3.89.79 \(\int \frac {-37500000+53750000 x-32000000 x^2+9800000 x^3-1400000 x^4-28000 x^5+44800 x^6-7360 x^7+544 x^8-16 x^9+(-3125000 x+3750000 x^2-1750000 x^3+350000 x^4-14000 x^6+2800 x^7-240 x^8+8 x^9+(-3125000+3750000 x-1750000 x^2+350000 x^3-14000 x^5+2800 x^6-240 x^7+8 x^8) \log (5+x)) \log (\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(9000000-9300000 x+3600000 x^2-540000 x^3-24000 x^4+18720 x^5-2304 x^6+96 x^7) \log ^2(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(750000 x-600000 x^2+150000 x^3-6000 x^5+960 x^6-48 x^7+(750000-600000 x+150000 x^2-6000 x^4+960 x^5-48 x^6) \log (5+x)) \log ^3(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(-720000+456000 x-76800 x^2-5760 x^3+2688 x^4-192 x^5) \log ^4(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(-60000 x+24000 x^2-960 x^4+96 x^5+(-60000+24000 x-960 x^3+96 x^4) \log (5+x)) \log ^5(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(19200-4480 x-512 x^2+128 x^3) \log ^6(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))+(1600 x-64 x^3+(1600-64 x^2) \log (5+x)) \log ^7(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))}{(5 x+x^2+(5+x) \log (5+x)) \log ^9(\frac {1}{4} (x^2+2 x \log (5+x)+\log ^2(5+x)))} \, dx\) [8879]
Optimal. Leaf size=26 \[ \left (2-\frac {(-5+x)^2}{\log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )}\right )^4 \]
[Out]
(2-(-5+x)^2/ln(1/4*(ln(5+x)+x)^2)^2)^4
________________________________________________________________________________________
Rubi [A]
time = 4.30, antiderivative size = 28, normalized size of antiderivative = 1.08, number of steps
used = 4, number of rules used = 4, integrand size = 533, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6820, 12,
6844, 32} \begin {gather*} \left (2-\frac {(5-x)^2}{\log ^2\left (\frac {1}{4} (x+\log (x+5))^2\right )}\right )^4 \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-37500000 + 53750000*x - 32000000*x^2 + 9800000*x^3 - 1400000*x^4 - 28000*x^5 + 44800*x^6 - 7360*x^7 + 54
4*x^8 - 16*x^9 + (-3125000*x + 3750000*x^2 - 1750000*x^3 + 350000*x^4 - 14000*x^6 + 2800*x^7 - 240*x^8 + 8*x^9
+ (-3125000 + 3750000*x - 1750000*x^2 + 350000*x^3 - 14000*x^5 + 2800*x^6 - 240*x^7 + 8*x^8)*Log[5 + x])*Log[
(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4] + (9000000 - 9300000*x + 3600000*x^2 - 540000*x^3 - 24000*x^4 + 18720
*x^5 - 2304*x^6 + 96*x^7)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^2 + (750000*x - 600000*x^2 + 150000*x^3
- 6000*x^5 + 960*x^6 - 48*x^7 + (750000 - 600000*x + 150000*x^2 - 6000*x^4 + 960*x^5 - 48*x^6)*Log[5 + x])*Lo
g[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^3 + (-720000 + 456000*x - 76800*x^2 - 5760*x^3 + 2688*x^4 - 192*x^5
)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^4 + (-60000*x + 24000*x^2 - 960*x^4 + 96*x^5 + (-60000 + 24000*
x - 960*x^3 + 96*x^4)*Log[5 + x])*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^5 + (19200 - 4480*x - 512*x^2 +
128*x^3)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^6 + (1600*x - 64*x^3 + (1600 - 64*x^2)*Log[5 + x])*Log[
(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^7)/((5*x + x^2 + (5 + x)*Log[5 + x])*Log[(x^2 + 2*x*Log[5 + x] + Log[
5 + x]^2)/4]^9),x]
[Out]
(2 - (5 - x)^2/Log[(x + Log[5 + x])^2/4]^2)^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 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 6820
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6844
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]] /; Fre
eQ[{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} &=\int \frac {8 (5-x) \left (2 \left (-30+x+x^2\right )-(5+x) (x+\log (5+x)) \log \left (\frac {1}{4} (x+\log (5+x))^2\right )\right ) \left ((-5+x)^2-2 \log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )\right )^3}{(5+x) (x+\log (5+x)) \log ^9\left (\frac {1}{4} (x+\log (5+x))^2\right )} \, dx\\ &=8 \int \frac {(5-x) \left (2 \left (-30+x+x^2\right )-(5+x) (x+\log (5+x)) \log \left (\frac {1}{4} (x+\log (5+x))^2\right )\right ) \left ((-5+x)^2-2 \log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )\right )^3}{(5+x) (x+\log (5+x)) \log ^9\left (\frac {1}{4} (x+\log (5+x))^2\right )} \, dx\\ &=4 \text {Subst}\left (\int (-2+x)^3 \, dx,x,\frac {(-5+x)^2}{\log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )}\right )\\ &=\left (2-\frac {(5-x)^2}{\log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )}\right )^4\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(26)=52\).
