3.8.29 \(\int \frac {-40000-65000 x-5000 x^2-5000 x^3-16250 x^4-12500 x^5+3750 x^7+3750 x^8+1250 x^9+e^2 (-1250 x^2-2500 x^3)+e (-15000 x-26250 x^2-1250 x^3-1250 x^5-1250 x^6)+(-3200-5200 x-400 x^2-400 x^3-1300 x^4-1000 x^5+300 x^7+300 x^8+100 x^9+e^2 (-100 x^2-200 x^3)+e (-1200 x-2100 x^2-100 x^3-100 x^5-100 x^6)) \log (4)+(-64-104 x-8 x^2-8 x^3-26 x^4-20 x^5+6 x^7+6 x^8+2 x^9+e^2 (-2 x^2-4 x^3)+e (-24 x-42 x^2-2 x^3-2 x^5-2 x^6)) \log ^2(4)+(15000 x+26250 x^2+1250 x^3+1250 x^5+1250 x^6+e (2500 x^2+5000 x^3)+(1200 x+2100 x^2+100 x^3+100 x^5+100 x^6+e (200 x^2+400 x^3)) \log (4)+(24 x+42 x^2+2 x^3+2 x^5+2 x^6+e (4 x^2+8 x^3)) \log ^2(4)) \log (x)+(-1250 x^2-2500 x^3+(-100 x^2-200 x^3) \log (4)+(-2 x^2-4 x^3) \log ^2(4)) \log ^2(x)}{x^5+3 x^6+3 x^7+x^8} \, dx\)

Optimal. Leaf size=31 \[ (25+\log (4))^2 \left (x-\frac {-e-\frac {4}{x}+\log (x)}{x+x^2}\right )^2 \]

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Rubi [B]  time = 2.53, antiderivative size = 778, normalized size of antiderivative = 25.10, number of steps used = 51, number of rules used = 18, integrand size = 439, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6688, 12, 6742, 44, 37, 43, 2357, 2304, 2301, 2319, 2314, 31, 2317, 2391, 2305, 2347, 2344, 2318} \begin {gather*} \frac {16 (25+\log (4))^2}{x^4}-\frac {8 (25+\log (4))^2 \log (x)}{x^3}+\frac {8 (13+3 e) (25+\log (4))^2}{3 x^3}-\frac {200 (25+\log (4))^2}{3 x^3}+\frac {(25+\log (4))^2 \log ^2(x)}{x^2}-\frac {e x^2 (25+\log (4))^2}{(x+1)^2}+x^2 (25+\log (4))^2+\frac {(15-2 e) (25+\log (4))^2 \log (x)}{x^2}+\frac {(25+\log (4))^2 \log (x)}{x^2}+\frac {\left (4+21 e+e^2\right ) (25+\log (4))^2}{x^2}-\frac {12 (13+3 e) (25+\log (4))^2}{x^2}+\frac {(15-2 e) (25+\log (4))^2}{2 x^2}+\frac {385 (25+\log (4))^2}{2 x^2}-\frac {2 x (25+\log (4))^2 \log ^2(x)}{x+1}+\frac {(25+\log (4))^2 \log ^2(x)}{(x+1)^2}+2 (25+\log (4))^2 \log ^2(x)-\frac {2 (25+\log (4))^2 \log ^2(x)}{x}-\frac {4 (6-e) x (25+\log (4))^2 \log (x)}{x+1}+\frac {2 x (25+\log (4))^2 \log (x)}{x+1}+\frac {2 (4-e) (25+\log (4))^2 \log (x)}{(x+1)^2}+6 \left (4+e+2 e^2\right ) (25+\log (4))^2 \log (x)-12 \left (4+21 e+e^2\right ) (25+\log (4))^2 \log (x)+80 (13+3 e) (25+\log (4))^2 \log (x)-2 (4-e) (25+\log (4))^2 \log (x)-986 (25+\log (4))^2 \log (x)-6 \left (4+e+2 e^2\right ) (25+\log (4))^2 \log (x+1)+12 \left (4+21 e+e^2\right ) (25+\log (4))^2 \log (x+1)-80 (13+3 e) (25+\log (4))^2 \log (x+1)+4 (6-e) (25+\log (4))^2 \log (x+1)+2 (4-e) (25+\log (4))^2 \log (x+1)+984 (25+\log (4))^2 \log (x+1)+\frac {4 \left (4+e+2 e^2\right ) (25+\log (4))^2}{x+1}-\frac {6 \left (4+21 e+e^2\right ) (25+\log (4))^2}{x+1}+\frac {32 (13+3 e) (25+\log (4))^2}{x+1}-\frac {2 (4-e) (25+\log (4))^2}{x+1}-\frac {344 (25+\log (4))^2}{x+1}+\frac {\left (4+e+2 e^2\right ) (25+\log (4))^2}{(x+1)^2}-\frac {\left (4+21 e+e^2\right ) (25+\log (4))^2}{(x+1)^2}+\frac {4 (13+3 e) (25+\log (4))^2}{(x+1)^2}+\frac {(10+e) (25+\log (4))^2}{(x+1)^2}-\frac {46 (25+\log (4))^2}{(x+1)^2}-\frac {4 (5-e) (25+\log (4))^2 \log (x)}{x}-\frac {4 (25+\log (4))^2 \log (x)}{x}+\frac {2 \left (4+e+2 e^2\right ) (25+\log (4))^2}{x}-\frac {6 \left (4+21 e+e^2\right ) (25+\log (4))^2}{x}+\frac {48 (13+3 e) (25+\log (4))^2}{x}-\frac {4 (5-e) (25+\log (4))^2}{x}-\frac {644 (25+\log (4))^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-40000 - 65000*x - 5000*x^2 - 5000*x^3 - 16250*x^4 - 12500*x^5 + 3750*x^7 + 3750*x^8 + 1250*x^9 + E^2*(-1
250*x^2 - 2500*x^3) + E*(-15000*x - 26250*x^2 - 1250*x^3 - 1250*x^5 - 1250*x^6) + (-3200 - 5200*x - 400*x^2 -
400*x^3 - 1300*x^4 - 1000*x^5 + 300*x^7 + 300*x^8 + 100*x^9 + E^2*(-100*x^2 - 200*x^3) + E*(-1200*x - 2100*x^2
- 100*x^3 - 100*x^5 - 100*x^6))*Log[4] + (-64 - 104*x - 8*x^2 - 8*x^3 - 26*x^4 - 20*x^5 + 6*x^7 + 6*x^8 + 2*x
^9 + E^2*(-2*x^2 - 4*x^3) + E*(-24*x - 42*x^2 - 2*x^3 - 2*x^5 - 2*x^6))*Log[4]^2 + (15000*x + 26250*x^2 + 1250
*x^3 + 1250*x^5 + 1250*x^6 + E*(2500*x^2 + 5000*x^3) + (1200*x + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6 + E*(2
00*x^2 + 400*x^3))*Log[4] + (24*x + 42*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + E*(4*x^2 + 8*x^3))*Log[4]^2)*Log[x] + (-1
250*x^2 - 2500*x^3 + (-100*x^2 - 200*x^3)*Log[4] + (-2*x^2 - 4*x^3)*Log[4]^2)*Log[x]^2)/(x^5 + 3*x^6 + 3*x^7 +
x^8),x]

