3.46.63 \(\int \frac {240 x^4+1713 x^5+3776 x^6+3328 x^7+1024 x^8+(-52 x^3+384 x^4+960 x^5+512 x^6) \log (4)+(-140 x^2-144 x^3) \log ^2(4)+(-44 x-32 x^2) \log ^3(4)-4 \log ^4(4)+(188 x^4+1136 x^5+1728 x^6+768 x^7+(-92 x^3+96 x^4+192 x^5) \log (4)+(-84 x^2-48 x^3) \log ^2(4)-12 x \log ^3(4)) \log (x)+(48 x^4+240 x^5+192 x^6-36 x^3 \log (4)-12 x^2 \log ^2(4)) \log ^2(x)+(4 x^4+16 x^5-4 x^3 \log (4)) \log ^3(x)}{x^5} \, dx\)

Optimal. Leaf size=24 \[ x+\left (-1+\left (\frac {4 x (1+x)+\log (4)}{x}+\log (x)\right )^2\right )^2 \]

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Rubi [B]  time = 0.62, antiderivative size = 288, normalized size of antiderivative = 12.00, number of steps used = 35, number of rules used = 9, integrand size = 216, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {14, 2357, 2304, 2295, 2301, 2296, 2302, 30, 2305} \begin {gather*} 256 x^4+\frac {\log ^4(4)}{x^4}+1024 x^3+\frac {4 \log ^3(4) \log (x)}{x^3}+\frac {16 \log ^3(4)}{x^3}+256 x^3 \log (x)-384 x^2+96 x^2 \log ^2(x)+\frac {6 \log ^2(4) \log ^2(x)}{x^2}+\frac {48 \log ^2(4) \log (x)}{x^2}+\frac {2 \log ^2(4) (35+8 \log (4))}{x^2}+\frac {24 \log ^2(4)}{x^2}+768 x^2 \log (x)+32 x^2 (59+8 \log (4))+384 x+\log ^4(x)+16 x \log ^3(x)+16 \log ^3(x)+\frac {4 \log (4) \log ^3(x)}{x}+192 x \log ^2(x)+2 (47+24 \log (4)) \log ^2(x)+\frac {48 \log (4) \log ^2(x)}{x}+16 x (71+12 \log (4)) \log (x)-384 x \log (x)+3 x (571+320 \log (4))-16 x (71+12 \log (4))+48 (5+8 \log (4)) \log (x)+\frac {4 \log (4) (23+12 \log (4)) \log (x)}{x}+\frac {96 \log (4) \log (x)}{x}+\frac {4 \log (4) (13+36 \log (4))}{x}+\frac {4 \log (4) (23+12 \log (4))}{x}+\frac {96 \log (4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(240*x^4 + 1713*x^5 + 3776*x^6 + 3328*x^7 + 1024*x^8 + (-52*x^3 + 384*x^4 + 960*x^5 + 512*x^6)*Log[4] + (-
140*x^2 - 144*x^3)*Log[4]^2 + (-44*x - 32*x^2)*Log[4]^3 - 4*Log[4]^4 + (188*x^4 + 1136*x^5 + 1728*x^6 + 768*x^
7 + (-92*x^3 + 96*x^4 + 192*x^5)*Log[4] + (-84*x^2 - 48*x^3)*Log[4]^2 - 12*x*Log[4]^3)*Log[x] + (48*x^4 + 240*
x^5 + 192*x^6 - 36*x^3*Log[4] - 12*x^2*Log[4]^2)*Log[x]^2 + (4*x^4 + 16*x^5 - 4*x^3*Log[4])*Log[x]^3)/x^5,x]

[Out]

384*x - 384*x^2 + 1024*x^3 + 256*x^4 + (96*Log[4])/x + (24*Log[4]^2)/x^2 + (16*Log[4]^3)/x^3 + Log[4]^4/x^4 +
(2*Log[4]^2*(35 + 8*Log[4]))/x^2 + 32*x^2*(59 + 8*Log[4]) + (4*Log[4]*(23 + 12*Log[4]))/x - 16*x*(71 + 12*Log[
4]) + (4*Log[4]*(13 + 36*Log[4]))/x + 3*x*(571 + 320*Log[4]) - 384*x*Log[x] + 768*x^2*Log[x] + 256*x^3*Log[x]
+ (96*Log[4]*Log[x])/x + (48*Log[4]^2*Log[x])/x^2 + (4*Log[4]^3*Log[x])/x^3 + 48*(5 + 8*Log[4])*Log[x] + (4*Lo
g[4]*(23 + 12*Log[4])*Log[x])/x + 16*x*(71 + 12*Log[4])*Log[x] + 192*x*Log[x]^2 + 96*x^2*Log[x]^2 + (48*Log[4]
*Log[x]^2)/x + (6*Log[4]^2*Log[x]^2)/x^2 + 2*(47 + 24*Log[4])*Log[x]^2 + 16*Log[x]^3 + 16*x*Log[x]^3 + (4*Log[
4]*Log[x]^3)/x + Log[x]^4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