time = 0.60, size = 93, normalized size = 3.58 \begin {gather*} 8 \left (\frac {(-5+x)^8}{8 \log ^8\left (\frac {1}{4} (x+\log (5+x))^2\right )}-\frac {(-5+x)^6}{\log ^6\left (\frac {1}{4} (x+\log (5+x))^2\right )}+\frac {3 (-5+x)^4}{\log ^4\left (\frac {1}{4} (x+\log (5+x))^2\right )}-\frac {4 (-5+x)^2}{\log ^2\left (\frac {1}{4} (x+\log (5+x))^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-37500000 + 53750000*x - 32000000*x^2 + 9800000*x^3 - 1400000*x^4 - 28000*x^5 + 44800*x^6 - 7360*x^
7 + 544*x^8 - 16*x^9 + (-3125000*x + 3750000*x^2 - 1750000*x^3 + 350000*x^4 - 14000*x^6 + 2800*x^7 - 240*x^8 +
8*x^9 + (-3125000 + 3750000*x - 1750000*x^2 + 350000*x^3 - 14000*x^5 + 2800*x^6 - 240*x^7 + 8*x^8)*Log[5 + x]
)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4] + (9000000 - 9300000*x + 3600000*x^2 - 540000*x^3 - 24000*x^4 +
18720*x^5 - 2304*x^6 + 96*x^7)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^2 + (750000*x - 600000*x^2 + 1500
00*x^3 - 6000*x^5 + 960*x^6 - 48*x^7 + (750000 - 600000*x + 150000*x^2 - 6000*x^4 + 960*x^5 - 48*x^6)*Log[5 +
x])*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^3 + (-720000 + 456000*x - 76800*x^2 - 5760*x^3 + 2688*x^4 - 1
92*x^5)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^4 + (-60000*x + 24000*x^2 - 960*x^4 + 96*x^5 + (-60000 +
24000*x - 960*x^3 + 96*x^4)*Log[5 + x])*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^5 + (19200 - 4480*x - 512
*x^2 + 128*x^3)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^6 + (1600*x - 64*x^3 + (1600 - 64*x^2)*Log[5 + x]
)*Log[(x^2 + 2*x*Log[5 + x] + Log[5 + x]^2)/4]^7)/((5*x + x^2 + (5 + x)*Log[5 + x])*Log[(x^2 + 2*x*Log[5 + x]
+ Log[5 + x]^2)/4]^9),x]
[Out]
8*((-5 + x)^8/(8*Log[(x + Log[5 + x])^2/4]^8) - (-5 + x)^6/Log[(x + Log[5 + x])^2/4]^6 + (3*(-5 + x)^4)/Log[(x
+ Log[5 + x])^2/4]^4 - (4*(-5 + x)^2)/Log[(x + Log[5 + x])^2/4]^2)
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 16.20, size = 28461, normalized size = 1094.65
| | |
| method |
result |
size |
| | |
| risch |
\(\text {Expression too large to display}\) |
\(27279\) |
| default |
\(\text {Expression too large to display}\) |
\(28461\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-64*x^2+1600)*ln(5+x)-64*x^3+1600*x)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^7+(128*x^3-512*x^2-4480*x+
19200)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^6+((96*x^4-960*x^3+24000*x-60000)*ln(5+x)+96*x^5-960*x^4+24000*
x^2-60000*x)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^5+(-192*x^5+2688*x^4-5760*x^3-76800*x^2+456000*x-720000)*
ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^4+((-48*x^6+960*x^5-6000*x^4+150000*x^2-600000*x+750000)*ln(5+x)-48*x^
7+960*x^6-6000*x^5+150000*x^3-600000*x^2+750000*x)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^3+(96*x^7-2304*x^6+
18720*x^5-24000*x^4-540000*x^3+3600000*x^2-9300000*x+9000000)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^2+((8*x^
8-240*x^7+2800*x^6-14000*x^5+350000*x^3-1750000*x^2+3750000*x-3125000)*ln(5+x)+8*x^9-240*x^8+2800*x^7-14000*x^
6+350000*x^4-1750000*x^3+3750000*x^2-3125000*x)*ln(1/4*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)-16*x^9+544*x^8-7360*x^
7+44800*x^6-28000*x^5-1400000*x^4+9800000*x^3-32000000*x^2+53750000*x-37500000)/((5+x)*ln(5+x)+x^2+5*x)/ln(1/4
*ln(5+x)^2+1/2*x*ln(5+x)+1/4*x^2)^9,x,method=_RETURNVERBOSE)
[Out]
result too large to display
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 630 vs.
