Integrand size = 25, antiderivative size = 165 \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}} \]
I*arctan(x*b^(1/2)/a^(1/2))^2/a^(1/2)/b^(1/2)+arctan(x*b^(1/2)/a^(1/2))*ln (c*x^2/(b*x^2+a))/a^(1/2)/b^(1/2)-2*arctan(x*b^(1/2)/a^(1/2))*ln(2-2*a^(1/ 2)/(a^(1/2)-I*x*b^(1/2)))/a^(1/2)/b^(1/2)+I*polylog(2,-1+2*a^(1/2)/(a^(1/2 )-I*x*b^(1/2)))/a^(1/2)/b^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(373\) vs. \(2(165)=330\).
Time = 0.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.26 \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\frac {-4 \log \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )+\log ^2\left (\sqrt {-a}-\sqrt {b} x\right )+4 \log \left (\frac {a \sqrt {b} x}{(-a)^{3/2}}\right ) \log \left (\sqrt {-a}+\sqrt {b} x\right )-\log ^2\left (\sqrt {-a}+\sqrt {b} x\right )+2 \log \left (\sqrt {-a}-\sqrt {b} x\right ) \log \left (\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {a+\sqrt {-a} \sqrt {b} x}{2 a}\right )+2 \log \left (\sqrt {-a}-\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )+4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {b} x}{\sqrt {-a}}\right )-2 \operatorname {PolyLog}\left (2,\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right )+2 \operatorname {PolyLog}\left (2,\frac {a+\sqrt {-a} \sqrt {b} x}{2 a}\right )-4 \operatorname {PolyLog}\left (2,1+\frac {a \sqrt {b} x}{(-a)^{3/2}}\right )}{4 \sqrt {-a} \sqrt {b}} \]
(-4*Log[(Sqrt[b]*x)/Sqrt[-a]]*Log[Sqrt[-a] - Sqrt[b]*x] + Log[Sqrt[-a] - S qrt[b]*x]^2 + 4*Log[(a*Sqrt[b]*x)/(-a)^(3/2)]*Log[Sqrt[-a] + Sqrt[b]*x] - Log[Sqrt[-a] + Sqrt[b]*x]^2 + 2*Log[Sqrt[-a] - Sqrt[b]*x]*Log[(a - Sqrt[-a ]*Sqrt[b]*x)/(2*a)] - 2*Log[Sqrt[-a] + Sqrt[b]*x]*Log[(a + Sqrt[-a]*Sqrt[b ]*x)/(2*a)] + 2*Log[Sqrt[-a] - Sqrt[b]*x]*Log[(c*x^2)/(a + b*x^2)] - 2*Log [Sqrt[-a] + Sqrt[b]*x]*Log[(c*x^2)/(a + b*x^2)] + 4*PolyLog[2, 1 + (Sqrt[b ]*x)/Sqrt[-a]] - 2*PolyLog[2, (a - Sqrt[-a]*Sqrt[b]*x)/(2*a)] + 2*PolyLog[ 2, (a + Sqrt[-a]*Sqrt[b]*x)/(2*a)] - 4*PolyLog[2, 1 + (a*Sqrt[b]*x)/(-a)^( 3/2)])/(4*Sqrt[-a]*Sqrt[b])
Time = 0.53 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3006, 27, 5459, 27, 5403, 27, 2897}
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.
| \(\displaystyle \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 3006 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\int \frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} x \left (b x^2+a\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \int \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x \left (b x^2+a\right )}dx}{\sqrt {b}}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \left (\frac {i \int \frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x \left (\sqrt {b} x+i \sqrt {a}\right )}dx}{a}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 a}\right )}{\sqrt {b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \left (\frac {i \int \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x \left (\sqrt {b} x+i \sqrt {a}\right )}dx}{\sqrt {a}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 a}\right )}{\sqrt {b}}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \left (\frac {i \left (\frac {i \sqrt {b} \int \frac {a \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{b x^2+a}dx}{a}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 a}\right )}{\sqrt {b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \left (\frac {i \left (i \sqrt {b} \int \frac {\log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{b x^2+a}dx-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 a}\right )}{\sqrt {b}}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \sqrt {a} \left (\frac {i \left (-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a}}-\frac {\operatorname {PolyLog}\left (2,\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}-1\right )}{2 \sqrt {a}}\right )}{\sqrt {a}}-\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{2 a}\right )}{\sqrt {b}}\) |
(ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(c*x^2)/(a + b*x^2)])/(Sqrt[a]*Sqrt[b]) - (2*Sqrt[a]*(((-1/2*I)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a + (I*(((-I)*ArcTan [(Sqrt[b]*x)/Sqrt[a]]*Log[2 - (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b]*x)])/Sqrt[a ] - PolyLog[2, -1 + (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b]*x)]/(2*Sqrt[a])))/Sqr t[a]))/Sqrt[b]
3.3.79.3.1 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[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[Log[(c_.)*(RFx_)^(n_.)]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = I ntHide[1/(d + e*x^2), x]}, Simp[u*Log[c*RFx^n], x] - Simp[n Int[SimplifyI ntegrand[u*(D[RFx, x]/RFx), x], x], x]] /; FreeQ[{c, d, e, n}, x] && Ration alFunctionQ[RFx, x] && !PolynomialQ[RFx, x]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si mp[b*c*(p/d) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* d^2 + e^2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[I/d Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \,x^{2}}{b \,x^{2}+a}\right )+b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )-4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 b}\) | \(121\) |
| risch | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \,x^{2}}{b \,x^{2}+a}\right )+b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )-4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 b}\) | \(121\) |
1/4/b*sum(1/_alpha*(2*ln(x-_alpha)*ln(c*x^2/(b*x^2+a))+b*(1/_alpha/b*ln(x- _alpha)^2+2*_alpha/a*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+2*_alpha/a*dil og(1/2*(x+_alpha)/_alpha))-4*dilog(x/_alpha)-4*ln(x-_alpha)*ln(x/_alpha)), _alpha=RootOf(_Z^2*b+a))
\[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\int { \frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a} \,d x } \]
\[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\int \frac {\log {\left (\frac {c x^{2}}{a + b x^{2}} \right )}}{a + b x^{2}}\, dx \]
\[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\int { \frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a} \,d x } \]
\[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\int { \frac {\log \left (\frac {c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a} \,d x } \]
Timed out. \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\int \frac {\ln \left (\frac {c\,x^2}{b\,x^2+a}\right )}{b\,x^2+a} \,d x \]