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

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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3328 x^7+1024 x^8+3776 x^6 \left (1+\frac {8 \log (4)}{59}\right )+1713 x^5 \left (1+\frac {320 \log (4)}{571}\right )-140 x^2 \left (1+\frac {8 \log (4)}{35}\right ) \log ^2(4)-44 x \log ^3(4)-4 \log ^4(4)+240 x^4 \left (1+\frac {8 \log (4)}{5}\right )-52 x^3 \log (4) \left (1+\frac {36 \log (4)}{13}\right )}{x^5}+\frac {4 \left (x+4 x^2-\log (4)\right ) \left (96 x^3+48 x^4+24 x \log (4)+3 \log ^2(4)+x^2 (47+24 \log (4))\right ) \log (x)}{x^4}+\frac {12 \left (x+4 x^2-\log (4)\right ) \left (4 x+4 x^2+\log (4)\right ) \log ^2(x)}{x^3}+\frac {4 \left (x+4 x^2-\log (4)\right ) \log ^3(x)}{x^2}\right ) \, dx\\ &=4 \int \frac {\left (x+4 x^2-\log (4)\right ) \left (96 x^3+48 x^4+24 x \log (4)+3 \log ^2(4)+x^2 (47+24 \log (4))\right ) \log (x)}{x^4} \, dx+4 \int \frac {\left (x+4 x^2-\log (4)\right ) \log ^3(x)}{x^2} \, dx+12 \int \frac {\left (x+4 x^2-\log (4)\right ) \left (4 x+4 x^2+\log (4)\right ) \log ^2(x)}{x^3} \, dx+\int \frac {3328 x^7+1024 x^8+3776 x^6 \left (1+\frac {8 \log (4)}{59}\right )+1713 x^5 \left (1+\frac {320 \log (4)}{571}\right )-140 x^2 \left (1+\frac {8 \log (4)}{35}\right ) \log ^2(4)-44 x \log ^3(4)-4 \log ^4(4)+240 x^4 \left (1+\frac {8 \log (4)}{5}\right )-52 x^3 \log (4) \left (1+\frac {36 \log (4)}{13}\right )}{x^5} \, dx\\ &=4 \int \left (432 x \log (x)+192 x^2 \log (x)-\frac {21 \log ^2(4) \log (x)}{x^3}-\frac {3 \log ^3(4) \log (x)}{x^4}-\frac {\log (4) (23+12 \log (4)) \log (x)}{x^2}+4 (71+12 \log (4)) \log (x)+\frac {(47+24 \log (4)) \log (x)}{x}\right ) \, dx+4 \int \left (4 \log ^3(x)+\frac {\log ^3(x)}{x}-\frac {\log (4) \log ^3(x)}{x^2}\right ) \, dx+12 \int \left (20 \log ^2(x)+\frac {4 \log ^2(x)}{x}+16 x \log ^2(x)-\frac {3 \log (4) \log ^2(x)}{x^2}-\frac {\log ^2(4) \log ^2(x)}{x^3}\right ) \, dx+\int \left (3328 x^2+1024 x^3-\frac {44 \log ^3(4)}{x^4}-\frac {4 \log ^4(4)}{x^5}+\frac {48 (5+8 \log (4))}{x}-\frac {4 \log ^2(4) (35+8 \log (4))}{x^3}+64 x (59+8 \log (4))-\frac {4 \log (4) (13+36 \log (4))}{x^2}+3 (571+320 \log (4))\right ) \, dx\\ &=\frac {3328 x^3}{3}+256 x^4+\frac {44 \log ^3(4)}{3 x^3}+\frac {\log ^4(4)}{x^4}+\frac {2 \log ^2(4) (35+8 \log (4))}{x^2}+32 x^2 (59+8 \log (4))+\frac {4 \log (4) (13+36 \log (4))}{x}+3 x (571+320 \log (4))+48 (5+8 \log (4)) \log (x)+4 \int \frac {\log ^3(x)}{x} \, dx+16 \int \log ^3(x) \, dx+48 \int \frac {\log ^2(x)}{x} \, dx+192 \int x \log ^2(x) \, dx+240 \int \log ^2(x) \, dx+768 \int x^2 \log (x) \, dx+1728 \int x \log (x) \, dx-(4 \log (4)) \int \frac {\log ^3(x)}{x^2} \, dx-(36 \log (4)) \int \frac {\log ^2(x)}{x^2} \, dx-\left (12 \log ^2(4)\right ) \int \frac {\log ^2(x)}{x^3} \, dx-\left (84 \log ^2(4)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (12 \log ^3(4)\right ) \int \frac {\log (x)}{x^4} \, dx-(4 \log (4) (23+12 \log (4))) \int \frac {\log (x)}{x^2} \, dx+(16 (71+12 \log (4))) \int \log (x) \, dx+(4 (47+24 \log (4))) \int \frac {\log (x)}{x} \, dx\\ &=-432 x^2+1024 x^3+256 x^4+\frac {21 \log ^2(4)}{x^2}+\frac {16 \log ^3(4)}{x^3}+\frac {\log ^4(4)}{x^4}+\frac {2 \log ^2(4) (35+8 \log (4))}{x^2}+32 x^2 (59+8 \log (4))+\frac {4 \log (4) (23+12 \log (4))}{x}-16 x (71+12 \log (4))+\frac {4 \log (4) (13+36 \log (4))}{x}+3 x (571+320 \log (4))+864 x^2 \log (x)+256 x^3 \log (x)+\frac {42 \log ^2(4) \log (x)}{x^2}+\frac {4 \log ^3(4) \log (x)}{x^3}+48 (5+8 \log (4)) \log (x)+\frac {4 \log (4) (23+12 \log (4)) \log (x)}{x}+16 x (71+12 \log (4)) \log (x)+240 x \log ^2(x)+96 x^2 \log ^2(x)+\frac {36 \log (4) \log ^2(x)}{x}+\frac {6 \log ^2(4) \log ^2(x)}{x^2}+2 (47+24 \log (4)) \log ^2(x)+16 x \log ^3(x)+\frac {4 \log (4) \log ^3(x)}{x}+4 \operatorname {Subst}\left (\int x^3 \, dx,x,\log (x)\right )-48 \int \log ^2(x) \, dx+48 \operatorname {Subst}\left (\int x^2 \, dx,x,\log (x)\right )-192 \int x \log (x) \, dx-480 \int \log (x) \, dx-(12 \log (4)) \int \frac {\log ^2(x)}{x^2} \, dx-(72 \log (4)) \int \frac {\log (x)}{x^2} \, dx-\left (12 \log ^2(4)\right ) \int \frac {\log (x)}{x^3} \, dx\\ &=480 x-384 x^2+1024 x^3+256 x^4+\frac {72 \log (4)}{x}+\frac {24 \log ^2(4)}{x^2}+\frac {16 \log ^3(4)}{x^3}+\frac {\log ^4(4)}{x^4}+\frac {2 \log ^2(4) (35+8 \log (4))}{x^2}+32 x^2 (59+8 \log (4))+\frac {4 \log (4) (23+12 \log (4))}{x}-16 x (71+12 \log (4))+\frac {4 \log (4) (13+36 \log (4))}{x}+3 x (571+320 \log (4))-480 x \log (x)+768 x^2 \log (x)+256 x^3 \log (x)+\frac {72 \log (4) \log (x)}{x}+\frac {48 \log ^2(4) \log (x)}{x^2}+\frac {4 \log ^3(4) \log (x)}{x^3}+48 (5+8 \log (4)) \log (x)+\frac {4 \log (4) (23+12 \log (4)) \log (x)}{x}+16 x (71+12 \log (4)) \log (x)+192 x \log ^2(x)+96 x^2 \log ^2(x)+\frac {48 \log (4) \log ^2(x)}{x}+\frac {6 \log ^2(4) \log ^2(x)}{x^2}+2 (47+24 \log (4)) \log ^2(x)+16 \log ^3(x)+16 x \log ^3(x)+\frac {4 \log (4) \log ^3(x)}{x}+\log ^4(x)+96 \int \log (x) \, dx-(24 \log (4)) \int \frac {\log (x)}{x^2} \, dx\\ &=384 x-384 x^2+1024 x^3+256 x^4+\frac {96 \log (4)}{x}+\frac {24 \log ^2(4)}{x^2}+\frac {16 \log ^3(4)}{x^3}+\frac {\log ^4(4)}{x^4}+\frac {2 \log ^2(4) (35+8 \log (4))}{x^2}+32 x^2 (59+8 \log (4))+\frac {4 \log (4) (23+12 \log (4))}{x}-16 x (71+12 \log (4))+\frac {4 \log (4) (13+36 \log (4))}{x}+3 x (571+320 \log (4))-384 x \log (x)+768 x^2 \log (x)+256 x^3 \log (x)+\frac {96 \log (4) \log (x)}{x}+\frac {48 \log ^2(4) \log (x)}{x^2}+\frac {4 \log ^3(4) \log (x)}{x^3}+48 (5+8 \log (4)) \log (x)+\frac {4 \log (4) (23+12 \log (4)) \log (x)}{x}+16 x (71+12 \log (4)) \log (x)+192 x \log ^2(x)+96 x^2 \log ^2(x)+\frac {48 \log (4) \log ^2(x)}{x}+\frac {6 \log ^2(4) \log ^2(x)}{x^2}+2 (47+24 \log (4)) \log ^2(x)+16 \log ^3(x)+16 x \log ^3(x)+\frac {4 \log (4) \log ^3(x)}{x}+\log ^4(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 242, normalized size = 10.08 \begin {gather*} 961 x+1504 x^2+1024 x^3+256 x^4+\frac {240 \log (4)}{x}+768 x \log (4)+256 x^2 \log (4)+\frac {94 \log ^2(4)}{x^2}+\frac {192 \log ^2(4)}{x}+\frac {16 \log ^3(4)}{x^3}+\frac {16 \log ^3(4)}{x^2}+\frac {\log ^4(4)}{x^4}+240 \log (x)+752 x \log (x)+768 x^2 \log (x)+256 x^3 \log (x)+384 \log (4) \log (x)+\frac {188 \log (4) \log (x)}{x}+192 x \log (4) \log (x)+\frac {48 \log ^2(4) \log (x)}{x^2}+\frac {48 \log ^2(4) \log (x)}{x}+\frac {4 \log ^3(4) \log (x)}{x^3}+94 \log ^2(x)+192 x \log ^2(x)+96 x^2 \log ^2(x)+48 \log (4) \log ^2(x)+\frac {48 \log (4) \log ^2(x)}{x}+\frac {6 \log ^2(4) \log ^2(x)}{x^2}+16 \log ^3(x)+16 x \log ^3(x)+\frac {4 \log (4) \log ^3(x)}{x}+\log ^4(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(240*x^4 + 1713*x^5 + 3776*x^6 + 3328*x^7 + 1024*x^8 + (-52*x^3 + 384*x^4 + 960*x^5 + 512*x^6)*Log[4
] + (-140*x^2 - 144*x^3)*Log[4]^2 + (-44*x - 32*x^2)*Log[4]^3 - 4*Log[4]^4 + (188*x^4 + 1136*x^5 + 1728*x^6 +
768*x^7 + (-92*x^3 + 96*x^4 + 192*x^5)*Log[4] + (-84*x^2 - 48*x^3)*Log[4]^2 - 12*x*Log[4]^3)*Log[x] + (48*x^4
+ 240*x^5 + 192*x^6 - 36*x^3*Log[4] - 12*x^2*Log[4]^2)*Log[x]^2 + (4*x^4 + 16*x^5 - 4*x^3*Log[4])*Log[x]^3)/x^
5,x]

