3.1.86 \(\int \frac {\log ^2(c (a+b x)^n)}{x^4} \, dx\) [86]

Optimal. Leaf size=177 \[ -\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {2 b^3 n \log \left (c (a+b x)^n\right ) \log \left (1-\frac {a}{a+b x}\right )}{3 a^3}-\frac {2 b^3 n^2 \text {Li}_2\left (\frac {a}{a+b x}\right )}{3 a^3} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\frac {2 b^3 n^2 \text {PolyLog}\left (2,\frac {a}{a+b x}\right )}{3 a^3}+\frac {2 b^3 n \log \left (1-\frac {a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {b^2 n^2}{3 a^2 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^n]^2/x^4,x]

[Out]

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

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*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 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 186, normalized size = 1.05 \begin {gather*} -\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n \log \left (c (a+b x)^n\right )}{3 a^2 x}+\frac {2 b^3 n \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {2 b^3 n^2 \text {Li}_2\left (\frac {a+b x}{a}\right )}{3 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^n]^2/x^4,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.28, size = 1102, normalized size = 6.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(1102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^n)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*I*b^2*n/a^2/x*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)+2/3*b^2*n*ln((b*x+a)^n)/a^2/x-2/3*b^3*n*
ln((b*x+a)^n)/a^3*ln(b*x+a)-2/3*b^3*n^2/a^3*ln(x)*ln(1/a*(b*x+a))-1/3/x^3*ln((b*x+a)^n)^2-1/6*I*b*n/a/x^2*Pi*c
sgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)-1/6*I*b*n/a/x^
2*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)+1/3*I*b^2*n/a^2/x*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^2*n/
a^2/x*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)-1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)-
2/3*b^3*n^2/a^3*dilog(1/a*(b*x+a))+1/3*b^3*n^2/a^3*ln(b*x+a)^2+1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*c*(b*x+a)^n)^2*
csgn(I*c)-1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)-1/3*I/x^3*ln((b*x+a)^n)*Pi*csgn
(I*c*(b*x+a)^n)^2*csgn(I*c)+1/3*I*b^3*n/a^3*ln(b*x+a)*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)-2/3/x
^3*ln((b*x+a)^n)*ln(c)-1/3*b*n/a/x^2*ln(c)+1/6*I*b*n/a/x^2*Pi*csgn(I*c*(b*x+a)^n)^3+1/3*I*b^3*n/a^3*ln(b*x+a)*
Pi*csgn(I*c*(b*x+a)^n)^3+1/6*I*b*n/a/x^2*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)-1/3*b*n*ln((b*x+a)
^n)/a/x^2+2/3*b^3*n*ln((b*x+a)^n)/a^3*ln(x)+1/3*I/x^3*ln((b*x+a)^n)*Pi*csgn(I*c*(b*x+a)^n)^3-1/3*I*b^2*n/a^2/x
*Pi*csgn(I*c*(b*x+a)^n)^3+2/3*b^2*n/a^2/x*ln(c)+2/3*b^3*n/a^3*ln(x)*ln(c)-2/3*b^3*n/a^3*ln(b*x+a)*ln(c)+1/3*I/
x^3*ln((b*x+a)^n)*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)-1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*c*(b*x+a)
^n)^3-1/12*(-I*Pi*csgn(I*c*(b*x+a)^n)^3+I*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)+I*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I
*(b*x+a)^n)-I*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)+2*ln(c))^2/x^3-1/3*I/x^3*ln((b*x+a)^n)*Pi*csg
n(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)-1/3*I*b^3*n/a^3*ln(x)*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*c)*csgn(I*(b*x+a)^n)-
1/3*b^2*n^2/a^2/x-b^3*n^2*ln(x)/a^3+b^3*n^2*ln(b*x+a)/a^3

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Maxima [A]
time = 0.27, size = 150, normalized size = 0.85 \begin {gather*} -\frac {1}{3} \, b^{2} n^{2} {\left (\frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x + a\right )}{a^{3}} - \frac {b x \log \left (b x + a\right )^{2} - 3 \, b x \log \left (x\right ) - a}{a^{3} x}\right )} - \frac {1}{3} \, b n {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) - \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^2/x^4,x, algorithm="maxima")

[Out]

-1/3*b^2*n^2*(2*(log(b*x/a + 1)*log(x) + dilog(-b*x/a))*b/a^3 - 3*b*log(b*x + a)/a^3 - (b*x*log(b*x + a)^2 - 3
*b*x*log(x) - a)/(a^3*x)) - 1/3*b*n*(2*b^2*log(b*x + a)/a^3 - 2*b^2*log(x)/a^3 - (2*b*x - a)/(a^2*x^2))*log((b
*x + a)^n*c) - 1/3*log((b*x + a)^n*c)^2/x^3

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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(log(c*(b*x+a)^n)^2/x^4,x, algorithm="fricas")

[Out]

integral(log((b*x + a)^n*c)^2/x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x+a)**n)**2/x**4,x)

[Out]

Integral(log(c*(a + b*x)**n)**2/x**4, 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(log(c*(b*x+a)^n)^2/x^4,x, algorithm="giac")

[Out]

integrate(log((b*x + a)^n*c)^2/x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x)^n)^2/x^4,x)

[Out]

int(log(c*(a + b*x)^n)^2/x^4, x)

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