Integrand size = 16, antiderivative size = 177 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {2 b^3 n \log \left (c (a+b x)^n\right ) \log \left (1-\frac {a}{a+b x}\right )}{3 a^3}-\frac {2 b^3 n^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x}\right )}{3 a^3} \]
-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*polyl og(2,a/(b*x+a))/a^3
Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {a b^2 n^2 x^2+3 b^3 n^2 x^3 \log (x)-3 b^3 n^2 x^3 \log (a+b x)+a^2 b n x \log \left (c (a+b x)^n\right )-2 a b^2 n x^2 \log \left (c (a+b x)^n\right )-2 b^3 n x^3 \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )+a^3 \log ^2\left (c (a+b x)^n\right )+b^3 x^3 \log ^2\left (c (a+b x)^n\right )-2 b^3 n^2 x^3 \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{3 a^3 x^3} \]
-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)
Time = 0.76 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {2845, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {2}{3} b n \int \frac {\log \left (c (a+b x)^n\right )}{x^3 (a+b x)}dx-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {2}{3} n \int \frac {\log \left (c (a+b x)^n\right )}{x^3 (a+b x)}d(a+b x)-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{3} n \int -\frac {\log \left (c (a+b x)^n\right )}{x^3 (a+b x)}d(a+b x)-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} b^3 n \int -\frac {\log \left (c (a+b x)^n\right )}{b^3 x^3 (a+b x)}d(a+b x)-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\int -\frac {\log \left (c (a+b x)^n\right )}{b^3 x^3}d(a+b x)}{a}+\frac {\int \frac {\log \left (c (a+b x)^n\right )}{b^2 x^2 (a+b x)}d(a+b x)}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \int \frac {1}{b^2 x^2 (a+b x)}d(a+b x)}{a}+\frac {\int \frac {\log \left (c (a+b x)^n\right )}{b^2 x^2 (a+b x)}d(a+b x)}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \int \left (\frac {1}{b^2 x^2 a}-\frac {1}{b x a^2}+\frac {1}{(a+b x) a^2}\right )d(a+b x)}{a}+\frac {\int \frac {\log \left (c (a+b x)^n\right )}{b^2 x^2 (a+b x)}d(a+b x)}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\int \frac {\log \left (c (a+b x)^n\right )}{b^2 x^2 (a+b x)}d(a+b x)}{a}+\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\int \frac {\log \left (c (a+b x)^n\right )}{b^2 x^2}d(a+b x)}{a}+\frac {\int -\frac {\log \left (c (a+b x)^n\right )}{b x (a+b x)}d(a+b x)}{a}}{a}+\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {-\frac {n \int -\frac {1}{b x}d(a+b x)}{a}-\frac {(a+b x) \log \left (c (a+b x)^n\right )}{a b x}}{a}+\frac {\int -\frac {\log \left (c (a+b x)^n\right )}{b x (a+b x)}d(a+b x)}{a}}{a}+\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\int -\frac {\log \left (c (a+b x)^n\right )}{b x (a+b x)}d(a+b x)}{a}+\frac {\frac {n \log (-b x)}{a}-\frac {(a+b x) \log \left (c (a+b x)^n\right )}{a b x}}{a}}{a}+\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\frac {n \int \frac {\log \left (1-\frac {a}{a+b x}\right )}{a+b x}d(a+b x)}{a}-\frac {\log \left (1-\frac {a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{a}}{a}+\frac {\frac {n \log (-b x)}{a}-\frac {(a+b x) \log \left (c (a+b x)^n\right )}{a b x}}{a}}{a}+\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2}{3} b^3 n \left (\frac {\frac {\log \left (c (a+b x)^n\right )}{2 b^2 x^2}-\frac {1}{2} n \left (-\frac {\log (-b x)}{a^2}+\frac {\log (a+b x)}{a^2}-\frac {1}{a b x}\right )}{a}+\frac {\frac {\frac {n \operatorname {PolyLog}\left (2,\frac {a}{a+b x}\right )}{a}-\frac {\log \left (1-\frac {a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{a}}{a}+\frac {\frac {n \log (-b x)}{a}-\frac {(a+b x) \log \left (c (a+b x)^n\right )}{a b x}}{a}}{a}\right )-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}\) |
-1/3*Log[c*(a + b*x)^n]^2/x^3 - (2*b^3*n*((-1/2*(n*(-(1/(a*b*x)) - Log[-(b *x)]/a^2 + Log[a + b*x]/a^2)) + Log[c*(a + b*x)^n]/(2*b^2*x^2))/a + (((n*L og[-(b*x)])/a - ((a + b*x)*Log[c*(a + b*x)^n])/(a*b*x))/a + (-((Log[c*(a + b*x)^n]*Log[1 - a/(a + b*x)])/a) + (n*PolyLog[2, a/(a + b*x)])/a)/a)/a))/ 3
3.1.86.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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] - Simp[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]
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] - Simp[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] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
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] + Simp[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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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]
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] - Simp[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] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(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]
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}}\) | \(483\) |
-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
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{x^{4}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=-\frac {1}{3} \, b^{2} n^{2} {\left (\frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x + a\right )}{a^{3}} - \frac {b x \log \left (b x + a\right )^{2} - 3 \, b x \log \left (x\right ) - a}{a^{3} x}\right )} - \frac {1}{3} \, b n {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) - \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{3 \, x^{3}} \]
-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
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{x^4} \,d x \]