[Out]

961*x + 1504*x^2 + 1024*x^3 + 256*x^4 + (240*Log[4])/x + 768*x*Log[4] + 256*x^2*Log[4] + (94*Log[4]^2)/x^2 + (
192*Log[4]^2)/x + (16*Log[4]^3)/x^3 + (16*Log[4]^3)/x^2 + Log[4]^4/x^4 + 240*Log[x] + 752*x*Log[x] + 768*x^2*L
og[x] + 256*x^3*Log[x] + 384*Log[4]*Log[x] + (188*Log[4]*Log[x])/x + 192*x*Log[4]*Log[x] + (48*Log[4]^2*Log[x]
)/x^2 + (48*Log[4]^2*Log[x])/x + (4*Log[4]^3*Log[x])/x^3 + 94*Log[x]^2 + 192*x*Log[x]^2 + 96*x^2*Log[x]^2 + 48
*Log[4]*Log[x]^2 + (48*Log[4]*Log[x]^2)/x + (6*Log[4]^2*Log[x]^2)/x^2 + 16*Log[x]^3 + 16*x*Log[x]^3 + (4*Log[4
]*Log[x]^3)/x + Log[x]^4

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fricas [B]  time = 0.70, size = 216, normalized size = 9.00 \begin {gather*} \frac {256 \, x^{8} + x^{4} \log \relax (x)^{4} + 1024 \, x^{7} + 1504 \, x^{6} + 961 \, x^{5} + 128 \, {\left (x^{2} + x\right )} \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + 8 \, {\left (2 \, x^{5} + 2 \, x^{4} + x^{3} \log \relax (2)\right )} \log \relax (x)^{3} + 8 \, {\left (96 \, x^{3} + 47 \, x^{2}\right )} \log \relax (2)^{2} + 2 \, {\left (48 \, x^{6} + 96 \, x^{5} + 47 \, x^{4} + 12 \, x^{2} \log \relax (2)^{2} + 48 \, {\left (x^{4} + x^{3}\right )} \log \relax (2)\right )} \log \relax (x)^{2} + 32 \, {\left (16 \, x^{6} + 48 \, x^{5} + 15 \, x^{3}\right )} \log \relax (2) + 8 \, {\left (32 \, x^{7} + 96 \, x^{6} + 94 \, x^{5} + 30 \, x^{4} + 4 \, x \log \relax (2)^{3} + 24 \, {\left (x^{3} + x^{2}\right )} \log \relax (2)^{2} + {\left (48 \, x^{5} + 96 \, x^{4} + 47 \, x^{3}\right )} \log \relax (2)\right )} \log \relax (x)}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3*log(2)+16*x^5+4*x^4)*log(x)^3+(-48*x^2*log(2)^2-72*x^3*log(2)+192*x^6+240*x^5+48*x^4)*log(x
)^2+(-96*x*log(2)^3+4*(-48*x^3-84*x^2)*log(2)^2+2*(192*x^5+96*x^4-92*x^3)*log(2)+768*x^7+1728*x^6+1136*x^5+188
*x^4)*log(x)-64*log(2)^4+8*(-32*x^2-44*x)*log(2)^3+4*(-144*x^3-140*x^2)*log(2)^2+2*(512*x^6+960*x^5+384*x^4-52
*x^3)*log(2)+1024*x^8+3328*x^7+3776*x^6+1713*x^5+240*x^4)/x^5,x, algorithm="fricas")

[Out]

