3.43 \(\int \frac {(a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2))}{x^3} \, dx\)

Optimal. Leaf size=425 \[ \frac {3}{4} b^2 d f n^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b^2 d f n^2 \text {Li}_3\left (-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b^2 d f n^2 \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}+\frac {3}{4} b d f n \text {Li}_2\left (-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b d f n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{2} d f \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {3}{8} b^3 d f n^3 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {Li}_3\left (-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {Li}_4\left (-\frac {1}{d f x^2}\right )-\frac {3}{8} b^3 d f n^3 \log \left (d f x^2+1\right )-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}+\frac {3}{4} b^3 d f n^3 \log (x) \]

[Out]

3/4*b^3*d*f*n^3*ln(x)-3/4*b^2*d*f*n^2*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))-3/4*b*d*f*n*ln(1+1/d/f/x^2)*(a+b*ln(c*x^
n))^2-1/2*d*f*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))^3-3/8*b^3*d*f*n^3*ln(d*f*x^2+1)-3/8*b^3*n^3*ln(d*f*x^2+1)/x^2-3/
4*b^2*n^2*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)/x^2-1/2*(a+b*ln(c*x^n))^3*
ln(d*f*x^2+1)/x^2+3/8*b^3*d*f*n^3*polylog(2,-1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b*ln(c*x^n))*polylog(2,-1/d/f/x^2)+
3/4*b*d*f*n*(a+b*ln(c*x^n))^2*polylog(2,-1/d/f/x^2)+3/8*b^3*d*f*n^3*polylog(3,-1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b
*ln(c*x^n))*polylog(3,-1/d/f/x^2)+3/8*b^3*d*f*n^3*polylog(4,-1/d/f/x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.58, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2305, 2304, 2378, 266, 36, 29, 31, 2345, 2391, 2374, 6589, 2383} \[ \frac {3}{4} b^2 d f n^2 \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b^2 d f n^2 \text {PolyLog}\left (3,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} b d f n \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3}{8} b^3 d f n^3 \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {PolyLog}\left (3,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {PolyLog}\left (4,-\frac {1}{d f x^2}\right )-\frac {3}{4} b^2 d f n^2 \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{4} b d f n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{2} d f \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3}{8} b^3 d f n^3 \log \left (d f x^2+1\right )-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}+\frac {3}{4} b^3 d f n^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^3,x]

[Out]

(3*b^3*d*f*n^3*Log[x])/4 - (3*b^2*d*f*n^2*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n]))/4 - (3*b*d*f*n*Log[1 + 1/(d
*f*x^2)]*(a + b*Log[c*x^n])^2)/4 - (d*f*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^3)/2 - (3*b^3*d*f*n^3*Log[1 +
d*f*x^2])/8 - (3*b^3*n^3*Log[1 + d*f*x^2])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(4*x^2) -
(3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(2*x^2) + (3*
b^3*d*f*n^3*PolyLog[2, -(1/(d*f*x^2))])/8 + (3*b^2*d*f*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(d*f*x^2))])/4 +
(3*b*d*f*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(d*f*x^2))])/4 + (3*b^3*d*f*n^3*PolyLog[3, -(1/(d*f*x^2))])/8 +
(3*b^2*d*f*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(d*f*x^2))])/4 + (3*b^3*d*f*n^3*PolyLog[4, -(1/(d*f*x^2))])/
8

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2345

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

Rule 2374

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

Rule 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[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 6589

