3.1.15 \(\int \frac {(a+b \log (c x^n))^2 \log (1+e x)}{x^2} \, dx\) [15]

Optimal. Leaf size=203 \[ 2 b^2 e n^2 \log (x)-2 b e n \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-e \log \left (1+\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 e n^2 \log (1+e x)-\frac {2 b^2 n^2 \log (1+e x)}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}+2 b^2 e n^2 \text {Li}_2\left (-\frac {1}{e x}\right )+2 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {1}{e x}\right )+2 b^2 e n^2 \text {Li}_3\left (-\frac {1}{e x}\right ) \]

[Out]

2*b^2*e*n^2*ln(x)-2*b*e*n*ln(1+1/e/x)*(a+b*ln(c*x^n))-e*ln(1+1/e/x)*(a+b*ln(c*x^n))^2-2*b^2*e*n^2*ln(e*x+1)-2*
b^2*n^2*ln(e*x+1)/x-2*b*n*(a+b*ln(c*x^n))*ln(e*x+1)/x-(a+b*ln(c*x^n))^2*ln(e*x+1)/x+2*b^2*e*n^2*polylog(2,-1/e
/x)+2*b*e*n*(a+b*ln(c*x^n))*polylog(2,-1/e/x)+2*b^2*e*n^2*polylog(3,-1/e/x)

