x )-12500\right ) \ln \left (\ln \relax (x )+4\right )^{3}+\left (-150 \ln \relax (x )^{2}-7500 \ln \relax (x )-93750\right ) \ln \left (\ln \relax (x )+4\right )^{2}+\left (-20 \ln \relax (x )^{3}-1500 \ln \relax (x )^{2}-37500 \ln \relax (x )\right ) \ln \left (\ln \relax (x )+4\right )-\ln \relax (x )^{4}-100 \ln \relax (x )^{3}-3750 \ln \relax (x )^{2}-62500 \ln \relax (x )-312500 \ln \left (\ln \relax (x )+4\right )\)	\(97\)
default	\(-\ln \relax (x )^{4}-100 \ln \relax (x )^{3}-3750 \ln \relax (x )^{2}-62500 \ln \relax (x )-185220 \ln \left (\ln \relax (x )+4\right )-625 \ln \left (\ln \relax (x )+4\right )^{4}-500 \left (\ln \relax (x )+4\right ) \ln \left (\ln \relax (x )+4\right )^{3}-6300 \ln \left (\ln \relax (x )+4\right )^{2} \left (\ln \relax (x )+4\right )-26460 \ln \left (\ln \relax (x )+4\right ) \left (\ln \relax (x )+4\right )+\frac {349040}{3}-150 \left (\ln \relax (x )+4\right )^{2} \ln \left (\ln \relax (x )+4\right )^{2}-1260 \ln \left (\ln \relax (x )+4\right ) \left (\ln \relax (x )+4\right )^{2}-10500 \ln \left (\ln \relax (x )+4\right )^{3}-20 \ln \left (\ln \relax (x )+4\right ) \left (\ln \relax (x )+4\right )^{3}-66150 \ln \left (\ln \relax (x )+4\right )^{2}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-500*ln(x)-4500)*ln(ln(x)+4)^3+(-300*ln(x)^2-10200*ln(x)-67500)*ln(ln(x)+4)^2+(-60*ln(x)^3-3540*ln(x)^2-
64500*ln(x)-337500)*ln(ln(x)+4)-4*ln(x)^4-336*ln(x)^3-10200*ln(x)^2-130000*ln(x)-562500)/(x*ln(x)+4*x),x,metho
d=_RETURNVERBOSE)

[Out]

-625*ln(ln(x)+4)^4+(-500*ln(x)-12500)*ln(ln(x)+4)^3+(-150*ln(x)^2-7500*ln(x)-93750)*ln(ln(x)+4)^2+(-20*ln(x)^3
-1500*ln(x)^2-37500*ln(x))*ln(ln(x)+4)-ln(x)^4-100*ln(x)^3-3750*ln(x)^2-62500*ln(x)-312500*ln(ln(x)+4)

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maxima [B]  time = 0.38, size = 330, normalized size = 15.00 \begin {gather*} -30 \, \log \relax (x)^{3} \log \left (\log \relax (x) + 4\right )^{2} - 100 \, \log \relax (x)^{2} \log \left (\log \relax (x) + 4\right )^{3} - 125 \, \log \relax (x) \log \left (\log \relax (x) + 4\right )^{4} + \frac {10}{3} \, {\left (9 \, \log \left (\log \relax (x) + 4\right )^{2} - 6 \, \log \left (\log \relax (x) + 4\right ) + 2\right )} {\left (\log \relax (x) + 4\right )}^{3} - \log \relax (x)^{4} - 1770 \, \log \relax (x)^{2} \log \left (\log \relax (x) + 4\right )^{2} - 3400 \, \log \relax (x) \log \left (\log \relax (x) + 4\right )^{3} - 1125 \, \log \left (\log \relax (x) + 4\right )^{4} + 25 \, {\left (4 \, \log \left (\log \relax (x) + 4\right )^{3} - 6 \, \log \left (\log \relax (x) + 4\right )^{2} + 6 \, \log \left (\log \relax (x) + 4\right ) - 3\right )} {\left (\log \relax (x) + 4\right )}^{2} + 705 \, {\left (2 \, \log \left (\log \relax (x) + 4\right )^{2} - 2 \, \log \left (\log \relax (x) + 4\right ) + 1\right )} {\left (\log \relax (x) + 4\right )}^{2} - \frac {320}{3} \, \log \relax (x)^{3} - 32250 \, \log \relax (x) \log \left (\log \relax (x) + 4\right )^{2} - 22500 \, \log \left (\log \relax (x) + 4\right )^{3} + 125 \, {\left (\log \left (\log \relax (x) + 4\right )^{4} - 4 \, \log \left (\log \relax (x) + 4\right )^{3} + 12 \, \log \left (\log \relax (x) + 4\right )^{2} - 24 \, \log \left (\log \relax (x) + 4\right ) + 24\right )} {\left (\log \relax (x) + 4\right )} + 2600 \, {\left (\log \left (\log \relax (x) + 4\right )^{3} - 3 \, \log \left (\log \relax (x) + 4\right )^{2} + 6 \, \log \left (\log \relax (x) + 4\right ) - 6\right )} {\left (\log \relax (x) + 4\right )} + 19530 \, {\left (\log \left (\log \relax (x) + 4\right )^{2} - 2 \, \log \left (\log \relax (x) + 4\right ) + 2\right )} {\left (\log \relax (x) + 4\right )} - 4460 \, \log \relax (x)^{2} + 130000 \, {\left (\log \relax (x) + 4\right )} \log \left (\log \relax (x) + 4\right ) - 130000 \, \log \relax (x) \log \left (\log \relax (x) + 4\right ) - 168750 \, \log \left (\log \relax (x) + 4\right )^{2} - 94320 \, \log \relax (x) - 705220 \, \log \left (\log \relax (x) + 4\right ) - 520000 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*log(x)-4500)*log(log(x)+4)^3+(-300*log(x)^2-10200*log(x)-67500)*log(log(x)+4)^2+(-60*log(x)^3
-3540*log(x)^2-64500*log(x)-337500)*log(log(x)+4)-4*log(x)^4-336*log(x)^3-10200*log(x)^2-130000*log(x)-562500)
/(x*log(x)+4*x),x, algorithm="maxima")