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx &=-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}-(2 f) \int \left (-\frac {3 b^3 d n^3}{8 x \left (1+d f x^2\right )}-\frac {3 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x \left (1+d f x^2\right )}-\frac {3 b d n \left (a+b \log \left (c x^n\right )\right )^2}{4 x \left (1+d f x^2\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (1+d f x^2\right )}\right ) \, dx\\ &=-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+(d f) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x \left (1+d f x^2\right )} \, dx+\frac {1}{2} (3 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (1+d f x^2\right )} \, dx+\frac {1}{2} \left (3 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (1+d f x^2\right )} \, dx+\frac {1}{4} \left (3 b^3 d f n^3\right ) \int \frac {1}{x \left (1+d f x^2\right )} \, dx\\ &=-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {1}{2} (3 b d f n) \int \frac {\log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{2} \left (3 b^2 d f n^2\right ) \int \frac {\log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 d f n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+d f x)} \, dx,x,x^2\right )+\frac {1}{4} \left (3 b^3 d f n^3\right ) \int \frac {\log \left (1+\frac {1}{d f x^2}\right )}{x} \, dx\\ &=-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {3}{8} b^3 d f n^3 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b d f n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )-\frac {1}{2} \left (3 b^2 d f n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )}{x} \, dx+\frac {1}{8} \left (3 b^3 d f n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{4} \left (3 b^3 d f n^3\right ) \int \frac {\text {Li}_2\left (-\frac {1}{d f x^2}\right )}{x} \, dx-\frac {1}{8} \left (3 b^3 d^2 f^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+d f x} \, dx,x,x^2\right )\\ &=\frac {3}{4} b^3 d f n^3 \log (x)-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 d f n^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {3}{8} b^3 d f n^3 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b d f n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {Li}_3\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {1}{d f x^2}\right )-\frac {1}{4} \left (3 b^3 d f n^3\right ) \int \frac {\text {Li}_3\left (-\frac {1}{d f x^2}\right )}{x} \, dx\\ &=\frac {3}{4} b^3 d f n^3 \log (x)-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 d f n^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {3}{8} b^3 d f n^3 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b d f n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {Li}_3\left (-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \text {Li}_4\left (-\frac {1}{d f x^2}\right )\\ \end {align*}

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Mathematica [C]  time = 0.39, size = 940, normalized size = 2.21 \[ \frac {1}{8} \left (2 b^3 d f \left (\log ^4(x)-2 \log \left (1-i \sqrt {d} \sqrt {f} x\right ) \log ^3(x)-2 \log \left (i \sqrt {d} \sqrt {f} x+1\right ) \log ^3(x)-6 \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)-6 \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)+12 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right ) \log (x)+12 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right ) \log (x)-12 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )-12 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right ) n^3+12 b^2 d f \left (2 a+b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right ) \left (\frac {\log ^3(x)}{3}-\frac {1}{2} \log \left (1-i \sqrt {d} \sqrt {f} x\right ) \log ^2(x)-\frac {1}{2} \log \left (i \sqrt {d} \sqrt {f} x+1\right ) \log ^2(x)-\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right ) \log (x)-\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right ) \log (x)+\text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right ) n^2+6 b d f \left (2 a^2+2 b n a+4 b \left (\log \left (c x^n\right )-n \log (x)\right ) a+b^2 n^2+2 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (\log (x) \left (\log (x)-\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (i \sqrt {d} \sqrt {f} x+1\right )\right )-\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right ) n+2 d f \log (x) \left (4 a^3+6 b n a^2+12 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+12 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+12 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right ) a+3 b^3 n^3+4 b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3+6 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )-\frac {\left (4 a^3+6 b n a^2+6 b^2 n^2 a+3 b^3 n^3+4 b^3 \log ^3\left (c x^n\right )+6 b^2 (2 a+b n) \log ^2\left (c x^n\right )+6 b \left (2 a^2+2 b n a+b^2 n^2\right ) \log \left (c x^n\right )\right ) \log \left (d f x^2+1\right )}{x^2}-d f \left (4 a^3+6 b n a^2+12 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+12 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+12 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right ) a+3 b^3 n^3+4 b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3+6 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \log \left (d f x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^3,x]

[Out]

(2*d*f*Log[x]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*
(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*
b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - ((4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 +
3*b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^2*n^2)*Log[c*x^n] + 6*b^2*(2*a + b*n)*Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3)
*Log[1 + d*f*x^2])/x^2 - d*f*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]
) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log
[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3)*Log[1 + d*f*x^2] + 6*b
*d*f*n*(2*a^2 + 2*a*b*n + b^2*n^2 + 4*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 2*
b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(Log[x]*(Log[x] - Log[1 - I*Sqrt[d]*Sqrt[f]*x] - Log[1 + I*Sqrt[d]*Sqrt[f]*x
]) - PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]) + 12*b^2*d*f*n^2*(2*a + b*n - 2*b*n
*Log[x] + 2*b*Log[c*x^n])*(Log[x]^3/3 - (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x])/2 - (Log[x]^2*Log[1 + I*Sqrt[d
]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + PolyLog
[3, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + 2*b^3*d*f*n^3*(Log[x]^4 - 2*Log[x]^3*Log[1 -
I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]^3*Log[1 + I*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]
- 6*Log[x]^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + 12*Log[x]*PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] + 12*Log[x]*PolyL
og[3, I*Sqrt[d]*Sqrt[f]*x] - 12*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x] - 12*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x]))/8

