3.1.35 \(\int \frac {(a+b \log (c x^n))^2 \log (d (\frac {1}{d}+f x^2))}{x^3} \, dx\) [35]

Optimal. Leaf size=257 \[ \frac {1}{2} b^2 d f n^2 \log (x)-\frac {1}{2} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} b^2 d f n^2 \log \left (1+d f x^2\right )-\frac {b^2 n^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{2 x^2}+\frac {1}{4} b^2 d f n^2 \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{2} b d f n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{d f x^2}\right )+\frac {1}{4} b^2 d f n^2 \text {Li}_3\left (-\frac {1}{d f x^2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2342, 2341, 2425, 272, 36, 29, 31, 2379, 2438, 2421, 6724} \begin {gather*} \frac {1}{2} 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 )+\frac {1}{4} b^2 d f n^2 \text {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {1}{4} b^2 d f n^2 \text {PolyLog}\left (3,-\frac {1}{d f x^2}\right )-\frac {1}{2} b d f n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 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 )^2-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {1}{4} b^2 d f n^2 \log \left (d f x^2+1\right )-\frac {b^2 n^2 \log \left (d f x^2+1\right )}{4 x^2}+\frac {1}{2} b^2 d f n^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 272

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 2341

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 2342

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 2379

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2425

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 2438

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 6724

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 488, normalized size = 1.90 \begin {gather*} \frac {1}{4} \left (2 d f \log (x) \left (2 a^2+2 a b n+b^2 n^2+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-\frac {\left (2 a^2+2 a b n+b^2 n^2+2 b (2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x^2}-d f \left (2 a^2+2 a b n+b^2 n^2+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \log \left (1+d f x^2\right )-2 b d f n \left (-2 a-b n+2 b n \log (x)-2 b \log \left (c x^n\right )\right ) \left (\log (x) \left (\log (x)-\log \left (1-i \sqrt {d} \sqrt {f} x\right )-\log \left (1+i \sqrt {d} \sqrt {f} x\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 )+\frac {2}{3} b^2 d f n^2 \left (2 \log ^3(x)-3 \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-3 \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 3493, normalized size = 13.59

