3.43.19 \(\int \frac {(-14+64 x-24 x^2) \log (-112+28 x-64 x^2+16 x^3)+(-7-4 x^2) \log ^2(-112+28 x-64 x^2+16 x^3)}{7+4 x^2} \, dx\)

Optimal. Leaf size=20 \[ (4-x) \log ^2\left ((-4+x) \left (28+16 x^2\right )\right ) \]

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Rubi [C]  time = 2.29, antiderivative size = 1101, normalized size of antiderivative = 55.05, number of steps used = 86, number of rules used = 14, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6725, 2528, 2523, 2075, 203, 2524, 2418, 2394, 2393, 2391, 260, 2416, 2390, 2301} \begin {gather*} -\frac {1}{2} \left (8-i \sqrt {7}\right ) \log ^2\left (i \sqrt {7}-2 x\right )-\frac {1}{2} i \sqrt {7} \log ^2\left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )+\left (8-i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )+i \sqrt {7} \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )-4 \log ^2(x-4)-\frac {1}{2} \left (8+i \sqrt {7}\right ) \log ^2\left (2 x+i \sqrt {7}\right )+\frac {1}{2} i \sqrt {7} \log ^2\left (2 x+i \sqrt {7}\right )-x \log ^2\left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \log \left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right ) \log (x-4)-\left (8+i \sqrt {7}\right ) \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-8 \log (x-4) \log \left (\frac {2 x+i \sqrt {7}}{8+i \sqrt {7}}\right )+8 \log (x-4) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-i \sqrt {7} \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \text {Li}_2\left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \text {Li}_2\left (\frac {2 (4-x)}{8+i \sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \text {Li}_2\left (-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )+i \sqrt {7} \text {Li}_2\left (-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \text {Li}_2\left (\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-i \sqrt {7} \text {Li}_2\left (\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \text {Li}_2\left (\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )+i \sqrt {7} \text {Li}_2\left (\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \text {Li}_2\left (\frac {i x}{\sqrt {7}}+\frac {1}{2}\right )-i \sqrt {7} \text {Li}_2\left (\frac {i x}{\sqrt {7}}+\frac {1}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-14 + 64*x - 24*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3] + (-7 - 4*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3
]^2)/(7 + 4*x^2),x]

[Out]

(-1/2*I)*Sqrt[7]*Log[I*Sqrt[7] - 2*x]^2 - ((8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]^2)/2 - I*Sqrt[7]*Log[I*Sqrt[7]
- 2*x]*Log[(2*(4 - x))/(8 - I*Sqrt[7])] - (8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]*Log[(2*(4 - x))/(8 - I*Sqrt[7]
)] - 8*Log[-((I*Sqrt[7] - 2*x)/(8 - I*Sqrt[7]))]*Log[-4 + x] - 4*Log[-4 + x]^2 + I*Sqrt[7]*Log[((-1/2*I)*(I*Sq
rt[7] - 2*x))/Sqrt[7]]*Log[I*Sqrt[7] + 2*x] - (8 + I*Sqrt[7])*Log[((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]]*Log[I*
Sqrt[7] + 2*x] + I*Sqrt[7]*Log[(2*(4 - x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2*x] - (8 + I*Sqrt[7])*Log[(2*(4 -
x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2*x] + (I/2)*Sqrt[7]*Log[I*Sqrt[7] + 2*x]^2 - ((8 + I*Sqrt[7])*Log[I*Sqr
t[7] + 2*x]^2)/2 - I*Sqrt[7]*Log[I*Sqrt[7] - 2*x]*Log[((-1/2*I)*(I*Sqrt[7] + 2*x))/Sqrt[7]] - (8 - I*Sqrt[7])*
Log[I*Sqrt[7] - 2*x]*Log[((-1/2*I)*(I*Sqrt[7] + 2*x))/Sqrt[7]] - 8*Log[-4 + x]*Log[(I*Sqrt[7] + 2*x)/(8 + I*Sq
rt[7])] + I*Sqrt[7]*Log[I*Sqrt[7] - 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + (8 - I*Sqrt[7])*Log[I*Sqrt[7] -
2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + 8*Log[-4 + x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] - I*Sqrt[7]*Log[
I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + (8 + I*Sqrt[7])*Log[I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x +
16*x^2 - 4*x^3)] - x*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)]^2 - 8*PolyLog[2, (2*(4 - x))/(8 - I*Sqrt[7])] - 8*Po
lyLog[2, (2*(4 - x))/(8 + I*Sqrt[7])] + I*Sqrt[7]*PolyLog[2, -((Sqrt[7] - (2*I)*x)/(8*I - Sqrt[7]))] - (8 + I*
Sqrt[7])*PolyLog[2, -((Sqrt[7] - (2*I)*x)/(8*I - Sqrt[7]))] - I*Sqrt[7]*PolyLog[2, (Sqrt[7] + (2*I)*x)/(8*I +
Sqrt[7])] - (8 - I*Sqrt[7])*PolyLog[2, (Sqrt[7] + (2*I)*x)/(8*I + Sqrt[7])] + I*Sqrt[7]*PolyLog[2, 1/2 - (I*x)
/Sqrt[7]] - (8 + I*Sqrt[7])*PolyLog[2, 1/2 - (I*x)/Sqrt[7]] - I*Sqrt[7]*PolyLog[2, 1/2 + (I*x)/Sqrt[7]] - (8 -
I*Sqrt[7])*PolyLog[2, 1/2 + (I*x)/Sqrt[7]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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