\(2 (23) = 46\).
time = 1.33, size = 630, normalized size = 24.23 \begin {gather*} \frac {x^{8} - 4 \, {\left (8 \, \log \left (2\right )^{2} - 175\right )} x^{6} - 40 \, x^{7} - 2048 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x + \log \left (x + 5\right )\right )^{6} + 40 \, {\left (24 \, \log \left (2\right )^{2} - 175\right )} x^{5} - 51200 \, \log \left (2\right )^{6} + 12288 \, {\left (x^{2} \log \left (2\right ) - 10 \, x \log \left (2\right ) + 25 \, \log \left (2\right )\right )} \log \left (x + \log \left (x + 5\right )\right )^{5} + 2 \, {\left (192 \, \log \left (2\right )^{4} - 6000 \, \log \left (2\right )^{2} + 21875\right )} x^{4} + 384 \, {\left (x^{4} - 10 \, {\left (8 \, \log \left (2\right )^{2} - 15\right )} x^{2} - 20 \, x^{3} + 100 \, {\left (8 \, \log \left (2\right )^{2} - 5\right )} x - 2000 \, \log \left (2\right )^{2} + 625\right )} \log \left (x + \log \left (x + 5\right )\right )^{4} - 40 \, {\left (192 \, \log \left (2\right )^{4} - 2000 \, \log \left (2\right )^{2} + 4375\right )} x^{3} + 240000 \, \log \left (2\right )^{4} - 512 \, {\left (3 \, x^{4} \log \left (2\right ) - 60 \, x^{3} \log \left (2\right ) - 10 \, {\left (8 \, \log \left (2\right )^{3} - 45 \, \log \left (2\right )\right )} x^{2} - 2000 \, \log \left (2\right )^{3} + 100 \, {\left (8 \, \log \left (2\right )^{3} - 15 \, \log \left (2\right )\right )} x + 1875 \, \log \left (2\right )\right )} \log \left (x + \log \left (x + 5\right )\right )^{3} - 4 \, {\left (512 \, \log \left (2\right )^{6} - 14400 \, \log \left (2\right )^{4} + 75000 \, \log \left (2\right )^{2} - 109375\right )} x^{2} - 32 \, {\left (x^{6} - 3 \, {\left (24 \, \log \left (2\right )^{2} - 125\right )} x^{4} - 30 \, x^{5} + 20 \, {\left (72 \, \log \left (2\right )^{2} - 125\right )} x^{3} + 24000 \, \log \left (2\right )^{4} + 15 \, {\left (64 \, \log \left (2\right )^{4} - 720 \, \log \left (2\right )^{2} + 625\right )} x^{2} - 150 \, {\left (64 \, \log \left (2\right )^{4} - 240 \, \log \left (2\right )^{2} + 125\right )} x - 45000 \, \log \left (2\right )^{2} + 15625\right )} \log \left (x + \log \left (x + 5\right )\right )^{2} + 40 \, {\left (512 \, \log \left (2\right )^{6} - 4800 \, \log \left (2\right )^{4} + 15000 \, \log \left (2\right )^{2} - 15625\right )} x - 500000 \, \log \left (2\right )^{2} + 64 \, {\left (x^{6} \log \left (2\right ) - 30 \, x^{5} \log \left (2\right ) - 3 \, {\left (8 \, \log \left (2\right )^{3} - 125 \, \log \left (2\right )\right )} x^{4} + 4800 \, \log \left (2\right )^{5} + 20 \, {\left (24 \, \log \left (2\right )^{3} - 125 \, \log \left (2\right )\right )} x^{3} + 3 \, {\left (64 \, \log \left (2\right )^{5} - 1200 \, \log \left (2\right )^{3} + 3125 \, \log \left (2\right )\right )} x^{2} - 15000 \, \log \left (2\right )^{3} - 30 \, {\left (64 \, \log \left (2\right )^{5} - 400 \, \log \left (2\right )^{3} + 625 \, \log \left (2\right )\right )} x + 15625 \, \log \left (2\right )\right )} \log \left (x + \log \left (x + 5\right )\right ) + 390625}{256 \, {\left (\log \left (2\right )^{8} - 8 \, \log \left (2\right )^{7} \log \left (x + \log \left (x + 5\right )\right ) + 28 \, \log \left (2\right )^{6} \log \left (x + \log \left (x + 5\right )\right )^{2} - 56 \, \log \left (2\right )^{5} \log \left (x + \log \left (x + 5\right )\right )^{3} + 70 \, \log \left (2\right )^{4} \log \left (x + \log \left (x + 5\right )\right )^{4} - 56 \, \log \left (2\right )^{3} \log \left (x + \log \left (x + 5\right )\right )^{5} + 28 \, \log \left (2\right )^{2} \log \left (x + \log \left (x + 5\right )\right )^{6} - 8 \, \log \left (2\right ) \log \left (x + \log \left (x + 5\right )\right )^{7} + \log \left (x + \log \left (x + 5\right )\right )^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-64*x^2+1600)*log(5+x)-64*x^3+1600*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^7+(128*x^3-512*x
^2-4480*x+19200)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^6+((96*x^4-960*x^3+24000*x-60000)*log(5+x)+96*x^5-
960*x^4+24000*x^2-60000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^5+(-192*x^5+2688*x^4-5760*x^3-76800*x^2+
456000*x-720000)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^4+((-48*x^6+960*x^5-6000*x^4+150000*x^2-600000*x+7
50000)*log(5+x)-48*x^7+960*x^6-6000*x^5+150000*x^3-600000*x^2+750000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*
x^2)^3+(96*x^7-2304*x^6+18720*x^5-24000*x^4-540000*x^3+3600000*x^2-9300000*x+9000000)*log(1/4*log(5+x)^2+1/2*x
*log(5+x)+1/4*x^2)^2+((8*x^8-240*x^7+2800*x^6-14000*x^5+350000*x^3-1750000*x^2+3750000*x-3125000)*log(5+x)+8*x
^9-240*x^8+2800*x^7-14000*x^6+350000*x^4-1750000*x^3+3750000*x^2-3125000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+
1/4*x^2)-16*x^9+544*x^8-7360*x^7+44800*x^6-28000*x^5-1400000*x^4+9800000*x^3-32000000*x^2+53750000*x-37500000)
/((5+x)*log(5+x)+x^2+5*x)/log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^9,x, algorithm="maxima")
[Out]
1/256*(x^8 - 4*(8*log(2)^2 - 175)*x^6 - 40*x^7 - 2048*(x^2 - 10*x + 25)*log(x + log(x + 5))^6 + 40*(24*log(2)^
2 - 175)*x^5 - 51200*log(2)^6 + 12288*(x^2*log(2) - 10*x*log(2) + 25*log(2))*log(x + log(x + 5))^5 + 2*(192*lo
g(2)^4 - 6000*log(2)^2 + 21875)*x^4 + 384*(x^4 - 10*(8*log(2)^2 - 15)*x^2 - 20*x^3 + 100*(8*log(2)^2 - 5)*x -
2000*log(2)^2 + 625)*log(x + log(x + 5))^4 - 40*(192*log(2)^4 - 2000*log(2)^2 + 4375)*x^3 + 240000*log(2)^4 -
512*(3*x^4*log(2) - 60*x^3*log(2) - 10*(8*log(2)^3 - 45*log(2))*x^2 - 2000*log(2)^3 + 100*(8*log(2)^3 - 15*log
(2))*x + 1875*log(2))*log(x + log(x + 5))^3 - 4*(512*log(2)^6 - 14400*log(2)^4 + 75000*log(2)^2 - 109375)*x^2
- 32*(x^6 - 3*(24*log(2)^2 - 125)*x^4 - 30*x^5 + 20*(72*log(2)^2 - 125)*x^3 + 24000*log(2)^4 + 15*(64*log(2)^4
- 720*log(2)^2 + 625)*x^2 - 150*(64*log(2)^4 - 240*log(2)^2 + 125)*x - 45000*log(2)^2 + 15625)*log(x + log(x
+ 5))^2 + 40*(512*log(2)^6 - 4800*log(2)^4 + 15000*log(2)^2 - 15625)*x - 500000*log(2)^2 + 64*(x^6*log(2) - 30
*x^5*log(2) - 3*(8*log(2)^3 - 125*log(2))*x^4 + 4800*log(2)^5 + 20*(24*log(2)^3 - 125*log(2))*x^3 + 3*(64*log(
2)^5 - 1200*log(2)^3 + 3125*log(2))*x^2 - 15000*log(2)^3 - 30*(64*log(2)^5 - 400*log(2)^3 + 625*log(2))*x + 15
625*log(2))*log(x + log(x + 5)) + 390625)/(log(2)^8 - 8*log(2)^7*log(x + log(x + 5)) + 28*log(2)^6*log(x + log
(x + 5))^2 - 56*log(2)^5*log(x + log(x + 5))^3 + 70*log(2)^4*log(x + log(x + 5))^4 - 56*log(2)^3*log(x + log(x
+ 5))^5 + 28*log(2)^2*log(x + log(x + 5))^6 - 8*log(2)*log(x + log(x + 5))^7 + log(x + log(x + 5))^8)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs.
