∫ (a+b log(cxn))3 log(1+ex)_________________

3.1.21 \(\int \frac {(a+b \log (c x^n))^3 \log (1+e x)}{x} \, dx\) [21]

Optimal. Leaf size=81 \[ -\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)+6 b^3 n^3 \text {Li}_5(-e x) \]

[Out]

-(a+b*ln(c*x^n))^3*polylog(2,-e*x)+3*b*n*(a+b*ln(c*x^n))^2*polylog(3,-e*x)-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(4

,-e*x)+6*b^3*n^3*polylog(5,-e*x)

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Rubi [A]
time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2421, 2430, 6724} \begin {gather*} -6 b^2 n^2 \text {PolyLog}(4,-e x) \left (a+b \log \left (c x^n\right )\right )+3 b n \text {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )^2-\text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^3 \text {PolyLog}(5,-e x) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((a + b*Log[c*x^n])^3*PolyLog[2, -(e*x)]) + 3*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(e*x)] - 6*b^2*n^2*(a + b*

Log[c*x^n])*PolyLog[4, -(e*x)] + 6*b^3*n^3*PolyLog[5, -(e*x)]

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo

g[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 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 )^3 \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)+\left (6 b^3 n^3\right ) \int \frac {\text {Li}_4(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)+6 b^3 n^3 \text {Li}_5(-e x)\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 77, normalized size = 0.95 \begin {gather*} -\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)+2 b n \left (-\left (\left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)\right )+b n \text {Li}_5(-e x)\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Log[c*x^n])*PolyLog[4, -(e*x)]) + b*n*PolyLog[5, -(e*x)]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 4058, normalized size = 50.10


method	result	size

risch	\(\text {Expression too large to display}\)	\(4058\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c

*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^3*dilog(e*x+1)-3/2*I*Pi*ln(x^n)^2*dilog(e*x+1)*b^3*csgn(I*c)*csg

n(I*c*x^n)^2+6*ln(c)*ln(x)*ln(x^n)*dilog(e*x+1)*b^3*n+3/2*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*x^n)*csgn(I*c*x

^n)^5-3/4*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+3/2*Pi^2*polylog(3,-e*x)*b^3*n*

csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-3*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+3*P

i^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+3/4*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c*

x^n)^6-3/4*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*c*x^n)^6+3/2*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*c)^2*csgn(I*x^n

)*csgn(I*c*x^n)^3-3/2*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3+3*I*ln(c)*Pi*po

lylog(3,-e*x)*b^3*n*csgn(I*c)*csgn(I*c*x^n)^2+3/4*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)^2*csgn(I*c*x^n)^4-3/

2*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)*csgn(I*c*x^n)^5-6*ln(c)*ln(x^n)*dilog(e*x+1)*a*b^2+6*ln(c)*polylog(3

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

*x+1)*b^3+3*polylog(3,-e*x)*a^2*b*n+3*ln(c)^2*polylog(3,-e*x)*b^3*n-6*ln(c)*polylog(4,-e*x)*b^3*n^2+3*I*ln(x)*

Pi*ln(x^n)*dilog(e*x+1)*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*Pi*polylog(3,-e*x)*ln(x^n)*b^3*n*csgn(I*c)*csgn(

I*c*x^n)^2+3*I*Pi*polylog(4,-e*x)*b^3*n^2*csgn(I*c*x^n)^3-3/4*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*c)^2*csgn(I*c*

x^n)^4+3*I*ln(c)*Pi*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3*I*ln(x)*Pi*polylog(2,-e*x)*

a*b^2*n*csgn(I*c)*csgn(I*c*x^n)^2+3/4*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-

3*I*ln(x)*Pi*dilog(e*x+1)*a*b^2*n*csgn(I*c*x^n)^3-6*ln(x)*polylog(2,-e*x)*ln(x^n)*a*b^2*n+6*ln(x)*ln(x^n)*dilo

g(e*x+1)*a*b^2*n-6*ln(c)*ln(x)*polylog(2,-e*x)*ln(x^n)*b^3*n+3/2*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*c)*csgn(

I*x^n)^2*csgn(I*c*x^n)^3-3/2*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-3*I*ln(x

)*Pi*polylog(2,-e*x)*ln(x^n)*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-3*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*c)*csgn(

I*x^n)*csgn(I*c*x^n)^4-3/4*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+3/4*ln(x)*P

i^2*polylog(2,-e*x)*b^3*n*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-3/2*I*ln(x)^2*Pi*dilog(e*x+1)*b^3*n^2*csgn