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Rubi [A]
time = 0.18, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2342, 2341, 2425, 36, 29, 31, 2379, 2438, 2421, 6724} \begin {gather*} 2 b e n \text {PolyLog}\left (2,-\frac {1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+2 b^2 e n^2 \text {PolyLog}\left (2,-\frac {1}{e x}\right )+2 b^2 e n^2 \text {PolyLog}\left (3,-\frac {1}{e x}\right )-2 b e n \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-e \log \left (\frac {1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}+2 b^2 e n^2 \log (x)-2 b^2 e n^2 \log (e x+1)-\frac {2 b^2 n^2 \log (e x+1)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^2*Log[1 + e*x])/x^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.13, size = 183, normalized size = 0.90 \begin {gather*} \frac {1}{3} b^2 e n^2 \log ^3(x)-b e n \log ^2(x) \left (a+b n+b \log \left (c x^n\right )\right )+e \log (x) \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right )-\frac {(1+e x) \left (a^2+2 a b n+2 b^2 n^2+2 b (a+b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \log (1+e x)}{x}-2 b e n \left (a+b n+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+2 b^2 e n^2 \text {Li}_3(-e x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^2*Log[1 + e*x])/x^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.24, size = 3402, normalized size = 16.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(3402\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(e*x+1)/x*a^2-I*e*ln(e*x+1)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*c*x^n)^2+I*ln(e*x+1)*ln(x^n)*e*b^2*Pi*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)+I/x*ln(e*x+1)*ln(x^n)*b^2*Pi*csgn(I*c*x^n)^3-1/2*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)^2*csg
n(I*x^n)*csgn(I*c*x^n)^3-2*e*ln(e*x+1)*ln(c)*a*b-2*ln(e*x+1)/x*ln(c)*a*b-2*n/x*ln(e*x+1)*b^2*ln(c)+2*n*ln(x)*e
*b^2*ln(c)-2*n*e*ln(e*x+1)*b^2*ln(c)+n*e*ln(x)^2*b^2*ln(c)-2*n*e*polylog(2,-e*x)*b^2*ln(c)+2*e*ln(e*x)*ln(c)*a
*b+b^2*ln(e*x)*ln(x)^2*e*n^2-I*n*e*polylog(2,-e*x)*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e*ln(e*x+1)*Pi*a*b*csg
n(I*c)*csgn(I*c*x^n)^2-ln(e*x+1)/x*ln(c)^2*b^2+e*ln(e*x)*ln(c)^2*b^2-e*ln(e*x+1)*ln(c)^2*b^2+1/2*I*n*e*ln(x)^2
*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-2*b*ln(e*x)*ln(x)*e*n*a+I*ln(e*x+1)/x*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*x^n)*csg
n(I*c*x^n)+2*b*ln(e*x)*ln(x^n)*e*a+1/4*ln(e*x+1)/x*Pi^2*b^2*csgn(I*c*x^n)^6-1/4*e*ln(e*x)*Pi^2*b^2*csgn(I*c*x^
n)^6+1/4*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c*x^n)^6-2*b^2*n/x*ln(e*x+1)*ln(x^n)+I*ln(e*x+1)/x*Pi*a*b*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)-I*ln(e*x)*ln(x^n)*e*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*n*e*ln(x)^2*b^2*Pi*c
sgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*ln(e*x)*ln(x)*e*n*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e*ln(e*x+1)*Pi*a*b
*csgn(I*x^n)*csgn(I*c*x^n)^2+e*a^2*ln(e*x)-e*a^2*ln(e*x+1)+2*ln(e*x)*ln(x^n)*e*b^2*ln(c)-2*ln(e*x+1)*ln(x^n)*e
*b^2*ln(c)-2/x*ln(e*x+1)*ln(x^n)*b^2*ln(c)+2*b^2*ln(x)*ln(x^n)*e*n-2*b^2*ln(e*x+1)*ln(x^n)*e*n+b^2*ln(x)^2*ln(
x^n)*e*n-2*b^2*ln(x^n)*polylog(2,-e*x)*e*n-I*ln(e*x+1)*ln(x^n)*e*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*n*e*ln(e*x
+1)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*n*e*ln(x)^2*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*b*n/x*ln
(e*x+1)*a+2*b*n*ln(x)*e*a-2*b*n*e*ln(e*x+1)*a+b*n*e*ln(x)^2*a-2*b*n*e*polylog(2,-e*x)*a-b^2*n^2*e*ln(x)^2-2*b^
2*n^2*e*polylog(2,-e*x)-2/3*b^2*n^2*e*ln(x)^3+2*b^2*n^2*e*polylog(3,-e*x)-I*n*e*ln(e*x+1)*b^2*Pi*csgn(I*c)*csg
n(I*c*x^n)^2-I*n*e*polylog(2,-e*x)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*ln(e*x+1)/x*Pi*ln(c)*b^2*csgn(I*x^n)*csg
n(I*c*x^n)^2-I*e*ln(e*x+1)*Pi*ln(c)*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I*e*ln(e*x)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*
c*x^n)^2-2*b*ln(e*x+1)*ln(x^n)*e*a-2*b/x*ln(e*x+1)*ln(x^n)*a+I*ln(e*x)*ln(x)*e*n*b^2*Pi*csgn(I*c*x^n)^3+I*ln(e
*x)*ln(x^n)*e*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-b^2*ln(e*x+1)*ln(x^n)^2*e-I/x*ln(e*x+1)*ln(x^n)*b^2*Pi*csgn(I
*c)*csgn(I*c*x^n)^2-I*ln(e*x+1)/x*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I/x*ln(e*x+1)*ln(x^n)*b^2*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2-I*n/x*ln(e*x+1)*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*ln(e*x+1)/x*Pi^2*b^2*csgn(I*c)*csgn(I*c*x
^n)^5+I*e*ln(e*x+1)*Pi*a*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n*ln(x)*e*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*
n*ln(x)*e*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-b^2*ln(e*x+1)/x*ln(x^n)^2+b^2*ln(e*x)*ln(x^n)^2*e-I*n*e*ln(e*x+1)
*b^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*ln(e*x+1)/x*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2-I*ln(e*x+1)*ln(x^n)*e*b^2*P
i*csgn(I*x^n)*csgn(I*c*x^n)^2+I*e*ln(e*x)*Pi*a*b*csgn(I*c)*csgn(I*c*x^n)^2+I*e*ln(e*x)*Pi*ln(c)*b^2*csgn(I*x^n
)*csgn(I*c*x^n)^2-I*ln(e*x+1)/x*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*c*x^n)^2-I*n/x*ln(e*x+1)*b^2*Pi*csgn(I*x^n)*csgn
(I*c*x^n)^2+I*e*ln(e*x)*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+I*ln(e*x)*ln(x^n)*e*b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^
2+I*e*ln(e*x+1)*Pi*a*b*csgn(I*c*x^n)^3+I*ln(e*x+1)*ln(x^n)*e*b^2*Pi*csgn(I*c*x^n)^3-I*e*ln(e*x)*Pi*a*b*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)-I*e*ln(e*x)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*ln(e*x)*ln(x)*e*n*
b^2*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*b^2*e*n^2*ln(x)-2*b^2*e*n^2*ln(e*x+1)-2*b^2*n^2*ln(e*x+1)/x-e*ln(e*x)*Pi^2*
b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-1/2*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)*csgn(I*c*x^n)^5+1/4*e*ln(e*x+1)*P
i^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-1/2*e*ln(e*x+1)*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5+I*ln(e*x)*ln(x)*e*n
*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n/x*ln(e*x+1)*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*n/x*l
n(e*x+1)*b^2*Pi*csgn(I*c*x^n)^3+I*e*ln(e*x+1)*Pi*ln(c)*b^2*csgn(I*c*x^n)^3+I/x*ln(e*x+1)*ln(x^n)*b^2*Pi*csgn(I
*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-1/2*ln(e*x+1)/x
*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-1/4*e*ln(e*x)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^
n)^2-I*n*ln(x)*e*b^2*Pi*csgn(I*c*x^n)^3+1/4*ln(e*x+1)/x*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+1/2
*e*ln(e*x)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+I*n*e*polylog(2,-e*x)*b^2*Pi*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)+e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4-2*ln(e*x)*ln(x)*e*n*b^2*ln(c)+I*n*e*
ln(e*x+1)*b^2*Pi*csgn(I*c*x^n)^3+I*n*e*polylog(2,-e*x)*b^2*Pi*csgn(I*c*x^n)^3+1/2*e*ln(e*x)*Pi^2*b^2*csgn(I*c)
^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/4*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-1/2*ln(e*x+1
)/x*Pi^2*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+1/4*ln(e*x+1)/x*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/
4*e*ln(e*x+1)*Pi^2*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-I*n*ln(x)*e*b^2*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*e*
ln(e*x+1)*Pi*ln(c)*b^2*csgn(I*c)*csgn(I*x^n)*cs...

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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(e*x+1)/x^2,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2} \log {\left (e x + 1 \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2*ln(e*x+1)/x**2,x)

[Out]

Integral((a + b*log(c*x**n))**2*log(e*x + 1)/x**2, x)

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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(e*x+1)/x^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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