method	result	size
risch	\(\text {Expression too large to display}\)	\(3493\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d*f*ln(x)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+1/8*d*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)^2*csgn
(I*x^n)^2*csgn(I*c*x^n)^2-1/4*d*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/2*I*d*f*ln(
d*f*x^2+1)*Pi*ln(c)*b^2*csgn(I*c*x^n)^3+2*d*f*ln(x)*ln(c)*a*b+1/4*I*n*d*f*polylog(2,-d*f*x^2)*b^2*Pi*csgn(I*c*
x^n)^3-1/2*I/x^2*ln(d*f*x^2+1)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/4*I*n*d*f*ln(d*f*x^2+1)*b^2*Pi*csgn(
I*c*x^n)^3-1/2*I*n*d*f*ln(x)*b^2*Pi*csgn(I*c*x^n)^3+1/2*I/x^2*ln(d*f*x^2+1)*Pi*ln(c)*b^2*csgn(I*c*x^n)^3+1/2*I
/x^2*ln(d*f*x^2+1)*Pi*a*b*csgn(I*c*x^n)^3+1/2*I/x^2*ln(d*f*x^2+1)*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^3-1/2*b*n*d*f*p
olylog(2,-d*f*x^2)*a+1/2/x^2*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-1/2*I/x^2*ln(d*f*x^2
+1)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I/x^2*ln(d*f*x^2+1)*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+1/
2*I*d*f*ln(d*f*x^2+1)*Pi*a*b*csgn(I*c*x^n)^3-1/2*I/x^2*ln(d*f*x^2+1)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I/x^
2*ln(d*f*x^2+1)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*d*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I
*c*x^n)^4-1/4*d*f*ln(x)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-1/2*a^2/x^2*ln(d*f*x^2+1)-1/2*b^2*n
^2*d*f*ln(x)^2-1/4*b^2*n^2*d*f*polylog(2,-d*f*x^2)+1/3*b^2*n^2*d*f*ln(x)^3+1/4*b^2*n^2*d*f*polylog(3,-d*f*x^2)
+1/2*I/x^2*ln(d*f*x^2+1)*ln(x^n)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*ln(d*f*x^2+1)*ln(x^n)*d*f*b^
2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I/x^2*ln(d*f*x^2+1)*ln(x^n)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*ln(d*f
*x^2+1)*ln(x^n)*d*f*b^2*Pi*csgn(I*c*x^n)^3+1/2*I*n*d*f*ln(x)^2*b^2*Pi*csgn(I*c*x^n)^3-b*n*d*f*ln(x)^2*a-1/2*b*
n*d*f*ln(d*f*x^2+1)*a+b*n*d*f*ln(x)*a-1/2*b^2/x^2*ln(d*f*x^2+1)*ln(x^n)^2-1/2*a^2*d*f*ln(d*f*x^2+1)+a^2*d*f*ln
(x)-I*d*f*ln(x)*Pi*a*b*csgn(I*c*x^n)^3+d*f*ln(x)*ln(c)^2*b^2-1/x^2*ln(d*f*x^2+1)*ln(c)*a*b-1/2*n/x^2*ln(d*f*x^
2+1)*b^2*ln(c)-1/2*d*f*ln(d*f*x^2+1)*ln(c)^2*b^2-I*d*f*ln(x)*Pi*ln(c)*b^2*csgn(I*c*x^n)^3+1/2*d*f*ln(x)*Pi^2*b
^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-d*f*ln(x)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-1/4*d*f*ln
(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-I*ln(x)*ln(x^n)*d*f*b^2*Pi*csgn(I*c*x^n)^3-1/4*I*
n/x^2*ln(d*f*x^2+1)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*I*n/x^2*ln(d*f*x^2+1)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n
)^2-1/2/x^2*ln(d*f*x^2+1)*ln(c)^2*b^2-1/4*I*n*d*f*ln(d*f*x^2+1)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*n*d*f
*ln(d*f*x^2+1)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*d*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/8*d
*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-1/4*d*f*ln(x)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-1/2
*b*n/x^2*ln(d*f*x^2+1)*a-1/4*I*n*d*f*polylog(2,-d*f*x^2)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*b^2*n/x^2*ln(d
*f*x^2+1)*ln(x^n)-1/2*b^2*ln(d*f*x^2+1)*ln(x^n)^2*d*f+b^2*ln(x)*ln(x^n)^2*d*f-b^2*ln(x)^2*ln(x^n)*d*f*n-1/2*b^
2*ln(d*f*x^2+1)*ln(x^n)*d*f*n+b^2*ln(x)*ln(x^n)*d*f*n-1/2*b^2*ln(x^n)*polylog(2,-d*f*x^2)*d*f*n+1/8*d*f*ln(d*f
*x^2+1)*Pi^2*b^2*csgn(I*c*x^n)^6-1/4*d*f*ln(x)*Pi^2*b^2*csgn(I*c*x^n)^6+1/8/x^2*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*
x^n)^2*csgn(I*c*x^n)^4-1/2*I*d*f*ln(d*f*x^2+1)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+1/8/x^2*ln(d*f*x^2+1)*Pi^2*b^2
*csgn(I*c*x^n)^6-b/x^2*ln(d*f*x^2+1)*ln(x^n)*a-1/x^2*ln(d*f*x^2+1)*ln(x^n)*b^2*ln(c)-1/2*I*d*f*ln(d*f*x^2+1)*P
i*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*n*d*f*polylog(2,-d*f*x^2)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*b^2*n^2
*ln(d*f*x^2+1)/x^2+I*ln(x)*ln(x^n)*d*f*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*d*f*ln(x)*Pi^2*b^2*csgn(I*x^n)^2*c
sgn(I*c*x^n)^4+1/2*d*f*ln(x)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-ln(d*f*x^2+1)*ln(x^n)*d*f*b^2*ln(c)+2*ln(x)*
ln(x^n)*d*f*b^2*ln(c)+1/4*I*n*d*f*polylog(2,-d*f*x^2)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*ln(d*f*
x^2+1)*ln(x^n)*d*f*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*d*f*ln(x)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*c*x^n)
^2-1/2*I*ln(d*f*x^2+1)*ln(x^n)*d*f*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4/x^2*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*x^
n)*csgn(I*c*x^n)^5-b*ln(d*f*x^2+1)*ln(x^n)*d*f*a+2*b*ln(x)*ln(x^n)*d*f*a+1/2*I*n*d*f*ln(x)^2*b^2*Pi*csgn(I*c)*
csgn(I*x^n)*csgn(I*c*x^n)-I*d*f*ln(x)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*d*f*ln(d*f*x^2+1)*Pi*ln
(c)*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*n*d*f*ln(x)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*n/x^2
*ln(d*f*x^2+1)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*d*f*ln(d*f*x^2+1)*Pi*ln(c)*b^2*csgn(I*c)*csgn(
I*c*x^n)^2+1/8/x^2*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-1/4/x^2*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*c)
*csgn(I*c*x^n)^5+I*ln(x)*ln(x^n)*d*f*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/8*d*f*ln(d*f*x^2+1)*Pi^2*b^2*csgn(I*
c)^2*csgn(I*c*x^n)^4+I*d*f*ln(x)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*n*d*f*ln(x)*b^2*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2-n*d*f*ln(x)^2*b^2*ln(c)-1/2*n*d*f*ln(d*f*x^2+1)*b^2*ln(c)+n*d*f*ln(x)*b^2*ln(c)-1/2*n*d*f*polylog(2,-
d*f*x^2)*b^2*ln(c)-d*f*ln(d*f*x^2+1)*ln(c)*a*b+I*d*f*ln(x)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*d*f*ln(d*f*x
^2+1)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+1/2*...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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))^2)/x^3,x)

[Out]

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

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