[Out]

(16*(25 + Log[4])^2)/x^4 - (200*(25 + Log[4])^2)/(3*x^3) + (8*(13 + 3*E)*(25 + Log[4])^2)/(3*x^3) + (385*(25 +
Log[4])^2)/(2*x^2) + ((15 - 2*E)*(25 + Log[4])^2)/(2*x^2) - (12*(13 + 3*E)*(25 + Log[4])^2)/x^2 + ((4 + 21*E
+ E^2)*(25 + Log[4])^2)/x^2 - (644*(25 + Log[4])^2)/x - (4*(5 - E)*(25 + Log[4])^2)/x + (48*(13 + 3*E)*(25 + L
og[4])^2)/x - (6*(4 + 21*E + E^2)*(25 + Log[4])^2)/x + (2*(4 + E + 2*E^2)*(25 + Log[4])^2)/x + x^2*(25 + Log[4
])^2 - (46*(25 + Log[4])^2)/(1 + x)^2 + ((10 + E)*(25 + Log[4])^2)/(1 + x)^2 + (4*(13 + 3*E)*(25 + Log[4])^2)/
(1 + x)^2 - ((4 + 21*E + E^2)*(25 + Log[4])^2)/(1 + x)^2 + ((4 + E + 2*E^2)*(25 + Log[4])^2)/(1 + x)^2 - (E*x^
2*(25 + Log[4])^2)/(1 + x)^2 - (344*(25 + Log[4])^2)/(1 + x) - (2*(4 - E)*(25 + Log[4])^2)/(1 + x) + (32*(13 +
3*E)*(25 + Log[4])^2)/(1 + x) - (6*(4 + 21*E + E^2)*(25 + Log[4])^2)/(1 + x) + (4*(4 + E + 2*E^2)*(25 + Log[4
])^2)/(1 + x) - 986*(25 + Log[4])^2*Log[x] - 2*(4 - E)*(25 + Log[4])^2*Log[x] + 80*(13 + 3*E)*(25 + Log[4])^2*
Log[x] - 12*(4 + 21*E + E^2)*(25 + Log[4])^2*Log[x] + 6*(4 + E + 2*E^2)*(25 + Log[4])^2*Log[x] - (8*(25 + Log[
4])^2*Log[x])/x^3 + ((25 + Log[4])^2*Log[x])/x^2 + ((15 - 2*E)*(25 + Log[4])^2*Log[x])/x^2 - (4*(25 + Log[4])^
2*Log[x])/x - (4*(5 - E)*(25 + Log[4])^2*Log[x])/x + (2*(4 - E)*(25 + Log[4])^2*Log[x])/(1 + x)^2 + (2*x*(25 +
Log[4])^2*Log[x])/(1 + x) - (4*(6 - E)*x*(25 + Log[4])^2*Log[x])/(1 + x) + 2*(25 + Log[4])^2*Log[x]^2 + ((25
+ Log[4])^2*Log[x]^2)/x^2 - (2*(25 + Log[4])^2*Log[x]^2)/x + ((25 + Log[4])^2*Log[x]^2)/(1 + x)^2 - (2*x*(25 +
Log[4])^2*Log[x]^2)/(1 + x) + 984*(25 + Log[4])^2*Log[1 + x] + 2*(4 - E)*(25 + Log[4])^2*Log[1 + x] + 4*(6 -
E)*(25 + Log[4])^2*Log[1 + x] - 80*(13 + 3*E)*(25 + Log[4])^2*Log[1 + x] + 12*(4 + 21*E + E^2)*(25 + Log[4])^2
*Log[1 + x] - 6*(4 + E + 2*E^2)*(25 + Log[4])^2*Log[1 + 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, 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]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (25+\log (4))^2 \left (-32-4 (13+3 e) x-\left (4+21 e+e^2\right ) x^2-\left (4+e+2 e^2\right ) x^3-13 x^4-(10+e) x^5-e x^6+3 x^7+3 x^8+x^9+x \left (12+(21+2 e) x+(1+4 e) x^2+x^4+x^5\right ) \log (x)-x^2 (1+2 x) \log ^2(x)\right )}{x^5 (1+x)^3} \, dx\\ &=\left (2 (25+\log (4))^2\right ) \int \frac {-32-4 (13+3 e) x-\left (4+21 e+e^2\right ) x^2-\left (4+e+2 e^2\right ) x^3-13 x^4-(10+e) x^5-e x^6+3 x^7+3 x^8+x^9+x \left (12+(21+2 e) x+(1+4 e) x^2+x^4+x^5\right ) \log (x)-x^2 (1+2 x) \log ^2(x)}{x^5 (1+x)^3} \, dx\\ &=\left (2 (25+\log (4))^2\right ) \int \left (\frac {-10-e}{(1+x)^3}-\frac {32}{x^5 (1+x)^3}-\frac {4 (13+3 e)}{x^4 (1+x)^3}+\frac {-4-21 e-e^2}{x^3 (1+x)^3}+\frac {-4-e-2 e^2}{x^2 (1+x)^3}-\frac {13}{x (1+x)^3}-\frac {e x}{(1+x)^3}+\frac {3 x^2}{(1+x)^3}+\frac {3 x^3}{(1+x)^3}+\frac {x^4}{(1+x)^3}+\frac {\left (12+21 \left (1+\frac {2 e}{21}\right ) x+(1+4 e) x^2+x^4+x^5\right ) \log (x)}{x^4 (1+x)^3}-\frac {(1+2 x) \log ^2(x)}{x^3 (1+x)^3}\right ) \, dx\\ &=\frac {(10+e) (25+\log (4))^2}{(1+x)^2}+\left (2 (25+\log (4))^2\right ) \int \frac {x^4}{(1+x)^3} \, dx+\left (2 (25+\log (4))^2\right ) \int \frac {\left (12+21 \left (1+\frac {2 e}{21}\right ) x+(1+4 e) x^2+x^4+x^5\right ) \log (x)}{x^4 (1+x)^3} \, dx-\left (2 (25+\log (4))^2\right ) \int \frac {(1+2 x) \log ^2(x)}{x^3 (1+x)^3} \, dx+\left (6 (25+\log (4))^2\right ) \int \frac {x^2}{(1+x)^3} \, dx+\left (6 (25+\log (4))^2\right ) \int \frac {x^3}{(1+x)^3} \, dx-\left (26 (25+\log (4))^2\right ) \int \frac {1}{x (1+x)^3} \, dx-\left (64 (25+\log (4))^2\right ) \int \frac {1}{x^5 (1+x)^3} \, dx-\left (2 e (25+\log (4))^2\right ) \int \frac {x}{(1+x)^3} \, dx-\left (8 (13+3 e) (25+\log (4))^2\right ) \int \frac {1}{x^4 (1+x)^3} \, dx-\left (2 \left (4+21 e+e^2\right ) (25+\log (4))^2\right ) \int \frac {1}{x^3 (1+x)^3} \, dx-\left (2 \left (4+e+2 e^2\right ) (25+\log (4))^2\right ) \int \frac {1}{x^2 (1+x)^3} \, dx\\ &=\frac {(10+e) (25+\log (4))^2}{(1+x)^2}-\frac {e x^2 (25+\log (4))^2}{(1+x)^2}+\left (2 (25+\log (4))^2\right ) \int \left (-3+x+\frac {1}{(1+x)^3}-\frac {4}{(1+x)^2}+\frac {6}{1+x}\right ) \, dx+\left (2 (25+\log (4))^2\right ) \int \left (\frac {12 \log (x)}{x^4}+\frac {(-15+2 e) \log (x)}{x^3}-\frac {2 (-5+e) \log (x)}{x^2}+\frac {3 \log (x)}{x}+\frac {2 (-4+e) \log (x)}{(1+x)^3}+\frac {2 (-6+e) \log (x)}{(1+x)^2}-\frac {3 \log (x)}{1+x}\right ) \, dx-\left (2 (25+\log (4))^2\right ) \int \left (\frac {\log ^2(x)}{x^3}-\frac {\log ^2(x)}{x^2}+\frac {\log ^2(x)}{(1+x)^3}+\frac {\log ^2(x)}{(1+x)^2}\right ) \, dx+\left (6 (25+\log (4))^2\right ) \int \left (1-\frac {1}{(1+x)^3}+\frac {3}{(1+x)^2}-\frac {3}{1+x}\right ) \, dx+\left (6 (25+\log (4))^2\right ) \int \left (\frac {1}{(1+x)^3}-\frac {2}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx-\left (26 (25+\log (4))^2\right ) \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^3}-\frac {1}{(1+x)^2}\right ) \, dx-\left (64 (25+\log (4))^2\right ) \int \left (\frac {1}{x^5}-\frac {3}{x^4}+\frac {6}{x^3}-\frac {10}{x^2}+\frac {15}{x}-\frac {1}{(1+x)^3}-\frac {5}{(1+x)^2}-\frac {15}{1+x}\right ) \, dx-\left (8 (13+3 e) (25+\log (4))^2\right ) \int \left (\frac {1}{x^4}-\frac {3}{x^3}+\frac {6}{x^2}-\frac {10}{x}+\frac {1}{(1+x)^3}+\frac {4}{(1+x)^2}+\frac {10}{1+x}\right ) \, dx-\left (2 \left (4+21 e+e^2\right ) (25+\log (4))^2\right ) \int \left (\frac {1}{x^3}-\frac {3}{x^2}+\frac {6}{x}-\frac {1}{(1+x)^3}-\frac {3}{(1+x)^2}-\frac {6}{1+x}\right ) \, dx-\left (2 \left (4+e+2 e^2\right ) (25+\log (4))^2\right ) \int \left (\frac {1}{x^2}-\frac {3}{x}+\frac {1}{(1+x)^3}+\frac {2}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx\\ &=\frac {16 (25+\log (4))^2}{x^4}-\frac {64 (25+\log (4))^2}{x^3}+\frac {8 (13+3 e) (25+\log (4))^2}{3 x^3}+\frac {192 (25+\log (4))^2}{x^2}-\frac {12 (13+3 e) (25+\log (4))^2}{x^2}+\frac {\left (4+21 e+e^2\right ) (25+\log (4))^2}{x^2}-\frac {640 (25+\log (4))^2}{x}+\frac {48 (13+3 e) (25+\log (4))^2}{x}-\frac {6 \left (4+21 e+e^2\right ) (25+\log (4))^2}{x}+\frac {2 \left (4+e+2 e^2\right ) (25+\log (4))^2}{x}+x^2 (25+\log (4))^2-\frac {46 (25+\log (4))^2}{(1+x)^2}+\frac {(10+e) (25+\log (4))^2}{(1+x)^2}+\frac {4 (13+3 e) (25+\log (4))^2}{(1+x)^2}-\frac {\left (4+21 e+e^2\right ) (25+\log (4))^2}{(1+x)^2}+\frac {\left (4+e+2 e^2\right ) (25+\log (4))^2}{(1+x)^2}-\frac {e x^2 (25+\log (4))^2}{(1+x)^2}-\frac {344 (25+\log (4))^2}{1+x}+\frac {32 (13+3 e) (25+\log (4))^2}{1+x}-\frac {6 \left (4+21 e+e^2\right ) (25+\log (4))^2}{1+x}+\frac {4 \left (4+e+2 e^2\right ) (25+\log (4))^2}{1+x}-986 (25+\log (4))^2 \log (x)+80 (13+3 e) (25+\log (4))^2 \log (x)-12 \left (4+21 e+e^2\right ) (25+\log (4))^2 \log (x)+6 \left (4+e+2 e^2\right ) (25+\log (4))^2 \log (x)+986 (25+\log (4))^2 \log (1+x)-80 (13+3 e) (25+\log (4))^2 \log (1+x)+12 \left (4+21 e+e^2\right ) (25+\log (4))^2 \log (1+x)-6 \left (4+e+2 e^2\right ) (25+\log (4))^2 \log (1+x)-\left (2 (25+\log (4))^2\right ) \int \frac {\log ^2(x)}{x^3} \, dx+\left (2 (25+\log (4))^2\right ) \int \frac {\log ^2(x)}{x^2} \, dx-\left (2 (25+\log (4))^2\right ) \int \frac {\log ^2(x)}{(1+x)^3} \, dx-\left (2 (25+\log (4))^2\right ) \int \frac {\log ^2(x)}{(1+x)^2} \, dx+\left (6 (25+\log (4))^2\right ) \int \frac {\log (x)}{x} \, dx-\left (6 (25+\log (4))^2\right ) \int \frac {\log (x)}{1+x} \, dx+\left (24 (25+\log (4))^2\right ) \int \frac {\log (x)}{x^4} \, dx-\left (2 (15-2 e) (25+\log (4))^2\right ) \int \frac {\log (x)}{x^3} \, dx-\left (4 (4-e) (25+\log (4))^2\right ) \int \frac {\log (x)}{(1+x)^3} \, dx+\left (4 (5-e) (25+\log (4))^2\right ) \int \frac {\log (x)}{x^2} \, dx-\left (4 (6-e) (25+\log (4))^2\right ) \int \frac {\log (x)}{(1+x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 33, normalized size = 1.06 \begin {gather*} \frac {(25+\log (4))^2 \left (4+e x+x^3+x^4-x \log (x)\right )^2}{x^4 (1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40000 - 65000*x - 5000*x^2 - 5000*x^3 - 16250*x^4 - 12500*x^5 + 3750*x^7 + 3750*x^8 + 1250*x^9 + E
^2*(-1250*x^2 - 2500*x^3) + E*(-15000*x - 26250*x^2 - 1250*x^3 - 1250*x^5 - 1250*x^6) + (-3200 - 5200*x - 400*
x^2 - 400*x^3 - 1300*x^4 - 1000*x^5 + 300*x^7 + 300*x^8 + 100*x^9 + E^2*(-100*x^2 - 200*x^3) + E*(-1200*x - 21
00*x^2 - 100*x^3 - 100*x^5 - 100*x^6))*Log[4] + (-64 - 104*x - 8*x^2 - 8*x^3 - 26*x^4 - 20*x^5 + 6*x^7 + 6*x^8
+ 2*x^9 + E^2*(-2*x^2 - 4*x^3) + E*(-24*x - 42*x^2 - 2*x^3 - 2*x^5 - 2*x^6))*Log[4]^2 + (15000*x + 26250*x^2
+ 1250*x^3 + 1250*x^5 + 1250*x^6 + E*(2500*x^2 + 5000*x^3) + (1200*x + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6
+ E*(200*x^2 + 400*x^3))*Log[4] + (24*x + 42*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + E*(4*x^2 + 8*x^3))*Log[4]^2)*Log[x]
+ (-1250*x^2 - 2500*x^3 + (-100*x^2 - 200*x^3)*Log[4] + (-2*x^2 - 4*x^3)*Log[4]^2)*Log[x]^2)/(x^5 + 3*x^6 + 3
*x^7 + x^8),x]