[Out]

-30*log(x)^3*log(log(x) + 4)^2 - 100*log(x)^2*log(log(x) + 4)^3 - 125*log(x)*log(log(x) + 4)^4 + 10/3*(9*log(l
og(x) + 4)^2 - 6*log(log(x) + 4) + 2)*(log(x) + 4)^3 - log(x)^4 - 1770*log(x)^2*log(log(x) + 4)^2 - 3400*log(x
)*log(log(x) + 4)^3 - 1125*log(log(x) + 4)^4 + 25*(4*log(log(x) + 4)^3 - 6*log(log(x) + 4)^2 + 6*log(log(x) +
4) - 3)*(log(x) + 4)^2 + 705*(2*log(log(x) + 4)^2 - 2*log(log(x) + 4) + 1)*(log(x) + 4)^2 - 320/3*log(x)^3 - 3
2250*log(x)*log(log(x) + 4)^2 - 22500*log(log(x) + 4)^3 + 125*(log(log(x) + 4)^4 - 4*log(log(x) + 4)^3 + 12*lo
g(log(x) + 4)^2 - 24*log(log(x) + 4) + 24)*(log(x) + 4) + 2600*(log(log(x) + 4)^3 - 3*log(log(x) + 4)^2 + 6*lo
g(log(x) + 4) - 6)*(log(x) + 4) + 19530*(log(log(x) + 4)^2 - 2*log(log(x) + 4) + 2)*(log(x) + 4) - 4460*log(x)
^2 + 130000*(log(x) + 4)*log(log(x) + 4) - 130000*log(x)*log(log(x) + 4) - 168750*log(log(x) + 4)^2 - 94320*lo
g(x) - 705220*log(log(x) + 4) - 520000

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mupad [B]  time = 2.07, size = 99, normalized size = 4.50 \begin {gather*} -312500\,\ln \left (\ln \relax (x)+4\right )-62500\,\ln \relax (x)-625\,{\ln \left (\ln \relax (x)+4\right )}^4-{\ln \left (\ln \relax (x)+4\right )}^3\,\left (500\,\ln \relax (x)+12500\right )-3750\,{\ln \relax (x)}^2-100\,{\ln \relax (x)}^3-{\ln \relax (x)}^4-{\ln \left (\ln \relax (x)+4\right )}^2\,\left (150\,{\ln \relax (x)}^2+7500\,\ln \relax (x)+93750\right )-\ln \left (\ln \relax (x)+4\right )\,\left (20\,{\ln \relax (x)}^3+1500\,{\ln \relax (x)}^2+37500\,\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(130000*log(x) + log(log(x) + 4)^3*(500*log(x) + 4500) + 10200*log(x)^2 + 336*log(x)^3 + 4*log(x)^4 + log
(log(x) + 4)^2*(10200*log(x) + 300*log(x)^2 + 67500) + log(log(x) + 4)*(64500*log(x) + 3540*log(x)^2 + 60*log(
x)^3 + 337500) + 562500)/(4*x + x*log(x)),x)

[Out]

- 312500*log(log(x) + 4) - 62500*log(x) - 625*log(log(x) + 4)^4 - log(log(x) + 4)^3*(500*log(x) + 12500) - 375
0*log(x)^2 - 100*log(x)^3 - log(x)^4 - log(log(x) + 4)^2*(7500*log(x) + 150*log(x)^2 + 93750) - log(log(x) + 4
)*(37500*log(x) + 1500*log(x)^2 + 20*log(x)^3)

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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(((-500*ln(x)-4500)*ln(ln(x)+4)**3+(-300*ln(x)**2-10200*ln(x)-67500)*ln(ln(x)+4)**2+(-60*ln(x)**3-354
0*ln(x)**2-64500*ln(x)-337500)*ln(ln(x)+4)-4*ln(x)**4-336*ln(x)**3-10200*ln(x)**2-130000*ln(x)-562500)/(x*ln(x
)+4*x),x)

[Out]

Timed out

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