\(2 (23) = 46\).
time = 0.40, size = 195, normalized size = 7.50 \begin {gather*} \frac {x^{8} - 40 \, x^{7} - 32 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (\frac {1}{4} \, x^{2} + \frac {1}{2} \, x \log \left (x + 5\right ) + \frac {1}{4} \, \log \left (x + 5\right )^{2}\right )^{6} + 700 \, x^{6} - 7000 \, x^{5} + 24 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} \log \left (\frac {1}{4} \, x^{2} + \frac {1}{2} \, x \log \left (x + 5\right ) + \frac {1}{4} \, \log \left (x + 5\right )^{2}\right )^{4} + 43750 \, x^{4} - 175000 \, x^{3} - 8 \, {\left (x^{6} - 30 \, x^{5} + 375 \, x^{4} - 2500 \, x^{3} + 9375 \, x^{2} - 18750 \, x + 15625\right )} \log \left (\frac {1}{4} \, x^{2} + \frac {1}{2} \, x \log \left (x + 5\right ) + \frac {1}{4} \, \log \left (x + 5\right )^{2}\right )^{2} + 437500 \, x^{2} - 625000 \, x + 390625}{\log \left (\frac {1}{4} \, x^{2} + \frac {1}{2} \, x \log \left (x + 5\right ) + \frac {1}{4} \, \log \left (x + 5\right )^{2}\right )^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-64*x^2+1600)*log(5+x)-64*x^3+1600*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^7+(128*x^3-512*x
^2-4480*x+19200)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^6+((96*x^4-960*x^3+24000*x-60000)*log(5+x)+96*x^5-
960*x^4+24000*x^2-60000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^5+(-192*x^5+2688*x^4-5760*x^3-76800*x^2+
456000*x-720000)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^4+((-48*x^6+960*x^5-6000*x^4+150000*x^2-600000*x+7
50000)*log(5+x)-48*x^7+960*x^6-6000*x^5+150000*x^3-600000*x^2+750000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*
x^2)^3+(96*x^7-2304*x^6+18720*x^5-24000*x^4-540000*x^3+3600000*x^2-9300000*x+9000000)*log(1/4*log(5+x)^2+1/2*x
*log(5+x)+1/4*x^2)^2+((8*x^8-240*x^7+2800*x^6-14000*x^5+350000*x^3-1750000*x^2+3750000*x-3125000)*log(5+x)+8*x
^9-240*x^8+2800*x^7-14000*x^6+350000*x^4-1750000*x^3+3750000*x^2-3125000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+
1/4*x^2)-16*x^9+544*x^8-7360*x^7+44800*x^6-28000*x^5-1400000*x^4+9800000*x^3-32000000*x^2+53750000*x-37500000)
/((5+x)*log(5+x)+x^2+5*x)/log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^9,x, algorithm="fricas")
[Out]
(x^8 - 40*x^7 - 32*(x^2 - 10*x + 25)*log(1/4*x^2 + 1/2*x*log(x + 5) + 1/4*log(x + 5)^2)^6 + 700*x^6 - 7000*x^5
+ 24*(x^4 - 20*x^3 + 150*x^2 - 500*x + 625)*log(1/4*x^2 + 1/2*x*log(x + 5) + 1/4*log(x + 5)^2)^4 + 43750*x^4
- 175000*x^3 - 8*(x^6 - 30*x^5 + 375*x^4 - 2500*x^3 + 9375*x^2 - 18750*x + 15625)*log(1/4*x^2 + 1/2*x*log(x +
5) + 1/4*log(x + 5)^2)^2 + 437500*x^2 - 625000*x + 390625)/log(1/4*x^2 + 1/2*x*log(x + 5) + 1/4*log(x + 5)^2)^
8
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs.
\(2 (20) = 40\).