[Out]

((25 + Log[4])^2*(4 + E*x + x^3 + x^4 - x*Log[x])^2)/(x^4*(1 + x)^2)

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fricas [B]  time = 0.73, size = 253, normalized size = 8.16 \begin {gather*} \frac {625 \, x^{8} + 1250 \, x^{7} + 625 \, x^{6} + 5000 \, x^{4} + 5000 \, x^{3} + 625 \, x^{2} e^{2} + 4 \, {\left (x^{8} + 2 \, x^{7} + x^{6} + 8 \, x^{4} + 8 \, x^{3} + x^{2} e^{2} + 2 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 16\right )} \log \relax (2)^{2} + {\left (4 \, x^{2} \log \relax (2)^{2} + 100 \, x^{2} \log \relax (2) + 625 \, x^{2}\right )} \log \relax (x)^{2} + 1250 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 100 \, {\left (x^{8} + 2 \, x^{7} + x^{6} + 8 \, x^{4} + 8 \, x^{3} + x^{2} e^{2} + 2 \, {\left (x^{5} + x^{4} + 4 \, x\right )} e + 16\right )} \log \relax (2) - 2 \, {\left (625 \, x^{5} + 625 \, x^{4} + 625 \, x^{2} e + 4 \, {\left (x^{5} + x^{4} + x^{2} e + 4 \, x\right )} \log \relax (2)^{2} + 100 \, {\left (x^{5} + x^{4} + x^{2} e + 4 \, x\right )} \log \relax (2) + 2500 \, x\right )} \log \relax (x) + 10000}{x^{6} + 2 \, x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2
)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+
1200*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+15000*x)*log(x)+4*((-4*x^3-2*x^
2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(
2)^2+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1)+100*x^9+300*x^8+300*x^7-
1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x
^3-26250*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5000*x^2-65000*x-40000)/(
x^8+3*x^7+3*x^6+x^5),x, algorithm="fricas")

