Integrand size = 22, antiderivative size = 533 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {2 p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \]
arctan(x*g^(1/2)/f^(1/2))*ln(c*(e*x^2+d)^p)/f^(1/2)/g^(1/2)+2*p*arctan(x*g ^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-p*arct an(x*g^(1/2)/f^(1/2))*ln(-2*((-d)^(1/2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2 )-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2)))/f^(1/2)/g^(1/2)-p*a rctan(x*g^(1/2)/f^(1/2))*ln(2*((-d)^(1/2)+x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1 /2)-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2)))/f^(1/2)/g^(1/2)-I *p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*po lylog(2,1+2*((-d)^(1/2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/( I*e^(1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1 -2*((-d)^(1/2)+x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^(1/2) *f^(1/2)+(-d)^(1/2)*g^(1/2)))/f^(1/2)/g^(1/2)
Time = 0.24 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.06 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=-\frac {i \left (p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-p \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )+2 i \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )\right )}{2 \sqrt {f} \sqrt {g}} \]
((-1/2*I)*(p*Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqr t[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + p*Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g ]*x)/Sqrt[f]] - p*Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[ f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] - p*Log[(Sqrt[g]*(S qrt[-d] + Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*S qrt[g]*x)/Sqrt[f]] + (2*I)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p ] + p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sq rt[-d]*Sqrt[g])] + p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e] *Sqrt[f] + I*Sqrt[-d]*Sqrt[g])] - p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[ g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - p*PolyLog[2, (Sqrt[e]*(Sq rt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])]))/(Sqrt[f]*S qrt[g])
Time = 0.73 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2920, 27, 5463, 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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2920 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-2 e p \int \frac {x \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (e x^2+d\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {2 e p \int \frac {x \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e x^2+d}dx}{\sqrt {f} \sqrt {g}}\) |
| \(\Big \downarrow \) 5463 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {2 e p \int \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}\right )dx}{\sqrt {f} \sqrt {g}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {2 e p \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 e}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 e}-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 e}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 e}\right )}{\sqrt {f} \sqrt {g}}\) |
(ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(Sqrt[f]*Sqrt[g]) - (2* e*p*(-((ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x )])/e) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x ))])/(2*e) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g ]*x))])/(2*e) + ((I/2)*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)] )/e - ((I/4)*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I *Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e - ((I/4) *PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqr t[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e))/(Sqrt[f]*Sqrt[g])
3.3.62.3.1 Defintions of rubi rules used
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_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) *(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p Int[u*(x^(n - 1)/(d + e*x^n)), x ], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a + b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.34 (sec) , antiderivative size = 449, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(\frac {\left (\ln \left (\left (e \,x^{2}+d \right )^{p}\right )-p \ln \left (e \,x^{2}+d \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (g \,\textit {\_Z}^{2}+f \right )}{\sum }\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (e \,x^{2}+d \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} e g +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} e g +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} e g +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} e g +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} e g +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} e g +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} e g +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} e g +d g -e f , \operatorname {index} =2\right )}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 g}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}\) | \(449\) |
(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2))+1/2*p/ g*sum(1/_alpha*(ln(x-_alpha)*ln(e*x^2+d)-ln(x-_alpha)*(ln((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))+ln((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)))-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))),_al pha=RootOf(_Z^2*g+f))+(1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^ 2-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/2*I*Pi*cs gn(I*c*(e*x^2+d)^p)^3+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+ln(c))/(f *g)^(1/2)*arctan(g*x/(f*g)^(1/2))
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {\log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]