time = 2.26, size = 201, normalized size = 7.73 \begin {gather*} \frac {x^{8} - 40 x^{7} + 700 x^{6} - 7000 x^{5} + 43750 x^{4} - 175000 x^{3} + 437500 x^{2} - 625000 x + \left (- 32 x^{2} + 320 x - 800\right ) \log {\left (\frac {x^{2}}{4} + \frac {x \log {\left (x + 5 \right )}}{2} + \frac {\log {\left (x + 5 \right )}^{2}}{4} \right )}^{6} + \left (24 x^{4} - 480 x^{3} + 3600 x^{2} - 12000 x + 15000\right ) \log {\left (\frac {x^{2}}{4} + \frac {x \log {\left (x + 5 \right )}}{2} + \frac {\log {\left (x + 5 \right )}^{2}}{4} \right )}^{4} + \left (- 8 x^{6} + 240 x^{5} - 3000 x^{4} + 20000 x^{3} - 75000 x^{2} + 150000 x - 125000\right ) \log {\left (\frac {x^{2}}{4} + \frac {x \log {\left (x + 5 \right )}}{2} + \frac {\log {\left (x + 5 \right )}^{2}}{4} \right )}^{2} + 390625}{\log {\left (\frac {x^{2}}{4} + \frac {x \log {\left (x + 5 \right )}}{2} + \frac {\log {\left (x + 5 \right )}^{2}}{4} \right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-64*x**2+1600)*ln(5+x)-64*x**3+1600*x)*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)**7+(128*x**3-512
*x**2-4480*x+19200)*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)**6+((96*x**4-960*x**3+24000*x-60000)*ln(5+x)+96*
x**5-960*x**4+24000*x**2-60000*x)*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)**5+(-192*x**5+2688*x**4-5760*x**3-
76800*x**2+456000*x-720000)*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)**4+((-48*x**6+960*x**5-6000*x**4+150000*
x**2-600000*x+750000)*ln(5+x)-48*x**7+960*x**6-6000*x**5+150000*x**3-600000*x**2+750000*x)*ln(1/4*ln(5+x)**2+1
/2*x*ln(5+x)+1/4*x**2)**3+(96*x**7-2304*x**6+18720*x**5-24000*x**4-540000*x**3+3600000*x**2-9300000*x+9000000)
*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)**2+((8*x**8-240*x**7+2800*x**6-14000*x**5+350000*x**3-1750000*x**2+
3750000*x-3125000)*ln(5+x)+8*x**9-240*x**8+2800*x**7-14000*x**6+350000*x**4-1750000*x**3+3750000*x**2-3125000*
x)*ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2)-16*x**9+544*x**8-7360*x**7+44800*x**6-28000*x**5-1400000*x**4+980
0000*x**3-32000000*x**2+53750000*x-37500000)/((5+x)*ln(5+x)+x**2+5*x)/ln(1/4*ln(5+x)**2+1/2*x*ln(5+x)+1/4*x**2
)**9,x)
[Out]
(x**8 - 40*x**7 + 700*x**6 - 7000*x**5 + 43750*x**4 - 175000*x**3 + 437500*x**2 - 625000*x + (-32*x**2 + 320*x
- 800)*log(x**2/4 + x*log(x + 5)/2 + log(x + 5)**2/4)**6 + (24*x**4 - 480*x**3 + 3600*x**2 - 12000*x + 15000)
*log(x**2/4 + x*log(x + 5)/2 + log(x + 5)**2/4)**4 + (-8*x**6 + 240*x**5 - 3000*x**4 + 20000*x**3 - 75000*x**2
+ 150000*x - 125000)*log(x**2/4 + x*log(x + 5)/2 + log(x + 5)**2/4)**2 + 390625)/log(x**2/4 + x*log(x + 5)/2
+ log(x + 5)**2/4)**8
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Giac [F(-1)] Timed out
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((((-64*x^2+1600)*log(5+x)-64*x^3+1600*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^7+(128*x^3-512*x
^2-4480*x+19200)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^6+((96*x^4-960*x^3+24000*x-60000)*log(5+x)+96*x^5-
960*x^4+24000*x^2-60000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^5+(-192*x^5+2688*x^4-5760*x^3-76800*x^2+
456000*x-720000)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^4+((-48*x^6+960*x^5-6000*x^4+150000*x^2-600000*x+7
50000)*log(5+x)-48*x^7+960*x^6-6000*x^5+150000*x^3-600000*x^2+750000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*
x^2)^3+(96*x^7-2304*x^6+18720*x^5-24000*x^4-540000*x^3+3600000*x^2-9300000*x+9000000)*log(1/4*log(5+x)^2+1/2*x