[Out]

(625*x^8 + 1250*x^7 + 625*x^6 + 5000*x^4 + 5000*x^3 + 625*x^2*e^2 + 4*(x^8 + 2*x^7 + x^6 + 8*x^4 + 8*x^3 + x^2
*e^2 + 2*(x^5 + x^4 + 4*x)*e + 16)*log(2)^2 + (4*x^2*log(2)^2 + 100*x^2*log(2) + 625*x^2)*log(x)^2 + 1250*(x^5
+ x^4 + 4*x)*e + 100*(x^8 + 2*x^7 + x^6 + 8*x^4 + 8*x^3 + x^2*e^2 + 2*(x^5 + x^4 + 4*x)*e + 16)*log(2) - 2*(6
25*x^5 + 625*x^4 + 625*x^2*e + 4*(x^5 + x^4 + x^2*e + 4*x)*log(2)^2 + 100*(x^5 + x^4 + x^2*e + 4*x)*log(2) + 2
500*x)*log(x) + 10000)/(x^6 + 2*x^5 + x^4)

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giac [B]  time = 0.46, size = 375, normalized size = 12.10 \begin {gather*} \frac {4 \, x^{8} \log \relax (2)^{2} + 100 \, x^{8} \log \relax (2) + 8 \, x^{7} \log \relax (2)^{2} + 625 \, x^{8} + 200 \, x^{7} \log \relax (2) + 4 \, x^{6} \log \relax (2)^{2} + 8 \, x^{5} e \log \relax (2)^{2} - 8 \, x^{5} \log \relax (2)^{2} \log \relax (x) + 1250 \, x^{7} + 100 \, x^{6} \log \relax (2) + 200 \, x^{5} e \log \relax (2) + 8 \, x^{4} e \log \relax (2)^{2} - 200 \, x^{5} \log \relax (2) \log \relax (x) - 8 \, x^{4} \log \relax (2)^{2} \log \relax (x) + 625 \, x^{6} + 1250 \, x^{5} e + 200 \, x^{4} e \log \relax (2) + 32 \, x^{4} \log \relax (2)^{2} - 1250 \, x^{5} \log \relax (x) - 200 \, x^{4} \log \relax (2) \log \relax (x) - 8 \, x^{2} e \log \relax (2)^{2} \log \relax (x) + 4 \, x^{2} \log \relax (2)^{2} \log \relax (x)^{2} + 1250 \, x^{4} e + 800 \, x^{4} \log \relax (2) + 32 \, x^{3} \log \relax (2)^{2} + 4 \, x^{2} e^{2} \log \relax (2)^{2} - 1250 \, x^{4} \log \relax (x) - 200 \, x^{2} e \log \relax (2) \log \relax (x) + 100 \, x^{2} \log \relax (2) \log \relax (x)^{2} + 5000 \, x^{4} + 800 \, x^{3} \log \relax (2) + 100 \, x^{2} e^{2} \log \relax (2) + 32 \, x e \log \relax (2)^{2} - 1250 \, x^{2} e \log \relax (x) - 32 \, x \log \relax (2)^{2} \log \relax (x) + 625 \, x^{2} \log \relax (x)^{2} + 5000 \, x^{3} + 625 \, x^{2} e^{2} + 800 \, x e \log \relax (2) - 800 \, x \log \relax (2) \log \relax (x) + 5000 \, x e + 64 \, \log \relax (2)^{2} - 5000 \, x \log \relax (x) + 1600 \, \log \relax (2) + 10000}{x^{6} + 2 \, x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2
)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+
1200*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+15000*x)*log(x)+4*((-4*x^3-2*x^
2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(
2)^2+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1)+100*x^9+300*x^8+300*x^7-
1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x
^3-26250*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5000*x^2-65000*x-40000)/(
x^8+3*x^7+3*x^6+x^5),x, algorithm="giac")

[Out]

(4*x^8*log(2)^2 + 100*x^8*log(2) + 8*x^7*log(2)^2 + 625*x^8 + 200*x^7*log(2) + 4*x^6*log(2)^2 + 8*x^5*e*log(2)
^2 - 8*x^5*log(2)^2*log(x) + 1250*x^7 + 100*x^6*log(2) + 200*x^5*e*log(2) + 8*x^4*e*log(2)^2 - 200*x^5*log(2)*
log(x) - 8*x^4*log(2)^2*log(x) + 625*x^6 + 1250*x^5*e + 200*x^4*e*log(2) + 32*x^4*log(2)^2 - 1250*x^5*log(x) -
200*x^4*log(2)*log(x) - 8*x^2*e*log(2)^2*log(x) + 4*x^2*log(2)^2*log(x)^2 + 1250*x^4*e + 800*x^4*log(2) + 32*
x^3*log(2)^2 + 4*x^2*e^2*log(2)^2 - 1250*x^4*log(x) - 200*x^2*e*log(2)*log(x) + 100*x^2*log(2)*log(x)^2 + 5000
*x^4 + 800*x^3*log(2) + 100*x^2*e^2*log(2) + 32*x*e*log(2)^2 - 1250*x^2*e*log(x) - 32*x*log(2)^2*log(x) + 625*
x^2*log(x)^2 + 5000*x^3 + 625*x^2*e^2 + 800*x*e*log(2) - 800*x*log(2)*log(x) + 5000*x*e + 64*log(2)^2 - 5000*x
*log(x) + 1600*log(2) + 10000)/(x^6 + 2*x^5 + x^4)