Rule 2075

Int[(P_)^(p_)*(Qm_), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProdu
ctQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 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 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2524

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

Rule 2528

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}-\log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )\right ) \, dx\\ &=-\left (2 \int \frac {\left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx\right )-\int \log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+2 \int \frac {x \left (7-32 x+12 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3} \, dx-2 \int \left (3 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )-\frac {2 (7+16 x) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+2 \int \left (3 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )+\frac {2 \left (42-7 x+8 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3}\right ) \, dx+4 \int \frac {(7+16 x) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx-6 \int \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-6 x \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+4 \int \frac {\left (42-7 x+8 x^2\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-28+7 x-16 x^2+4 x^3} \, dx+4 \int \left (\frac {\left (-56+7 i \sqrt {7}\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{14 \left (i \sqrt {7}-2 x\right )}+\frac {\left (56+7 i \sqrt {7}\right ) \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{14 \left (i \sqrt {7}+2 x\right )}\right ) \, dx+6 \int \frac {x \left (7-32 x+12 x^2\right )}{-28+7 x-16 x^2+4 x^3} \, dx+6 \int \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right ) \, dx\\ &=-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+4 \int \left (\frac {2 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-4+x}-\frac {7 \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2}\right ) \, dx-6 \int \frac {x \left (7-32 x+12 x^2\right )}{-28+7 x-16 x^2+4 x^3} \, dx+6 \int \left (3+\frac {4}{-4+x}-\frac {14}{7+4 x^2}\right ) \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}-2 x} \, dx+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}+2 x} \, dx\\ &=18 x+24 \log (4-x)+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-6 \int \left (3+\frac {4}{-4+x}-\frac {14}{7+4 x^2}\right ) \, dx+8 \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{-4+x} \, dx-28 \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{7+4 x^2} \, dx-84 \int \frac {1}{7+4 x^2} \, dx+\left (-8-i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}+2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx-\left (8-i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}-2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx\\ &=-6 \sqrt {7} \tan ^{-1}\left (\frac {2 x}{\sqrt {7}}\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \frac {\left (7-32 x+12 x^2\right ) \log (-4+x)}{-28+7 x-16 x^2+4 x^3} \, dx-28 \int \left (\frac {i \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{2 \sqrt {7} \left (i \sqrt {7}-2 x\right )}+\frac {i \log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{2 \sqrt {7} \left (i \sqrt {7}+2 x\right )}\right ) \, dx+84 \int \frac {1}{7+4 x^2} \, dx+\left (-8-i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2}\right ) \, dx-\left (8-i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2}\right ) \, dx\\ &=\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \left (\frac {\log (-4+x)}{-4+x}+\frac {8 x \log (-4+x)}{7+4 x^2}\right ) \, dx-\left (2 i \sqrt {7}\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}-2 x} \, dx-\left (2 i \sqrt {7}\right ) \int \frac {\log \left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )}{i \sqrt {7}+2 x} \, dx+\left (-8-i \sqrt {7}\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x} \, dx-\left (8-i \sqrt {7}\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x} \, dx-\left (8 \left (8-i \sqrt {7}\right )\right ) \int \frac {x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2} \, dx-\left (8 \left (8+i \sqrt {7}\right )\right ) \int \frac {x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2} \, dx\\ &=-\left (\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )+i \sqrt {7} \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-i \sqrt {7} \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \int \frac {\log (-4+x)}{-4+x} \, dx-64 \int \frac {x \log (-4+x)}{7+4 x^2} \, dx-\left (i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}-2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx+\left (i \sqrt {7}\right ) \int \frac {\left (7-32 x+12 x^2\right ) \log \left (i \sqrt {7}+2 x\right )}{-28+7 x-16 x^2+4 x^3} \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (-\frac {2 (-4+x)}{8-i \sqrt {7}}\right )}{i \sqrt {7}-2 x} \, dx-\left (8 \left (8-i \sqrt {7}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {7}-2 x\right )}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log \left (i \sqrt {7}-2 x\right )}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (\frac {2 (-4+x)}{-8-i \sqrt {7}}\right )}{i \sqrt {7}+2 x} \, dx-\left (8 \left (8+i \sqrt {7}\right )\right ) \int \left (-\frac {\log \left (i \sqrt {7}+2 x\right )}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log \left (i \sqrt {7}+2 x\right )}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx\\ &=-\left (\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )+i \sqrt {7} \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8-i \sqrt {7}\right ) \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-i \sqrt {7} \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-4+x\right )-64 \int \left (-\frac {\log (-4+x)}{4 \left (i \sqrt {7}-2 x\right )}+\frac {\log (-4+x)}{4 \left (i \sqrt {7}+2 x\right )}\right ) \, dx-\left (i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}-2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}-2 x\right )}{7+4 x^2}\right ) \, dx+\left (i \sqrt {7}\right ) \int \left (\frac {\log \left (i \sqrt {7}+2 x\right )}{-4+x}+\frac {8 x \log \left (i \sqrt {7}+2 x\right )}{7+4 x^2}\right ) \, dx+\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{i \sqrt {7}-2 x} \, dx-\left (2 \left (8-i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}-2 x\right )}{i \sqrt {7}+2 x} \, dx-\left (-8+i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{8-i \sqrt {7}}\right )}{x} \, dx,x,i \sqrt {7}-2 x\right )+\left (8+i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{-8-i \sqrt {7}}\right )}{x} \, dx,x,i \sqrt {7}+2 x\right )+\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{i \sqrt {7}-2 x} \, dx-\left (2 \left (8+i \sqrt {7}\right )\right ) \int \frac {\log \left (i \sqrt {7}+2 x\right )}{i \sqrt {7}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.47, size = 552, normalized size = 27.60 \begin {gather*} -4 \log ^2\left (i \sqrt {7}-2 x\right )-8 \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \log \left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right ) \log (-4+x)-4 \log ^2(-4+x)-8 \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )-8 \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )-4 \log ^2\left (i \sqrt {7}+2 x\right )-8 \log \left (i \sqrt {7}-2 x\right ) \log \left (-\frac {i \left (i \sqrt {7}+2 x\right )}{2 \sqrt {7}}\right )-8 \log (-4+x) \log \left (\frac {i \sqrt {7}+2 x}{8+i \sqrt {7}}\right )+8 \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )-8 \text {Li}_2\left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right )-8 \text {Li}_2\left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right )-8 \text {Li}_2\left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \text {Li}_2\left (\frac {2 (4-x)}{8+i \sqrt {7}}\right )-8 \text {Li}_2\left (-\frac {i \left (i \sqrt {7}+2 x\right )}{2 \sqrt {7}}\right )-8 \text {Li}_2\left (\frac {i \sqrt {7}+2 x}{8+i \sqrt {7}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-14 + 64*x - 24*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3] + (-7 - 4*x^2)*Log[-112 + 28*x - 64*x^2 +
16*x^3]^2)/(7 + 4*x^2),x]

[Out]