(256*x^8 + x^4*log(x)^4 + 1024*x^7 + 1504*x^6 + 961*x^5 + 128*(x^2 + x)*log(2)^3 + 16*log(2)^4 + 8*(2*x^5 + 2*
x^4 + x^3*log(2))*log(x)^3 + 8*(96*x^3 + 47*x^2)*log(2)^2 + 2*(48*x^6 + 96*x^5 + 47*x^4 + 12*x^2*log(2)^2 + 48
*(x^4 + x^3)*log(2))*log(x)^2 + 32*(16*x^6 + 48*x^5 + 15*x^3)*log(2) + 8*(32*x^7 + 96*x^6 + 94*x^5 + 30*x^4 +
4*x*log(2)^3 + 24*(x^3 + x^2)*log(2)^2 + (48*x^5 + 96*x^4 + 47*x^3)*log(2))*log(x))/x^4

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1024 \, x^{8} + 3328 \, x^{7} + 3776 \, x^{6} + 1713 \, x^{5} + 240 \, x^{4} - 32 \, {\left (8 \, x^{2} + 11 \, x\right )} \log \relax (2)^{3} - 64 \, \log \relax (2)^{4} + 4 \, {\left (4 \, x^{5} + x^{4} - 2 \, x^{3} \log \relax (2)\right )} \log \relax (x)^{3} - 16 \, {\left (36 \, x^{3} + 35 \, x^{2}\right )} \log \relax (2)^{2} + 24 \, {\left (8 \, x^{6} + 10 \, x^{5} + 2 \, x^{4} - 3 \, x^{3} \log \relax (2) - 2 \, x^{2} \log \relax (2)^{2}\right )} \log \relax (x)^{2} + 8 \, {\left (128 \, x^{6} + 240 \, x^{5} + 96 \, x^{4} - 13 \, x^{3}\right )} \log \relax (2) + 4 \, {\left (192 \, x^{7} + 432 \, x^{6} + 284 \, x^{5} + 47 \, x^{4} - 24 \, x \log \relax (2)^{3} - 12 \, {\left (4 \, x^{3} + 7 \, x^{2}\right )} \log \relax (2)^{2} + 2 \, {\left (48 \, x^{5} + 24 \, x^{4} - 23 \, x^{3}\right )} \log \relax (2)\right )} \log \relax (x)}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3*log(2)+16*x^5+4*x^4)*log(x)^3+(-48*x^2*log(2)^2-72*x^3*log(2)+192*x^6+240*x^5+48*x^4)*log(x
)^2+(-96*x*log(2)^3+4*(-48*x^3-84*x^2)*log(2)^2+2*(192*x^5+96*x^4-92*x^3)*log(2)+768*x^7+1728*x^6+1136*x^5+188
*x^4)*log(x)-64*log(2)^4+8*(-32*x^2-44*x)*log(2)^3+4*(-144*x^3-140*x^2)*log(2)^2+2*(512*x^6+960*x^5+384*x^4-52
*x^3)*log(2)+1024*x^8+3328*x^7+3776*x^6+1713*x^5+240*x^4)/x^5,x, algorithm="giac")

[Out]

integrate((1024*x^8 + 3328*x^7 + 3776*x^6 + 1713*x^5 + 240*x^4 - 32*(8*x^2 + 11*x)*log(2)^3 - 64*log(2)^4 + 4*
(4*x^5 + x^4 - 2*x^3*log(2))*log(x)^3 - 16*(36*x^3 + 35*x^2)*log(2)^2 + 24*(8*x^6 + 10*x^5 + 2*x^4 - 3*x^3*log
(2) - 2*x^2*log(2)^2)*log(x)^2 + 8*(128*x^6 + 240*x^5 + 96*x^4 - 13*x^3)*log(2) + 4*(192*x^7 + 432*x^6 + 284*x
^5 + 47*x^4 - 24*x*log(2)^3 - 12*(4*x^3 + 7*x^2)*log(2)^2 + 2*(48*x^5 + 24*x^4 - 23*x^3)*log(2))*log(x))/x^5,
x)

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maple [B]  time = 0.05, size = 230, normalized size = 9.58