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maple [B]  time = 0.25, size = 357, normalized size = 11.52

method	result	size
risch	\(\frac {\left (4 \ln \relax (2)^{2}+100 \ln \relax (2)+625\right ) \ln \relax (x )^{2}}{x^{2} \left (x^{2}+2 x +1\right )}-\frac {2 \left (4 x^{4} \ln \relax (2)^{2}+4 x^{3} \ln \relax (2)^{2}+100 x^{4} \ln \relax (2)+4 \ln \relax (2)^{2} {\mathrm e} x +100 x^{3} \ln \relax (2)+625 x^{4}+100 x \,{\mathrm e} \ln \relax (2)+625 x^{3}+625 x \,{\mathrm e}+16 \ln \relax (2)^{2}+400 \ln \relax (2)+2500\right ) \ln \relax (x )}{x^{3} \left (x^{2}+2 x +1\right )}+\frac {10000+800 x \,{\mathrm e} \ln \relax (2)+4 x^{6} \ln \relax (2)^{2}+32 \ln \relax (2)^{2} {\mathrm e} x +64 \ln \relax (2)^{2}+1250 x^{7}+625 x^{8}+1600 \ln \relax (2)+625 x^{6}+5000 x^{4}+5000 x^{3}+100 x^{8} \ln \relax (2)+1250 x^{5} {\mathrm e}+8 x^{7} \ln \relax (2)^{2}+200 x^{7} \ln \relax (2)+32 x^{4} \ln \relax (2)^{2}+32 x^{3} \ln \relax (2)^{2}+800 x^{4} \ln \relax (2)+100 x^{6} \ln \relax (2)+625 x^{2} {\mathrm e}^{2}+5000 x \,{\mathrm e}+800 x^{3} \ln \relax (2)+1250 x^{4} {\mathrm e}+8 \,{\mathrm e} \ln \relax (2)^{2} x^{4}+200 \,{\mathrm e} \ln \relax (2) x^{5}+200 \,{\mathrm e} \ln \relax (2) x^{4}+4 \ln \relax (2)^{2} x^{8}+8 \,{\mathrm e} \ln \relax (2)^{2} x^{5}+4 x^{2} {\mathrm e}^{2} \ln \relax (2)^{2}+100 \,{\mathrm e}^{2} \ln \relax (2) x^{2}}{x^{4} \left (x^{2}+2 x +1\right )}\)	\(357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*(-4*x^3-2*x^2)*ln(2)^2+2*(-200*x^3-100*x^2)*ln(2)-2500*x^3-1250*x^2)*ln(x)^2+(4*((8*x^3+4*x^2)*exp(1)+
2*x^6+2*x^5+2*x^3+42*x^2+24*x)*ln(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+1200*x)*ln
(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+15000*x)*ln(x)+4*((-4*x^3-2*x^2)*exp(1)^2+
(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*ln(2)^2+2*((-200
*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1)+100*x^9+300*x^8+300*x^7-1000*x^5-1300
*x^4-400*x^3-400*x^2-5200*x-3200)*ln(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x^3-26250*x^2-1
5000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5000*x^2-65000*x-40000)/(x^8+3*x^7+3*x^
6+x^5),x,method=_RETURNVERBOSE)

[Out]

(4*ln(2)^2+100*ln(2)+625)/x^2/(x^2+2*x+1)*ln(x)^2-2*(4*x^4*ln(2)^2+4*x^3*ln(2)^2+100*x^4*ln(2)+4*ln(2)^2*exp(1
)*x+100*x^3*ln(2)+625*x^4+100*x*exp(1)*ln(2)+625*x^3+625*x*exp(1)+16*ln(2)^2+400*ln(2)+2500)/x^3/(x^2+2*x+1)*l
n(x)+(10000+800*x*exp(1)*ln(2)+4*x^6*ln(2)^2+32*ln(2)^2*exp(1)*x+64*ln(2)^2+1250*x^7+625*x^8+1600*ln(2)+625*x^
6+5000*x^4+5000*x^3+100*x^8*ln(2)+1250*x^5*exp(1)+8*x^7*ln(2)^2+200*x^7*ln(2)+32*x^4*ln(2)^2+32*x^3*ln(2)^2+80
0*x^4*ln(2)+100*x^6*ln(2)+625*x^2*exp(2)+5000*x*exp(1)+800*x^3*ln(2)+1250*x^4*exp(1)+8*exp(1)*ln(2)^2*x^4+200*
exp(1)*ln(2)*x^5+200*exp(1)*ln(2)*x^4+4*ln(2)^2*x^8+8*exp(1)*ln(2)^2*x^5+4*x^2*exp(2)*ln(2)^2+100*exp(2)*ln(2)
*x^2)/x^4/(x^2+2*x+1)

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maxima [B]  time = 0.89, size = 1988, normalized size = 64.13 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(-4*x^3-2*x^2)*log(2)^2+2*(-200*x^3-100*x^2)*log(2)-2500*x^3-1250*x^2)*log(x)^2+(4*((8*x^3+4*x^2
)*exp(1)+2*x^6+2*x^5+2*x^3+42*x^2+24*x)*log(2)^2+2*((400*x^3+200*x^2)*exp(1)+100*x^6+100*x^5+100*x^3+2100*x^2+
1200*x)*log(2)+(5000*x^3+2500*x^2)*exp(1)+1250*x^6+1250*x^5+1250*x^3+26250*x^2+15000*x)*log(x)+4*((-4*x^3-2*x^
2)*exp(1)^2+(-2*x^6-2*x^5-2*x^3-42*x^2-24*x)*exp(1)+2*x^9+6*x^8+6*x^7-20*x^5-26*x^4-8*x^3-8*x^2-104*x-64)*log(
2)^2+2*((-200*x^3-100*x^2)*exp(1)^2+(-100*x^6-100*x^5-100*x^3-2100*x^2-1200*x)*exp(1)+100*x^9+300*x^8+300*x^7-
1000*x^5-1300*x^4-400*x^3-400*x^2-5200*x-3200)*log(2)+(-2500*x^3-1250*x^2)*exp(1)^2+(-1250*x^6-1250*x^5-1250*x
^3-26250*x^2-15000*x)*exp(1)+1250*x^9+3750*x^8+3750*x^7-12500*x^5-16250*x^4-5000*x^3-5000*x^2-65000*x-40000)/(
x^8+3*x^7+3*x^6+x^5),x, algorithm="maxima")