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a^{3} \log \left (d f x^{2} + 1\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="fricas")

[Out]

integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)
*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + 1/d)*d)/x^3, x)

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)^3*ln((f*x^2+1/d)*d)/x^3,x)

[Out]

int((b*ln(c*x^n)+a)^3*ln((f*x^2+1/d)*d)/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (4 \, b^{3} \log \left (x^{n}\right )^{3} + 6 \, {\left (n^{2} + 2 \, n \log \relax (c) + 2 \, \log \relax (c)^{2}\right )} a b^{2} + {\left (3 \, n^{3} + 6 \, n^{2} \log \relax (c) + 6 \, n \log \relax (c)^{2} + 4 \, \log \relax (c)^{3}\right )} b^{3} + 6 \, a^{2} b {\left (n + 2 \, \log \relax (c)\right )} + 4 \, a^{3} + 6 \, {\left (b^{3} {\left (n + 2 \, \log \relax (c)\right )} + 2 \, a b^{2}\right )} \log \left (x^{n}\right )^{2} + 6 \, {\left ({\left (n^{2} + 2 \, n \log \relax (c) + 2 \, \log \relax (c)^{2}\right )} b^{3} + 2 \, a b^{2} {\left (n + 2 \, \log \relax (c)\right )} + 2 \, a^{2} b\right )} \log \left (x^{n}\right )\right )} \log \left (d f x^{2} + 1\right )}{8 \, x^{2}} + \int \frac {4 \, b^{3} d f \log \left (x^{n}\right )^{3} + 4 \, a^{3} d f + 6 \, {\left (d f n + 2 \, d f \log \relax (c)\right )} a^{2} b + 6 \, {\left (d f n^{2} + 2 \, d f n \log \relax (c) + 2 \, d f \log \relax (c)^{2}\right )} a b^{2} + {\left (3 \, d f n^{3} + 6 \, d f n^{2} \log \relax (c) + 6 \, d f n \log \relax (c)^{2} + 4 \, d f \log \relax (c)^{3}\right )} b^{3} + 6 \, {\left (2 \, a b^{2} d f + {\left (d f n + 2 \, d f \log \relax (c)\right )} b^{3}\right )} \log \left (x^{n}\right )^{2} + 6 \, {\left (2 \, a^{2} b d f + 2 \, {\left (d f n + 2 \, d f \log \relax (c)\right )} a b^{2} + {\left (d f n^{2} + 2 \, d f n \log \relax (c) + 2 \, d f \log \relax (c)^{2}\right )} b^{3}\right )} \log \left (x^{n}\right )}{4 \, {\left (d f x^{3} + x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="maxima")

[Out]

-1/8*(4*b^3*log(x^n)^3 + 6*(n^2 + 2*n*log(c) + 2*log(c)^2)*a*b^2 + (3*n^3 + 6*n^2*log(c) + 6*n*log(c)^2 + 4*lo
g(c)^3)*b^3 + 6*a^2*b*(n + 2*log(c)) + 4*a^3 + 6*(b^3*(n + 2*log(c)) + 2*a*b^2)*log(x^n)^2 + 6*((n^2 + 2*n*log
(c) + 2*log(c)^2)*b^3 + 2*a*b^2*(n + 2*log(c)) + 2*a^2*b)*log(x^n))*log(d*f*x^2 + 1)/x^2 + integrate(1/4*(4*b^
3*d*f*log(x^n)^3 + 4*a^3*d*f + 6*(d*f*n + 2*d*f*log(c))*a^2*b + 6*(d*f*n^2 + 2*d*f*n*log(c) + 2*d*f*log(c)^2)*
a*b^2 + (3*d*f*n^3 + 6*d*f*n^2*log(c) + 6*d*f*n*log(c)^2 + 4*d*f*log(c)^3)*b^3 + 6*(2*a*b^2*d*f + (d*f*n + 2*d
*f*log(c))*b^3)*log(x^n)^2 + 6*(2*a^2*b*d*f + 2*(d*f*n + 2*d*f*log(c))*a*b^2 + (d*f*n^2 + 2*d*f*n*log(c) + 2*d
*f*log(c)^2)*b^3)*log(x^n))/(d*f*x^3 + x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^3,x)

[Out]

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2))/x**3,x)

[Out]

Timed out

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