method	result	size
risch	\(\ln \relax (x )^{4}+\frac {8 \left (2 x^{2}+\ln \relax (2)+2 x \right ) \ln \relax (x )^{3}}{x}+\frac {2 \left (48 x^{4}+48 x^{2} \ln \relax (2)+96 x^{3}+12 \ln \relax (2)^{2}+48 x \ln \relax (2)+47 x^{2}\right ) \ln \relax (x )^{2}}{x^{2}}+\frac {8 \left (32 x^{6}+48 x^{4} \ln \relax (2)+96 x^{5}+24 x^{2} \ln \relax (2)^{2}+94 x^{4}+4 \ln \relax (2)^{3}+24 x \ln \relax (2)^{2}+47 x^{2} \ln \relax (2)\right ) \ln \relax (x )}{x^{3}}+\frac {256 x^{8}+512 x^{6} \ln \relax (2)+1024 x^{7}+768 \ln \relax (x ) \ln \relax (2) x^{4}+1536 x^{5} \ln \relax (2)+1504 x^{6}+240 x^{4} \ln \relax (x )+128 x^{2} \ln \relax (2)^{3}+768 x^{3} \ln \relax (2)^{2}+961 x^{5}+16 \ln \relax (2)^{4}+128 x \ln \relax (2)^{3}+376 x^{2} \ln \relax (2)^{2}+480 x^{3} \ln \relax (2)}{x^{4}}\)	\(230\)
default	\(961 x +768 x^{2} \ln \relax (x )+16 x \ln \relax (x )^{3}+96 x^{2} \ln \relax (x )^{2}+\ln \relax (x )^{4}+16 \ln \relax (x )^{3}+240 \ln \relax (x )+94 \ln \relax (x )^{2}+256 x^{4}+1024 x^{3}+1504 x^{2}+256 x^{3} \ln \relax (x )+\frac {576 \ln \relax (2)^{2}}{x}+\frac {280 \ln \relax (2)^{2}}{x^{2}}+768 \ln \relax (2) \ln \relax (x )+\frac {104 \ln \relax (2)}{x}+192 x \ln \relax (x )^{2}+1920 x \ln \relax (2)+512 x^{2} \ln \relax (2)+96 \ln \relax (2) \ln \relax (x )^{2}+752 x \ln \relax (x )+\frac {16 \ln \relax (2)^{4}}{x^{4}}+\frac {128 \ln \relax (2)^{3}}{x^{2}}-184 \ln \relax (2) \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )-8 \ln \relax (2) \left (-\frac {\ln \relax (x )^{3}}{x}-\frac {3 \ln \relax (x )^{2}}{x}-\frac {6 \ln \relax (x )}{x}-\frac {6}{x}\right )-48 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-72 \ln \relax (2) \left (-\frac {\ln \relax (x )^{2}}{x}-\frac {2 \ln \relax (x )}{x}-\frac {2}{x}\right )-192 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )-96 \ln \relax (2)^{3} \left (-\frac {\ln \relax (x )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-336 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+\frac {352 \ln \relax (2)^{3}}{3 x^{3}}+384 \ln \relax (2) \left (x \ln \relax (x )-x \right )\)	\(335\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^3*ln(2)+16*x^5+4*x^4)*ln(x)^3+(-48*x^2*ln(2)^2-72*x^3*ln(2)+192*x^6+240*x^5+48*x^4)*ln(x)^2+(-96*x*
ln(2)^3+4*(-48*x^3-84*x^2)*ln(2)^2+2*(192*x^5+96*x^4-92*x^3)*ln(2)+768*x^7+1728*x^6+1136*x^5+188*x^4)*ln(x)-64
*ln(2)^4+8*(-32*x^2-44*x)*ln(2)^3+4*(-144*x^3-140*x^2)*ln(2)^2+2*(512*x^6+960*x^5+384*x^4-52*x^3)*ln(2)+1024*x
^8+3328*x^7+3776*x^6+1713*x^5+240*x^4)/x^5,x,method=_RETURNVERBOSE)

[Out]

ln(x)^4+8*(2*x^2+ln(2)+2*x)/x*ln(x)^3+2*(48*x^4+48*x^2*ln(2)+96*x^3+12*ln(2)^2+48*x*ln(2)+47*x^2)/x^2*ln(x)^2+
8*(32*x^6+48*x^4*ln(2)+96*x^5+24*x^2*ln(2)^2+94*x^4+4*ln(2)^3+24*x*ln(2)^2+47*x^2*ln(2))/x^3*ln(x)+(256*x^8+51
2*x^6*ln(2)+1024*x^7+768*ln(x)*ln(2)*x^4+1536*x^5*ln(2)+1504*x^6+240*x^4*ln(x)+128*x^2*ln(2)^3+768*x^3*ln(2)^2
+961*x^5+16*ln(2)^4+128*x*ln(2)^3+376*x^2*ln(2)^2+480*x^3*ln(2))/x^4