[Out]

-4*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*e^2*log(2)^2 + 8*((6*x^2 + 9*
x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*e^2*log(2)^2 + 16*((60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x
^5 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*e*log(2)^2 - 84*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2
) - 12*log(x + 1) + 12*log(x))*e*log(2)^2 + 4*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*
e*log(2)^2 - 100*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*e^2*log(2) + 20
0*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*e^2*log(2) + 400*((60*x^4 + 90*x^3 + 20*x^2
- 5*x + 2)/(x^5 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*e*log(2) - 2100*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4
+ 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*e*log(2) + 100*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1)
+ 6*log(x))*e*log(2) + 4*(x^2 - 6*x + (8*x + 7)/(x^2 + 2*x + 1) + 12*log(x + 1))*log(2)^2 + 12*(2*x - (6*x +
5)/(x^2 + 2*x + 1) - 6*log(x + 1))*log(2)^2 - 64*((60*x^5 + 90*x^4 + 20*x^3 - 5*x^2 + 2*x - 1)/(x^6 + 2*x^5 +
x^4) - 60*log(x + 1) + 60*log(x))*log(2)^2 + 208/3*((60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x^5 + 2*x^4 + x^3) -
60*log(x + 1) + 60*log(x))*log(2)^2 - 16*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 1
2*log(x))*log(2)^2 + 16*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*log(2)^2 + 12*((4*x +
3)/(x^2 + 2*x + 1) + 2*log(x + 1))*log(2)^2 - 52*((2*x + 3)/(x^2 + 2*x + 1) - 2*log(x + 1) + 2*log(x))*log(2)^
2 + 4*(2*x + 1)*e*log(2)^2/(x^2 + 2*x + 1) + 625*x^2 - 625*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) -
12*log(x + 1) + 12*log(x))*e^2 + 1250*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*e^2 + 25
00*((60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x^5 + 2*x^4 + x^3) - 60*log(x + 1) + 60*log(x))*e - 13125*((12*x^3 +
18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x + 1) + 12*log(x))*e + 625*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 +
x) - 6*log(x + 1) + 6*log(x))*e + 100*(x^2 - 6*x + (8*x + 7)/(x^2 + 2*x + 1) + 12*log(x + 1))*log(2) + 300*(2*
x - (6*x + 5)/(x^2 + 2*x + 1) - 6*log(x + 1))*log(2) - 1600*((60*x^5 + 90*x^4 + 20*x^3 - 5*x^2 + 2*x - 1)/(x^6
+ 2*x^5 + x^4) - 60*log(x + 1) + 60*log(x))*log(2) + 5200/3*((60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x^5 + 2*x^
4 + x^3) - 60*log(x + 1) + 60*log(x))*log(2) - 400*((12*x^3 + 18*x^2 + 4*x - 1)/(x^4 + 2*x^3 + x^2) - 12*log(x
+ 1) + 12*log(x))*log(2) + 400*((6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) - 6*log(x + 1) + 6*log(x))*log(2) + 300*(
(4*x + 3)/(x^2 + 2*x + 1) + 2*log(x + 1))*log(2) - 1300*((2*x + 3)/(x^2 + 2*x + 1) - 2*log(x + 1) + 2*log(x))*
log(2) + 100*(2*x + 1)*e*log(2)/(x^2 + 2*x + 1) + 4*e*log(2)^2/(x^2 + 2*x + 1) - 6*((4*log(2)^2 + 100*log(2) +
625)*e - 20*log(2)^2 - 500*log(2) - 3125)*log(x + 1) + 6*((4*log(2)^2 + 100*log(2) + 625)*e - 20*log(2)^2 - 5
00*log(2) - 3125)*log(x) + 625*(2*x + 1)*e/(x^2 + 2*x + 1) + 100*e*log(2)/(x^2 + 2*x + 1) + 40*log(2)^2/(x^2 +
2*x + 1) - 10000*(60*x^5 + 90*x^4 + 20*x^3 - 5*x^2 + 2*x - 1)/(x^6 + 2*x^5 + x^4) + 1/3*(6*(3*(4*log(2)^2 + 1
00*log(2) + 625)*e - 64*log(2)^2 - 1600*log(2) - 10000)*x^4 + 9*(3*(4*log(2)^2 + 100*log(2) + 625)*e - 64*log(
2)^2 - 1600*log(2) - 10000)*x^3 + 3*(4*log(2)^2 + 100*log(2) + 625)*x*log(x)^2 + 2*(3*(4*log(2)^2 + 100*log(2)
+ 625)*e - 64*log(2)^2 - 1600*log(2) - 10000)*x^2 - (3*(4*log(2)^2 + 100*log(2) + 625)*e - 32*log(2)^2 - 800*
log(2) - 5000)*x - 32*log(2)^2 - 6*((4*log(2)^2 + 100*log(2) + 625)*x^4 + (4*log(2)^2 + 100*log(2) + 625)*x^3
+ (4*log(2)^2 + 100*log(2) + 625)*x*e + 16*log(2)^2 + 400*log(2) + 2500)*log(x) - 800*log(2) - 5000)/(x^5 + 2*
x^4 + x^3) + 32500/3*(60*x^4 + 90*x^3 + 20*x^2 - 5*x + 2)/(x^5 + 2*x^4 + x^3) - 2500*(12*x^3 + 18*x^2 + 4*x -
1)/(x^4 + 2*x^3 + x^2) + 2500*(6*x^2 + 9*x + 2)/(x^3 + 2*x^2 + x) + 625*(8*x + 7)/(x^2 + 2*x + 1) - 1875*(6*x
+ 5)/(x^2 + 2*x + 1) + 1875*(4*x + 3)/(x^2 + 2*x + 1) - 8125*(2*x + 3)/(x^2 + 2*x + 1) + 625*e/(x^2 + 2*x + 1)
+ 1000*log(2)/(x^2 + 2*x + 1) + 6250/(x^2 + 2*x + 1) - 18750*log(x + 1) + 18750*log(x)