*log(5+x)+1/4*x^2)^2+((8*x^8-240*x^7+2800*x^6-14000*x^5+350000*x^3-1750000*x^2+3750000*x-3125000)*log(5+x)+8*x
^9-240*x^8+2800*x^7-14000*x^6+350000*x^4-1750000*x^3+3750000*x^2-3125000*x)*log(1/4*log(5+x)^2+1/2*x*log(5+x)+
1/4*x^2)-16*x^9+544*x^8-7360*x^7+44800*x^6-28000*x^5-1400000*x^4+9800000*x^3-32000000*x^2+53750000*x-37500000)
/((5+x)*log(5+x)+x^2+5*x)/log(1/4*log(5+x)^2+1/2*x*log(5+x)+1/4*x^2)^9,x, algorithm="giac")
[Out]
Timed out
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Mupad [B]
time = 62.94, size = 2500, normalized size = 96.15 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)^2*(9300000*x - 3600000*x^2 + 540000*x^3 + 24000*x^4 - 187
20*x^5 + 2304*x^6 - 96*x^7 - 9000000) - log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)^5*(log(x + 5)*(24000*x
- 960*x^3 + 96*x^4 - 60000) - 60000*x + 24000*x^2 - 960*x^4 + 96*x^5) - log((x*log(x + 5))/2 + log(x + 5)^2/4
+ x^2/4)*(log(x + 5)*(3750000*x - 1750000*x^2 + 350000*x^3 - 14000*x^5 + 2800*x^6 - 240*x^7 + 8*x^8 - 3125000)
- 3125000*x + 3750000*x^2 - 1750000*x^3 + 350000*x^4 - 14000*x^6 + 2800*x^7 - 240*x^8 + 8*x^9) - 53750000*x +
log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)^6*(4480*x + 512*x^2 - 128*x^3 - 19200) + log((x*log(x + 5))/2
+ log(x + 5)^2/4 + x^2/4)^7*(log(x + 5)*(64*x^2 - 1600) - 1600*x + 64*x^3) + 32000000*x^2 - 9800000*x^3 + 1400
000*x^4 + 28000*x^5 - 44800*x^6 + 7360*x^7 - 544*x^8 + 16*x^9 + log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)
^3*(log(x + 5)*(600000*x - 150000*x^2 + 6000*x^4 - 960*x^5 + 48*x^6 - 750000) - 750000*x + 600000*x^2 - 150000
*x^3 + 6000*x^5 - 960*x^6 + 48*x^7) + log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)^4*(76800*x^2 - 456000*x +
5760*x^3 - 2688*x^4 + 192*x^5 + 720000) + 37500000)/(log((x*log(x + 5))/2 + log(x + 5)^2/4 + x^2/4)^9*(5*x +
log(x + 5)*(x + 5) + x^2)),x)
[Out]
(((x + log(x + 5))*(x + 5)*(55288627410240000*x*log(x + 5) - 8482988787840000*log(x + 5) - 12611376211092480*x
+ 156074108417136000*x*log(x + 5)^2 + 241862097005020800*x^2*log(x + 5) + 12988535943600000*x*log(x + 5)^3 +
219135555377746560*x^3*log(x + 5) - 17507453317987500*x*log(x + 5)^4 + 19466522946388980*x^4*log(x + 5) + 7291
21375968750*x*log(x + 5)^5 - 60344061522890850*x^5*log(x + 5) + 39741494343750*x*log(x + 5)^6 - 28649456555375
295*x^6*log(x + 5) - 1313381299352676*x^7*log(x + 5) + 2445280974950892*x^8*log(x + 5) + 697746384595466*x^9*l
og(x + 5) + 15680466699259*x^10*log(x + 5) - 27297800938912*x^11*log(x + 5) - 5542567190600*x^12*log(x + 5) -
222496240678*x^13*log(x + 5) + 77733390619*x^14*log(x + 5) + 15162170468*x^15*log(x + 5) + 1284414472*x^16*log
(x + 5) + 55729694*x^17*log(x + 5) + 1011465*x^18*log(x + 5) + 22571107060320000*log(x + 5)^2 + 33304821354000
000*log(x + 5)^3 - 7272248898600000*log(x + 5)^4 - 290759903437500*log(x + 5)^5 + 41697089296875*log(x + 5)^6
+ 41535961272656640*x^2 + 130170253695868800*x^3 + 113007583910675520*x^4 + 25148133338317092*x^5 - 1539663477
9859881*x^6 - 9899427748529457*x^7 - 1219881546528954*x^8 + 552233673573928*x^9 + 209267584170743*x^10 + 14293
906627893*x^11 - 5903632467837*x^12 - 1487104303273*x^13 - 91140964848*x^14 + 15380571264*x^15 + 3573331363*x^
16 + 318512409*x^17 + 14195254*x^18 + 262144*x^19 + 132121766566137600*x^2*log(x + 5)^2 - 53776497040875000*x^
2*log(x + 5)^3 - 40686820173067320*x^3*log(x + 5)^2 - 9308170433109375*x^2*log(x + 5)^4 - 48980321192227500*x^
3*log(x + 5)^3 - 83767094561874915*x^4*log(x + 5)^2 + 1239318661303125*x^2*log(x + 5)^5 + 1814561618619375*x^3
