Integrand size = 25, antiderivative size = 201 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=-\frac {e p \log \left (d+e x^2\right )}{2 f (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 f \left (f+g x^2\right )}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^2}+\frac {e p \log \left (f+g x^2\right )}{2 f (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f^2}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f^2}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f^2} \]
-1/2*e*p*ln(e*x^2+d)/f/(-d*g+e*f)+1/2*ln(c*(e*x^2+d)^p)/f/(g*x^2+f)+1/2*ln (-e*x^2/d)*ln(c*(e*x^2+d)^p)/f^2+1/2*e*p*ln(g*x^2+f)/f/(-d*g+e*f)-1/2*ln(c *(e*x^2+d)^p)*ln(e*(g*x^2+f)/(-d*g+e*f))/f^2-1/2*p*polylog(2,-g*(e*x^2+d)/ (-d*g+e*f))/f^2+1/2*p*polylog(2,1+e*x^2/d)/f^2
Time = 0.07 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\frac {\frac {e f p \log \left (d+e x^2\right )}{-e f+d g}+\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}+\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {e f p \log \left (f+g x^2\right )}{e f-d g}-\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )-p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f^2} \]
((e*f*p*Log[d + e*x^2])/(-(e*f) + d*g) + (f*Log[c*(d + e*x^2)^p])/(f + g*x ^2) + Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + (e*f*p*Log[f + g*x^2])/(e*f - d*g) - Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)] - p*PolyLo g[2, (g*(d + e*x^2))/(-(e*f) + d*g)] + p*PolyLog[2, 1 + (e*x^2)/d])/(2*f^2 )
Time = 0.44 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 2863, 2009}
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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {\log \left (c \left (e x^2+d\right )^p\right )}{x^2 \left (g x^2+f\right )^2}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {g \log \left (c \left (e x^2+d\right )^p\right )}{f^2 \left (g x^2+f\right )}+\frac {\log \left (c \left (e x^2+d\right )^p\right )}{f^2 x^2}-\frac {g \log \left (c \left (e x^2+d\right )^p\right )}{f \left (g x^2+f\right )^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{f^2}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^2}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f \left (f+g x^2\right )}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{f^2}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )}{f^2}-\frac {e p \log \left (d+e x^2\right )}{f (e f-d g)}+\frac {e p \log \left (f+g x^2\right )}{f (e f-d g)}\right )\) |
(-((e*p*Log[d + e*x^2])/(f*(e*f - d*g))) + Log[c*(d + e*x^2)^p]/(f*(f + g* x^2)) + (Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/f^2 + (e*p*Log[f + g*x^2] )/(f*(e*f - d*g)) - (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)] )/f^2 - (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/f^2 + (p*PolyLog[2, 1 + (e*x^2)/d])/f^2)/2
3.4.51.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.56 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.45
| method | result | size |
| parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (x \right )}{f^{2}}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 f \left (g \,x^{2}+f \right )}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 f^{2}}-p e \left (-\frac {\ln \left (e \,x^{2}+d \right )}{2 f \left (d g -e f \right )}+\frac {\ln \left (g \,x^{2}+f \right )}{2 f \left (d g -e f \right )}+\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e}}{f^{2}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )}{2 f^{2} e}\right )\) | \(492\) |
| risch | \(\text {Expression too large to display}\) | \(650\) |
ln(c*(e*x^2+d)^p)/f^2*ln(x)+1/2*ln(c*(e*x^2+d)^p)/f/(g*x^2+f)-1/2*ln(c*(e* x^2+d)^p)/f^2*ln(g*x^2+f)-p*e*(-1/2/f/(d*g-e*f)*ln(e*x^2+d)+1/2/f/(d*g-e*f )*ln(g*x^2+f)+2/f^2*(1/2*ln(x)*(ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+ln((e *x+(-d*e)^(1/2))/(-d*e)^(1/2)))/e+1/2*(dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1 /2))+dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2)))/e)-1/2/f^2/e*sum(ln(x-_alpha) *ln(g*x^2+f)-ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,ind ex=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))+ln((Root Of(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_ Z*_alpha*e*g-d*g+e*f,index=2)))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g +e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))- dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)-x+_alpha)/RootOf(_ Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=2)),_alpha=RootOf(_Z^2*e+d)))
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=-\frac {1}{2} \, e p {\left (\frac {\log \left (e x^{2} + d\right )}{e f^{2} - d f g} - \frac {\log \left (g x^{2} + f\right )}{e f^{2} - d f g} + \frac {2 \, \log \left (\frac {e x^{2}}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x^{2}}{d}\right )}{e f^{2}} - \frac {\log \left (g x^{2} + f\right ) \log \left (-\frac {e g x^{2} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{2} + e f}{e f - d g}\right )}{e f^{2}}\right )} + \frac {1}{2} \, {\left (\frac {1}{f g x^{2} + f^{2}} - \frac {\log \left (g x^{2} + f\right )}{f^{2}} + \frac {\log \left (x^{2}\right )}{f^{2}}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
-1/2*e*p*(log(e*x^2 + d)/(e*f^2 - d*f*g) - log(g*x^2 + f)/(e*f^2 - d*f*g) + (2*log(e*x^2/d + 1)*log(x) + dilog(-e*x^2/d))/(e*f^2) - (log(g*x^2 + f)* log(-(e*g*x^2 + e*f)/(e*f - d*g) + 1) + dilog((e*g*x^2 + e*f)/(e*f - d*g)) )/(e*f^2)) + 1/2*(1/(f*g*x^2 + f^2) - log(g*x^2 + f)/f^2 + log(x^2)/f^2)*l og((e*x^2 + d)^p*c)
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2} x} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x\,{\left (g\,x^2+f\right )}^2} \,d x \]