-4*Log[I*Sqrt[7] - 2*x]^2 - 8*Log[I*Sqrt[7] - 2*x]*Log[(2*(4 - x))/(8 - I*Sqrt[7])] - 8*Log[-((I*Sqrt[7] - 2*x
)/(8 - I*Sqrt[7]))]*Log[-4 + x] - 4*Log[-4 + x]^2 - 8*Log[((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]]*Log[I*Sqrt[7]
+ 2*x] - 8*Log[(2*(4 - x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2*x] - 4*Log[I*Sqrt[7] + 2*x]^2 - 8*Log[I*Sqrt[7]
- 2*x]*Log[((-1/2*I)*(I*Sqrt[7] + 2*x))/Sqrt[7]] - 8*Log[-4 + x]*Log[(I*Sqrt[7] + 2*x)/(8 + I*Sqrt[7])] + 8*Lo
g[I*Sqrt[7] - 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + 8*Log[-4 + x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + 8
*Log[I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] - x*Log[4*(-28 + 7*x - 16*x^2 + 4*x^3)]^2 - 8*PolyLo
g[2, ((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]] - 8*PolyLog[2, -((I*Sqrt[7] - 2*x)/(8 - I*Sqrt[7]))] - 8*PolyLog[2,
(2*(4 - x))/(8 - I*Sqrt[7])] - 8*PolyLog[2, (2*(4 - x))/(8 + I*Sqrt[7])] - 8*PolyLog[2, ((-1/2*I)*(I*Sqrt[7]
+ 2*x))/Sqrt[7]] - 8*PolyLog[2, (I*Sqrt[7] + 2*x)/(8 + I*Sqrt[7])]

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fricas [A]  time = 0.61, size = 23, normalized size = 1.15 \begin {gather*} -{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log(16*x^3-64*x^2+28*x-112))/(4*x^2+7),x
, algorithm="fricas")

[Out]

-(x - 4)*log(16*x^3 - 64*x^2 + 28*x - 112)^2

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giac [A]  time = 0.38, size = 23, normalized size = 1.15 \begin {gather*} -{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log(16*x^3-64*x^2+28*x-112))/(4*x^2+7),x
, algorithm="giac")

[Out]

-(x - 4)*log(16*x^3 - 64*x^2 + 28*x - 112)^2

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maple [A]  time = 0.08, size = 25, normalized size = 1.25

method	result	size
risch	\(\left (-x +4\right ) \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\)	\(25\)
norman	\(4 \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}-x \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2-7)*ln(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*ln(16*x^3-64*x^2+28*x-112))/(4*x^2+7),x,method=
_RETURNVERBOSE)

[Out]

(-x+4)*ln(16*x^3-64*x^2+28*x-112)^2

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maxima [B]  time = 0.81, size = 77, normalized size = 3.85 \begin {gather*} -4 \, x \log \relax (2)^{2} - {\left (x - 4\right )} \log \left (4 \, x^{2} + 7\right )^{2} - {\left (x - 4\right )} \log \left (x - 4\right )^{2} - 2 \, {\left (2 \, x \log \relax (2) + {\left (x - 4\right )} \log \left (x - 4\right ) - 8 \, \log \relax (2)\right )} \log \left (4 \, x^{2} + 7\right ) - 4 \, {\left (x \log \relax (2) - 4 \, \log \relax (2)\right )} \log \left (x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log(16*x^3-64*x^2+28*x-112))/(4*x^2+7),x
, algorithm="maxima")

[Out]

-4*x*log(2)^2 - (x - 4)*log(4*x^2 + 7)^2 - (x - 4)*log(x - 4)^2 - 2*(2*x*log(2) + (x - 4)*log(x - 4) - 8*log(2
))*log(4*x^2 + 7) - 4*(x*log(2) - 4*log(2))*log(x - 4)

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mupad [B]  time = 3.21, size = 23, normalized size = 1.15 \begin {gather*} -{\ln \left (16\,x^3-64\,x^2+28\,x-112\right )}^2\,\left (x-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(28*x - 64*x^2 + 16*x^3 - 112)^2*(4*x^2 + 7) + log(28*x - 64*x^2 + 16*x^3 - 112)*(24*x^2 - 64*x + 14)
)/(4*x^2 + 7),x)

[Out]

-log(28*x - 64*x^2 + 16*x^3 - 112)^2*(x - 4)

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sympy [A]  time = 0.19, size = 20, normalized size = 1.00 \begin {gather*} \left (4 - x\right ) \log {\left (16 x^{3} - 64 x^{2} + 28 x - 112 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2-7)*ln(16*x**3-64*x**2+28*x-112)**2+(-24*x**2+64*x-14)*ln(16*x**3-64*x**2+28*x-112))/(4*x**
2+7),x)

[Out]

(4 - x)*log(16*x**3 - 64*x**2 + 28*x - 112)**2

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