Integrand size = 18, antiderivative size = 129 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {b^2 p^2 \log (x)}{a^2}-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {b^2 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^2}+\frac {b^2 p^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{2 a^2} \]
b^2*p^2*ln(x)/a^2-1/2*b*p*(b*x^2+a)*ln(c*(b*x^2+a)^p)/a^2/x^2-1/4*ln(c*(b* x^2+a)^p)^2/x^4-1/2*b^2*p*ln(c*(b*x^2+a)^p)*ln(1-a/(b*x^2+a))/a^2+1/2*b^2* p^2*polylog(2,a/(b*x^2+a))/a^2
Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.09 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {-\log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {b x^2 \left (4 b p^2 x^2 \log (x)-2 b p^2 x^2 \log \left (a+b x^2\right )-2 a p \log \left (c \left (a+b x^2\right )^p\right )+b x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )-2 b p x^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )\right )\right )}{a^2}}{4 x^4} \]
(-Log[c*(a + b*x^2)^p]^2 + (b*x^2*(4*b*p^2*x^2*Log[x] - 2*b*p^2*x^2*Log[a + b*x^2] - 2*a*p*Log[c*(a + b*x^2)^p] + b*x^2*Log[c*(a + b*x^2)^p]^2 - 2*b *p*x^2*(Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p] + p*PolyLog[2, 1 + (b*x^2)/ a])))/a^2)/(4*x^4)
Time = 0.55 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2904, 2845, 2858, 27, 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.
| \(\displaystyle \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {1}{2} \int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{x^6}dx^2\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {1}{2} \left (b p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{x^4 \left (b x^2+a\right )}dx^2-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {1}{2} \left (p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{x^6}d\left (b x^2+a\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \left (\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \left (\frac {-\frac {p \int -\frac {1}{b x^2}d\left (b x^2+a\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \left (\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \left (\frac {\frac {p \int \frac {\log \left (1-\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {1}{2} \left (b^2 p \left (\frac {\frac {p \operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^4}\right )\) |
(-1/2*Log[c*(a + b*x^2)^p]^2/x^4 + b^2*p*(((p*Log[-(b*x^2)])/a - ((a + b*x ^2)*Log[c*(a + b*x^2)^p])/(a*b*x^2))/a + (-((Log[1 - a/x^2]*Log[c*(a + b*x ^2)^p])/a) + (p*PolyLog[2, a/x^2])/a)/a))/2
3.1.82.3.1 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_.) + 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_.)/((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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.64 (sec) , antiderivative size = 554, normalized size of antiderivative = 4.29
| method | result | size |
| risch | \(-\frac {{\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}^{2}}{4 x^{4}}-\frac {p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2 a \,x^{2}}-\frac {p \,b^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{a^{2}}+\frac {p \,b^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {p^{2} b^{2} \ln \left (b \,x^{2}+a \right )^{2}}{4 a^{2}}+\frac {b^{2} p^{2} \ln \left (x \right )}{a^{2}}-\frac {p^{2} b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{2}}+\frac {p^{2} b^{2} \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a^{2}}+\frac {p^{2} b^{2} \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a^{2}}+\frac {p^{2} b^{2} \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a^{2}}+\frac {p^{2} b^{2} \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a^{2}}+\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{4 x^{4}}+\frac {p b \left (-\frac {1}{2 a \,x^{2}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\right )}{2}\right )-\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{16 x^{4}}\) | \(554\) |
-1/4*ln((b*x^2+a)^p)^2/x^4-1/2*p*b*ln((b*x^2+a)^p)/a/x^2-p*b^2*ln((b*x^2+a )^p)/a^2*ln(x)+1/2*p*b^2*ln((b*x^2+a)^p)/a^2*ln(b*x^2+a)-1/4*p^2*b^2/a^2*l n(b*x^2+a)^2+b^2*p^2*ln(x)/a^2-1/2*p^2*b^2/a^2*ln(b*x^2+a)+p^2*b^2/a^2*ln( x)*ln((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+p^2*b^2/a^2*ln(x)*ln((b*x+(-a*b)^( 1/2))/(-a*b)^(1/2))+p^2*b^2/a^2*dilog((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+p^ 2*b^2/a^2*dilog((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+(I*Pi*csgn(I*(b*x^2+a)^p) *csgn(I*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*cs gn(I*c)-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c )+2*ln(c))*(-1/4/x^4*ln((b*x^2+a)^p)+1/2*p*b*(-1/2/a/x^2-1/a^2*b*ln(x)+1/2 *b/a^2*ln(b*x^2+a)))-1/16*(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^ 2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-I*Pi*csgn(I*c*( b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln(c))^2/x^4
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{5}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.10 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {1}{4} \, b^{2} p^{2} {\left (\frac {\log \left (b x^{2} + a\right )^{2}}{a^{2}} - \frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )}}{a^{2}} + \frac {2 \, \log \left (b x^{2} + a\right )}{a^{2}} - \frac {4 \, \log \left (x\right )}{a^{2}}\right )} + \frac {1}{2} \, b p {\left (\frac {b \log \left (b x^{2} + a\right )}{a^{2}} - \frac {b \log \left (x^{2}\right )}{a^{2}} - \frac {1}{a x^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{4 \, x^{4}} \]
-1/4*b^2*p^2*(log(b*x^2 + a)^2/a^2 - 2*(2*log(b*x^2/a + 1)*log(x) + dilog( -b*x^2/a))/a^2 + 2*log(b*x^2 + a)/a^2 - 4*log(x)/a^2) + 1/2*b*p*(b*log(b*x ^2 + a)/a^2 - b*log(x^2)/a^2 - 1/(a*x^2))*log((b*x^2 + a)^p*c) - 1/4*log(( b*x^2 + a)^p*c)^2/x^4
\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^5} \,d x \]