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

Optimal result

Integrand size = 16, antiderivative size = 177 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {2 b^3 n \log \left (c (a+b x)^n\right ) \log \left (1-\frac {a}{a+b x}\right )}{3 a^3}-\frac {2 b^3 n^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x}\right )}{3 a^3} \]

[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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\frac {2 b^3 n \log \left (1-\frac {a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {2 b^3 n^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x}\right )}{3 a^3}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {b^2 n^2}{3 a^2 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]

[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*} \text {integral}& = -\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 {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 {\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 {\left (2 b^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {a}{x}\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 {\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {a b^2 n^2 x^2+3 b^3 n^2 x^3 \log (x)-3 b^3 n^2 x^3 \log (a+b x)+a^2 b n x \log \left (c (a+b x)^n\right )-2 a b^2 n x^2 \log \left (c (a+b x)^n\right )-2 b^3 n x^3 \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )+a^3 \log ^2\left (c (a+b x)^n\right )+b^3 x^3 \log ^2\left (c (a+b x)^n\right )-2 b^3 n^2 x^3 \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{3 a^3 x^3} \]

[In]

Integrate[Log[c*(a + b*x)^n]^2/x^4,x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.38 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.73

method	result	size
risch	\(-\frac {\ln \left (\left (b x +a \right )^{n}\right )^{2}}{3 x^{3}}-\frac {2 b^{3} n \ln \left (\left (b x +a \right )^{n}\right ) \ln \left (b x +a \right )}{3 a^{3}}-\frac {b n \ln \left (\left (b x +a \right )^{n}\right )}{3 a \,x^{2}}+\frac {2 b^{3} n \ln \left (\left (b x +a \right )^{n}\right ) \ln \left (x \right )}{3 a^{3}}+\frac {2 b^{2} n \ln \left (\left (b x +a \right )^{n}\right )}{3 a^{2} x}+\frac {b^{3} n^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {b^{2} n^{2}}{3 a^{2} x}-\frac {b^{3} n^{2} \ln \left (x \right )}{a^{3}}-\frac {2 b^{3} n^{2} \operatorname {dilog}\left (\frac {b x +a}{a}\right )}{3 a^{3}}-\frac {2 b^{3} n^{2} \ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{3 a^{3}}+\frac {b^{3} n^{2} \ln \left (b x +a \right )^{2}}{3 a^{3}}+\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (b x +a \right )^{n}\right )}{3 x^{3}}+\frac {b n \left (-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {1}{2 a \,x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}+\frac {b}{a^{2} x}\right )}{3}\right )-\frac {{\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{12 x^{3}}\)	\(483\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}} \,d x } \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {1}{3} \, b^{2} n^{2} {\left (\frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x + a\right )}{a^{3}} - \frac {b x \log \left (b x + a\right )^{2} - 3 \, b x \log \left (x\right ) - a}{a^{3} x}\right )} - \frac {1}{3} \, b n {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) - \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{3 \, x^{3}} \]

[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

Giac [F]

\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{x^4} \,d x \]

[In]

int(log(c*(a + b*x)^n)^2/x^4,x)

[Out]

int(log(c*(a + b*x)^n)^2/x^4, x)