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maxima [B]  time = 0.36, size = 322, normalized size = 13.42 \begin {gather*} 256 \, x^{4} + \frac {32}{3} \, {\left (\frac {3 \, \log \relax (x)}{x^{3}} + \frac {1}{x^{3}}\right )} \log \relax (2)^{3} + 256 \, x^{3} \log \relax (x) + \log \relax (x)^{4} + 48 \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + 1024 \, x^{3} + 512 \, x^{2} \log \relax (2) + 192 \, {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (2)^{2} + 84 \, {\left (\frac {2 \, \log \relax (x)}{x^{2}} + \frac {1}{x^{2}}\right )} \log \relax (2)^{2} + 864 \, x^{2} \log \relax (x) + 96 \, \log \relax (2) \log \relax (x)^{2} + 16 \, \log \relax (x)^{3} + 16 \, {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 6\right )} x + 240 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 1456 \, x^{2} + 384 \, {\left (x \log \relax (x) - x\right )} \log \relax (2) + 1920 \, x \log \relax (2) + 184 \, {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (2) + 1136 \, x \log \relax (x) + 768 \, \log \relax (2) \log \relax (x) + 94 \, \log \relax (x)^{2} + 577 \, x + \frac {8 \, {\left (\log \relax (x)^{3} + 3 \, \log \relax (x)^{2} + 6 \, \log \relax (x) + 6\right )} \log \relax (2)}{x} + \frac {72 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 2\right )} \log \relax (2)}{x} + \frac {12 \, {\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} \log \relax (2)^{2}}{x^{2}} + \frac {576 \, \log \relax (2)^{2}}{x} + \frac {128 \, \log \relax (2)^{3}}{x^{2}} + \frac {104 \, \log \relax (2)}{x} + \frac {280 \, \log \relax (2)^{2}}{x^{2}} + \frac {352 \, \log \relax (2)^{3}}{3 \, x^{3}} + \frac {16 \, \log \relax (2)^{4}}{x^{4}} + 240 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^3*log(2)+16*x^5+4*x^4)*log(x)^3+(-48*x^2*log(2)^2-72*x^3*log(2)+192*x^6+240*x^5+48*x^4)*log(x
)^2+(-96*x*log(2)^3+4*(-48*x^3-84*x^2)*log(2)^2+2*(192*x^5+96*x^4-92*x^3)*log(2)+768*x^7+1728*x^6+1136*x^5+188
*x^4)*log(x)-64*log(2)^4+8*(-32*x^2-44*x)*log(2)^3+4*(-144*x^3-140*x^2)*log(2)^2+2*(512*x^6+960*x^5+384*x^4-52
*x^3)*log(2)+1024*x^8+3328*x^7+3776*x^6+1713*x^5+240*x^4)/x^5,x, algorithm="maxima")

[Out]

256*x^4 + 32/3*(3*log(x)/x^3 + 1/x^3)*log(2)^3 + 256*x^3*log(x) + log(x)^4 + 48*(2*log(x)^2 - 2*log(x) + 1)*x^
2 + 1024*x^3 + 512*x^2*log(2) + 192*(log(x)/x + 1/x)*log(2)^2 + 84*(2*log(x)/x^2 + 1/x^2)*log(2)^2 + 864*x^2*l
og(x) + 96*log(2)*log(x)^2 + 16*log(x)^3 + 16*(log(x)^3 - 3*log(x)^2 + 6*log(x) - 6)*x + 240*(log(x)^2 - 2*log
(x) + 2)*x + 1456*x^2 + 384*(x*log(x) - x)*log(2) + 1920*x*log(2) + 184*(log(x)/x + 1/x)*log(2) + 1136*x*log(x
) + 768*log(2)*log(x) + 94*log(x)^2 + 577*x + 8*(log(x)^3 + 3*log(x)^2 + 6*log(x) + 6)*log(2)/x + 72*(log(x)^2
+ 2*log(x) + 2)*log(2)/x + 12*(2*log(x)^2 + 2*log(x) + 1)*log(2)^2/x^2 + 576*log(2)^2/x + 128*log(2)^3/x^2 +
104*log(2)/x + 280*log(2)^2/x^2 + 352/3*log(2)^3/x^3 + 16*log(2)^4/x^4 + 240*log(x)

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mupad [B]  time = 4.15, size = 223, normalized size = 9.29 \begin {gather*} x\,\left (1536\,\ln \relax (2)+961\right )+\frac {128\,{\ln \relax (2)}^3}{x^3}+\frac {16\,{\ln \relax (2)}^4}{x^4}+192\,x\,{\ln \relax (x)}^2+768\,x^2\,\ln \relax (x)+16\,x\,{\ln \relax (x)}^3+256\,x^3\,\ln \relax (x)+16\,{\ln \relax (x)}^3+{\ln \relax (x)}^4+x^2\,\left (512\,\ln \relax (2)+1504\right )+\ln \relax (x)\,\left (768\,\ln \relax (2)+240\right )+96\,x^2\,{\ln \relax (x)}^2+{\ln \relax (x)}^2\,\left (96\,\ln \relax (2)+94\right )+1024\,x^3+256\,x^4+\frac {8\,{\ln \relax (2)}^2\,\left (16\,\ln \relax (2)+47\right )}{x^2}+\frac {24\,{\ln \relax (2)}^2\,{\ln \relax (x)}^2}{x^2}+\frac {96\,\ln \relax (2)\,\left (8\,\ln \relax (2)+5\right )}{x}+x\,\ln \relax (x)\,\left (384\,\ln \relax (2)+752\right )+\frac {96\,\ln \relax (2)\,{\ln \relax (x)}^2}{x}+\frac {192\,{\ln \relax (2)}^2\,\ln \relax (x)}{x^2}+\frac {32\,{\ln \relax (2)}^3\,\ln \relax (x)}{x^3}+\frac {\ln \left (256\right )\,{\ln \relax (x)}^3}{x}+\frac {8\,\ln \relax (2)\,\ln \relax (x)\,\left (24\,\ln \relax (2)+47\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(384*x^4 - 52*x^3 + 960*x^5 + 512*x^6) + log(x)^3*(4*x^4 - 8*x^3*log(2) + 16*x^5) + log(x)^2*(48
*x^4 - 72*x^3*log(2) - 48*x^2*log(2)^2 + 240*x^5 + 192*x^6) - 8*log(2)^3*(44*x + 32*x^2) - 64*log(2)^4 + 240*x
^4 + 1713*x^5 + 3776*x^6 + 3328*x^7 + 1024*x^8 + log(x)*(2*log(2)*(96*x^4 - 92*x^3 + 192*x^5) - 96*x*log(2)^3
+ 188*x^4 + 1136*x^5 + 1728*x^6 + 768*x^7 - 4*log(2)^2*(84*x^2 + 48*x^3)) - 4*log(2)^2*(140*x^2 + 144*x^3))/x^
5,x)