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mupad [B]  time = 0.99, size = 163, normalized size = 5.26 \begin {gather*} x^2\,{\left (\ln \relax (4)+25\right )}^2+\frac {16\,{\left (\ln \relax (4)+25\right )}^2}{x^4}-\frac {2\,{\left (\ln \relax (4)+25\right )}^2\,\left ({\mathrm {e}}^2-12\,\mathrm {e}+12\,\ln \relax (x)+{\ln \relax (x)}^2-2\,\mathrm {e}\,\ln \relax (x)+28\right )}{x}+\frac {{\left (\ln \relax (4)+25\right )}^2\,\left ({\mathrm {e}}^2-16\,\mathrm {e}+16\,\ln \relax (x)+{\ln \relax (x)}^2-2\,\mathrm {e}\,\ln \relax (x)+48\right )}{x^2}+\frac {{\left (\ln \relax (4)+25\right )}^2\,{\left (\ln \relax (x)-\mathrm {e}+4\right )}^2}{{\left (x+1\right )}^2}-\frac {8\,{\left (\ln \relax (4)+25\right )}^2\,\left (\ln \relax (x)-\mathrm {e}+4\right )}{x^3}+\frac {2\,{\left (\ln \relax (4)+25\right )}^2\,\left ({\mathrm {e}}^2-11\,\mathrm {e}+11\,\ln \relax (x)+{\ln \relax (x)}^2-2\,\mathrm {e}\,\ln \relax (x)+28\right )}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(65000*x + log(x)^2*(2*log(2)*(100*x^2 + 200*x^3) + 1250*x^2 + 2500*x^3 + 4*log(2)^2*(2*x^2 + 4*x^3)) - l
og(x)*(15000*x + 2*log(2)*(1200*x + exp(1)*(200*x^2 + 400*x^3) + 2100*x^2 + 100*x^3 + 100*x^5 + 100*x^6) + exp
(1)*(2500*x^2 + 5000*x^3) + 4*log(2)^2*(24*x + exp(1)*(4*x^2 + 8*x^3) + 42*x^2 + 2*x^3 + 2*x^5 + 2*x^6) + 2625
0*x^2 + 1250*x^3 + 1250*x^5 + 1250*x^6) + 4*log(2)^2*(104*x + exp(2)*(2*x^2 + 4*x^3) + exp(1)*(24*x + 42*x^2 +
2*x^3 + 2*x^5 + 2*x^6) + 8*x^2 + 8*x^3 + 26*x^4 + 20*x^5 - 6*x^7 - 6*x^8 - 2*x^9 + 64) + exp(2)*(1250*x^2 + 2
500*x^3) + exp(1)*(15000*x + 26250*x^2 + 1250*x^3 + 1250*x^5 + 1250*x^6) + 5000*x^2 + 5000*x^3 + 16250*x^4 + 1
2500*x^5 - 3750*x^7 - 3750*x^8 - 1250*x^9 + 2*log(2)*(5200*x + exp(2)*(100*x^2 + 200*x^3) + exp(1)*(1200*x + 2
100*x^2 + 100*x^3 + 100*x^5 + 100*x^6) + 400*x^2 + 400*x^3 + 1300*x^4 + 1000*x^5 - 300*x^7 - 300*x^8 - 100*x^9
+ 3200) + 40000)/(x^5 + 3*x^6 + 3*x^7 + x^8),x)

[Out]

x^2*(log(4) + 25)^2 + (16*(log(4) + 25)^2)/x^4 - (2*(log(4) + 25)^2*(exp(2) - 12*exp(1) + 12*log(x) + log(x)^2
- 2*exp(1)*log(x) + 28))/x + ((log(4) + 25)^2*(exp(2) - 16*exp(1) + 16*log(x) + log(x)^2 - 2*exp(1)*log(x) +
48))/x^2 + ((log(4) + 25)^2*(log(x) - exp(1) + 4)^2)/(x + 1)^2 - (8*(log(4) + 25)^2*(log(x) - exp(1) + 4))/x^3
+ (2*(log(4) + 25)^2*(exp(2) - 11*exp(1) + 11*log(x) + log(x)^2 - 2*exp(1)*log(x) + 28))/(x + 1)

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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(((4*(-4*x**3-2*x**2)*ln(2)**2+2*(-200*x**3-100*x**2)*ln(2)-2500*x**3-1250*x**2)*ln(x)**2+(4*((8*x**3
+4*x**2)*exp(1)+2*x**6+2*x**5+2*x**3+42*x**2+24*x)*ln(2)**2+2*((400*x**3+200*x**2)*exp(1)+100*x**6+100*x**5+10
0*x**3+2100*x**2+1200*x)*ln(2)+(5000*x**3+2500*x**2)*exp(1)+1250*x**6+1250*x**5+1250*x**3+26250*x**2+15000*x)*
ln(x)+4*((-4*x**3-2*x**2)*exp(1)**2+(-2*x**6-2*x**5-2*x**3-42*x**2-24*x)*exp(1)+2*x**9+6*x**8+6*x**7-20*x**5-2
6*x**4-8*x**3-8*x**2-104*x-64)*ln(2)**2+2*((-200*x**3-100*x**2)*exp(1)**2+(-100*x**6-100*x**5-100*x**3-2100*x*
*2-1200*x)*exp(1)+100*x**9+300*x**8+300*x**7-1000*x**5-1300*x**4-400*x**3-400*x**2-5200*x-3200)*ln(2)+(-2500*x
**3-1250*x**2)*exp(1)**2+(-1250*x**6-1250*x**5-1250*x**3-26250*x**2-15000*x)*exp(1)+1250*x**9+3750*x**8+3750*x
**7-12500*x**5-16250*x**4-5000*x**3-5000*x**2-65000*x-40000)/(x**8+3*x**7+3*x**6+x**5),x)

[Out]

Timed out

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