(I*x^n)*csgn(I*c*x^n)^2-3*ln(x)*polylog(2,-e*x)*ln(x^n)^2*b^3*n+3*ln(x)*ln(x^n)^2*dilog(e*x+1)*b^3*n+3*ln(x)^2

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

)*dilog(e*x+1)*a^2*b*n-3*ln(c)^2*ln(x)*polylog(2,-e*x)*b^3*n+3*ln(c)^2*ln(x)*dilog(e*x+1)*b^3*n+6*ln(c)*polylo

g(3,-e*x)*a*b^2*n+3*ln(c)*ln(x)^2*polylog(2,-e*x)*b^3*n^2-3*ln(c)*ln(x)^2*dilog(e*x+1)*b^3*n^2+3*I*Pi*polylog(

3,-e*x)*a*b^2*n*csgn(I*c)*csgn(I*c*x^n)^2+3*I*ln(x)*Pi*polylog(2,-e*x)*ln(x^n)*b^3*n*csgn(I*c*x^n)^3+6*polylog

(3,-e*x)*ln(x^n)*a*b^2*n+3/2*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5-3*I*ln(c)*Pi*ln(x^n)*dilog

(e*x+1)*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*ln(x)*Pi*polylog(2,-e*x)*a*b^2*n*csgn(I*c*x^n)^3+3*I*Pi*polylog(4,

-e*x)*b^3*n^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3*I*Pi*polylog(4,-e*x)*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+3

*I*Pi*polylog(3,-e*x)*ln(x^n)*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2+3*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*csgn(I*c)*c

sgn(I*x^n)*csgn(I*c*x^n)^4+3*I*Pi*polylog(3,-e*x)*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+3*I*ln(c)*ln(x)*Pi*polyl

og(2,-e*x)*b^3*n*csgn(I*c*x^n)^3+3/2*I*Pi*ln(x^n)^2*dilog(e*x+1)*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3*I*l

n(c)*ln(x)*Pi*dilog(e*x+1)*b^3*n*csgn(I*c*x^n)^3-3/2*I*ln(x)^2*Pi*dilog(e*x+1)*b^3*n^2*csgn(I*c)*csgn(I*c*x^n)

^2-3*I*Pi*ln(x^n)*dilog(e*x+1)*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-3/2*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)^2

*csgn(I*x^n)*csgn(I*c*x^n)^3+3/4*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4-6*ln(c)*ln(x)*pol

ylog(2,-e*x)*a*b^2*n+6*ln(c)*ln(x)*dilog(e*x+1)*a*b^2*n-ln(x^n)^3*dilog(e*x+1)*b^3+3*I*ln(x)*Pi*ln(x^n)*dilog(

e*x+1)*b^3*n*csgn(I*c)*csgn(I*c*x^n)^2-3*ln(x^n)^2*dilog(e*x+1)*a*b^2+3*polylog(3,-e*x)*ln(x^n)^2*b^3*n-6*poly

log(4,-e*x)*ln(x^n)*b^3*n^2-3*I*ln(c)*Pi*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*c)*csgn(I*c*x^n)^2-6*polylog(4,-e*x)*

a*b^2*n^2-3/2*Pi^2*ln(x^n)*dilog(e*x+1)*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+3/4*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*c

sgn(I*c*x^n)^6-3/4*ln(x)*Pi^2*dilog(e*x+1)*b^3*n*csgn(I*c*x^n)^6+3/2*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*c)*csgn

(I*c*x^n)^5-3/4*Pi^2*polylog(3,-e*x)*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4+3/2*I*Pi*ln(x^n)^2*dilog(e*x+1)*b^3*c

sgn(I*c*x^n)^3-3*I*Pi*ln(x^n)*dilog(e*x+1)*a*b^2*csgn(I*c)*csgn(I*c*x^n)^2+3*I*ln(c)*Pi*polylog(3,-e*x)*b^3*n*

csgn(I*x^n)*csgn(I*c*x^n)^2-3/2*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*csgn(I*c)*csgn(I*c*x^n)^5+3/4*ln(x)*Pi^2*poly

log(2,-e*x)*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4-3/2*ln(x)*Pi^2*polylog(2,-e*x)*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)

^5+3*ln(x)^2*polylog(2,-e*x)*a*b^2*n^2-3*ln(x)^...

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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))^3*log(e*x+1)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral((b^3*log(c*x^n)^3*log(x*e + 1) + 3*a*b^2*log(c*x^n)^2*log(x*e + 1) + 3*a^2*b*log(c*x^n)*log(x*e + 1)

+ a^3*log(x*e + 1))/x, 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))**3*ln(e*x+1)/x,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))^3*log(e*x+1)/x,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,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))^3)/x,x)

[Out]

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

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