[Out]

x*(1536*log(2) + 961) + (128*log(2)^3)/x^3 + (16*log(2)^4)/x^4 + 192*x*log(x)^2 + 768*x^2*log(x) + 16*x*log(x)
^3 + 256*x^3*log(x) + 16*log(x)^3 + log(x)^4 + x^2*(512*log(2) + 1504) + log(x)*(768*log(2) + 240) + 96*x^2*lo
g(x)^2 + log(x)^2*(96*log(2) + 94) + 1024*x^3 + 256*x^4 + (8*log(2)^2*(16*log(2) + 47))/x^2 + (24*log(2)^2*log
(x)^2)/x^2 + (96*log(2)*(8*log(2) + 5))/x + x*log(x)*(384*log(2) + 752) + (96*log(2)*log(x)^2)/x + (192*log(2)
^2*log(x))/x^2 + (32*log(2)^3*log(x))/x^3 + (log(256)*log(x)^3)/x + (8*log(2)*log(x)*(24*log(2) + 47))/x

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sympy [B]  time = 1.32, size = 226, normalized size = 9.42 \begin {gather*} 256 x^{4} + 1024 x^{3} + x^{2} \left (512 \log {\relax (2 )} + 1504\right ) + x \left (961 + 1536 \log {\relax (2 )}\right ) + \log {\relax (x )}^{4} + 48 \left (5 + 16 \log {\relax (2 )}\right ) \log {\relax (x )} + \frac {\left (16 x^{2} + 16 x + 8 \log {\relax (2 )}\right ) \log {\relax (x )}^{3}}{x} + \frac {\left (96 x^{4} + 192 x^{3} + 96 x^{2} \log {\relax (2 )} + 94 x^{2} + 96 x \log {\relax (2 )} + 24 \log {\relax (2 )}^{2}\right ) \log {\relax (x )}^{2}}{x^{2}} + \frac {\left (256 x^{6} + 768 x^{5} + 384 x^{4} \log {\relax (2 )} + 752 x^{4} + 192 x^{2} \log {\relax (2 )}^{2} + 376 x^{2} \log {\relax (2 )} + 192 x \log {\relax (2 )}^{2} + 32 \log {\relax (2 )}^{3}\right ) \log {\relax (x )}}{x^{3}} + \frac {x^{3} \left (480 \log {\relax (2 )} + 768 \log {\relax (2 )}^{2}\right ) + x^{2} \left (128 \log {\relax (2 )}^{3} + 376 \log {\relax (2 )}^{2}\right ) + 128 x \log {\relax (2 )}^{3} + 16 \log {\relax (2 )}^{4}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**3*ln(2)+16*x**5+4*x**4)*ln(x)**3+(-48*x**2*ln(2)**2-72*x**3*ln(2)+192*x**6+240*x**5+48*x**4)
*ln(x)**2+(-96*x*ln(2)**3+4*(-48*x**3-84*x**2)*ln(2)**2+2*(192*x**5+96*x**4-92*x**3)*ln(2)+768*x**7+1728*x**6+
1136*x**5+188*x**4)*ln(x)-64*ln(2)**4+8*(-32*x**2-44*x)*ln(2)**3+4*(-144*x**3-140*x**2)*ln(2)**2+2*(512*x**6+9
60*x**5+384*x**4-52*x**3)*ln(2)+1024*x**8+3328*x**7+3776*x**6+1713*x**5+240*x**4)/x**5,x)

[Out]

256*x**4 + 1024*x**3 + x**2*(512*log(2) + 1504) + x*(961 + 1536*log(2)) + log(x)**4 + 48*(5 + 16*log(2))*log(x
) + (16*x**2 + 16*x + 8*log(2))*log(x)**3/x + (96*x**4 + 192*x**3 + 96*x**2*log(2) + 94*x**2 + 96*x*log(2) + 2
4*log(2)**2)*log(x)**2/x**2 + (256*x**6 + 768*x**5 + 384*x**4*log(2) + 752*x**4 + 192*x**2*log(2)**2 + 376*x**
2*log(2) + 192*x*log(2)**2 + 32*log(2)**3)*log(x)/x**3 + (x**3*(480*log(2) + 768*log(2)**2) + x**2*(128*log(2)
**3 + 376*log(2)**2) + 128*x*log(2)**3 + 16*log(2)**4)/x**4

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