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

Optimal. Leaf size=203 \[ 2 b e n \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+2 b^2 e n^2 \text{PolyLog}\left (2,-\frac{1}{e x}\right )+2 b^2 e n^2 \text{PolyLog}\left (3,-\frac{1}{e x}\right )-2 b e n \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-e \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}+2 b^2 e n^2 \log (x)-2 b^2 e n^2 \log (e x+1)-\frac{2 b^2 n^2 \log (e x+1)}{x} \]

[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))]

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Rubi [A]  time = 0.342843, antiderivative size = 220, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {2305, 2304, 2378, 36, 29, 31, 2344, 2301, 2317, 2391, 2302, 30, 2374, 6589} \[ -2 b e n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )-2 b^2 e n^2 \text{PolyLog}(2,-e x)+2 b^2 e n^2 \text{PolyLog}(3,-e x)+\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}+e \left (a+b \log \left (c x^n\right )\right )^2-e \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-2 b e n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}+2 b^2 e n^2 \log (x)-2 b^2 e n^2 \log (e x+1)-\frac{2 b^2 n^2 \log (e x+1)}{x} \]

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

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 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 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 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 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 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[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 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[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 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 30

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

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 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 )^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 \operatorname{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*}

Mathematica [A]  time = 0.206215, size = 183, normalized size = 0.9 \[ -2 b e n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )+b n\right )+2 b^2 e n^2 \text{PolyLog}(3,-e x)+e \log (x) \left (a^2+2 b (a+b n) \log \left (c x^n\right )+2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )-\frac{(e x+1) \log (e x+1) \left (a^2+2 b (a+b n) \log \left (c x^n\right )+2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )}{x}-b e n \log ^2(x) \left (a+b \log \left (c x^n\right )+b n\right )+\frac{1}{3} b^2 e n^2 \log ^3(x) \]

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 [F]  time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( ex+1 \right ) }{{x}^{2}}}\, dx \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{2} e x \log \left (x\right ) -{\left (b^{2} e x + b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{x} + \int \frac{{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (e x + 1\right ) - 2 \,{\left (b^{2} e n x \log \left (x\right ) -{\left (b^{2} e n x + b^{2}{\left (n + \log \left (c\right )\right )} + a b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{x^{2}}\,{d x} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 2 \, a b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{2} \log \left (e x + 1\right )}{x^{2}}, x\right ) \end{align*}

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

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

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]

Timed out

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

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(e*x + 1)/x^2, x)