*log(x + 5)^4 - 9190433077215750*x^4*log(x + 5)^3 - 28586791150914420*x^5*log(x + 5)^2 + 2873005706250*x^2*log
(x + 5)^6 + 632382409012500*x^3*log(x + 5)^5 + 3219178738615050*x^4*log(x + 5)^4 + 5499781204260720*x^5*log(x
+ 5)^3 + 2801845098969750*x^6*log(x + 5)^2 - 11100325550000*x^3*log(x + 5)^6 + 93141158366250*x^4*log(x + 5)^5
+ 1033961584376202*x^5*log(x + 5)^4 + 3060039041255918*x^6*log(x + 5)^3 + 4020951913790706*x^7*log(x + 5)^2 -
6284174547500*x^4*log(x + 5)^6 - 34740639476400*x^5*log(x + 5)^5 + 25332401375684*x^6*log(x + 5)^4 + 40980267
9829896*x^7*log(x + 5)^3 + 844344418574619*x^8*log(x + 5)^2 - 1518428323200*x^5*log(x + 5)^6 - 18415281663060*
x^6*log(x + 5)^5 - 65769779274148*x^7*log(x + 5)^4 - 94482093218872*x^8*log(x + 5)^3 - 44209391300450*x^9*log(
x + 5)^2 - 125135260390*x^6*log(x + 5)^6 - 3242859291252*x^7*log(x + 5)^5 - 17881219224330*x^8*log(x + 5)^4 -
41916599054400*x^9*log(x + 5)^3 - 48543894115556*x^10*log(x + 5)^2 + 27869941784*x^7*log(x + 5)^6 - 5495874779
4*x^8*log(x + 5)^5 - 1468009329410*x^9*log(x + 5)^4 - 5225031281700*x^10*log(x + 5)^3 - 7895051508156*x^11*log
(x + 5)^2 + 10539948165*x^8*log(x + 5)^6 + 86745320790*x^9*log(x + 5)^5 + 230022129307*x^10*log(x + 5)^4 + 200
907498388*x^11*log(x + 5)^3 - 84260482357*x^12*log(x + 5)^2 + 1680654658*x^9*log(x + 5)^6 + 19027167183*x^10*l
og(x + 5)^5 + 77180408589*x^11*log(x + 5)^4 + 150867344946*x^12*log(x + 5)^3 + 153385529504*x^13*log(x + 5)^2
+ 160939684*x^10*log(x + 5)^6 + 2120019864*x^11*log(x + 5)^5 + 9849466112*x^12*log(x + 5)^4 + 21994333008*x^13
*log(x + 5)^3 + 25692736590*x^14*log(x + 5)^2 + 9582624*x^11*log(x + 5)^6 + 139189692*x^12*log(x + 5)^5 + 6983
77220*x^13*log(x + 5)^4 + 1665074990*x^14*log(x + 5)^3 + 2062206274*x^15*log(x + 5)^2 + 329652*x^12*log(x + 5)
^6 + 5143284*x^13*log(x + 5)^5 + 27206880*x^14*log(x + 5)^4 + 67659920*x^15*log(x + 5)^3 + 86801149*x^16*log(x
+ 5)^2 + 5040*x^13*log(x + 5)^6 + 83160*x^14*log(x + 5)^5 + 457800*x^15*log(x + 5)^4 + 1173900*x^16*log(x + 5
)^3 + 1543542*x^17*log(x + 5)^2 - 9111412243353600))/(645120*(x + 6)^13) + (log((x*log(x + 5))/2 + log(x + 5)^
2/4 + x^2/4)^2*(x + log(x + 5))*(x + 5)*(3496598752106880*x*log(x + 5)^2 - 2992418142336000*log(x + 5) - 10317
190480289280*x*log(x + 5) - 5778928094561280*x - 2900875740497280*x^2*log(x + 5) + 4589712200304000*x*log(x +
5)^3 + 11563154454902400*x^3*log(x + 5) - 431922535209900*x*log(x + 5)^4 + 11511875532819060*x^4*log(x + 5) -
50852531531250*x*log(x + 5)^5 + 3765818428055550*x^5*log(x + 5) + 2246637656250*x*log(x + 5)^6 - 2068887605582
43*x^6*log(x + 5) - 528846913485366*x^7*log(x + 5) - 165706969793283*x^8*log(x + 5) - 18272572093030*x^9*log(x
+ 5) + 2507094783530*x^10*log(x + 5) + 1236715322372*x^11*log(x + 5) + 216042611284*x^12*log(x + 5) + 2198642
9892*x^13*log(x + 5) + 1372430965*x^14*log(x + 5) + 49087670*x^15*log(x + 5) + 775775*x^16*log(x + 5) - 252232
8769900800*log(x + 5)^2 + 2043304477200000*log(x + 5)^3 + 105371139384000*log(x + 5)^4 - 49871087437500*log(x
+ 5)^5 + 1513694671875*log(x + 5)^6 - 10281021049877760*x^2 - 5630188856234112*x^3 + 2581826910152256*x^4 + 42
50478783664548*x^5 + 1796569876513215*x^6 + 171327528302103*x^7 - 120604069236993*x^8 - 51182672202475*x^9 - 7
733842487574*x^10 + 186884421810*x^11 + 290250401743*x^12 + 57446260969*x^13 + 6170456569*x^14 + 397586053*x^1
5 + 14529440*x^16 + 233280*x^17 + 1306520246739...
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