Integrand size = 20, antiderivative size = 164 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}} \]
1/2*x*(a+b*ln(c*x^n))/d/(e*x^2+d)-1/2*b*n*arctan(x*e^(1/2)/d^(1/2))/d^(3/2 )/e^(1/2)+1/2*arctan(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))/d^(3/2)/e^(1/2)-1/ 4*I*b*n*polylog(2,-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(1/2)+1/4*I*b*n*polylog( 2,I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(1/2)
Time = 0.33 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\frac {1}{4} \left (\frac {a+b \log \left (c x^n\right )}{d \left (\sqrt {-d} \sqrt {e}+e x\right )}+\frac {a+b \log \left (c x^n\right )}{(-d)^{3/2} \sqrt {e}+d e x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2} \sqrt {e}}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2} \sqrt {e}}\right ) \]
((a + b*Log[c*x^n])/(d*(Sqrt[-d]*Sqrt[e] + e*x)) + (a + b*Log[c*x^n])/((-d )^(3/2)*Sqrt[e] + d*e*x) + (b*d*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(( -d)^(5/2)*Sqrt[e]) + (b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/((-d)^(3/2 )*Sqrt[e]) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(3/2 )*Sqrt[e]) + (d*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d )^(5/2)*Sqrt[e]) + (b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(5/2)*Sq rt[e]) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(3/2)*Sqrt[e]))/ 4
Time = 0.40 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2760, 218, 2761, 27, 5355, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2760 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{e x^2+d}dx}{2 d}-\frac {b n \int \frac {1}{e x^2+d}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{e x^2+d}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\) |
| \(\Big \downarrow \) 2761 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x}dx}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\) |
| \(\Big \downarrow \) 5355 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{x}dx\right )}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d} \sqrt {e}}}{2 d}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}\) |
-1/2*(b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (x*(a + b*Log[c *x^n]))/(2*d*(d + e*x^2)) + ((ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n ]))/(Sqrt[d]*Sqrt[e]) - (b*n*((I/2)*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] - (I/2)*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]]))/(Sqrt[d]*Sqrt[e]))/(2*d)
3.3.28.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Sym bol] :> Simp[(-x)*(d + e*x^2)^(q + 1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x ] + (Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*x^ n]), x], x] + Simp[b*(n/(2*d*(q + 1))) Int[(d + e*x^2)^(q + 1), x], x]) / ; FreeQ[{a, b, c, d, e, n}, x] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*(a + b*Log[c*x^n]), x] - Si mp[b*n Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]
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_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[I*(b/2) Int[Log[1 - I*c*x]/x, x], x] - Simp[I*(b/2) Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.52 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.74
| method | result | size |
| risch | \(\frac {b x \ln \left (x^{n}\right )}{2 d \left (e \,x^{2}+d \right )}-\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{2 d \sqrt {d e}}+\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 d \sqrt {d e}}-\frac {b n \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, d}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, d}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{2 d \left (e \,x^{2}+d \right )}+\frac {\arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}\right )\) | \(449\) |
1/2*b*x/d/(e*x^2+d)*ln(x^n)-1/2*b/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n* ln(x)+1/2*b/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln(x^n)-1/2*b*n/d/(d*e)^ (1/2)*arctan(x*e/(d*e)^(1/2))+1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln((- e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2*e-1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/ 2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2*e+1/4*b*n*ln(x)/(e*x^2+d)/(-d*e )^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n*ln(x)/(e*x^2+d)/(-d*e )^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/4*b*n/(-d*e)^(1/2)/d*dilog(( -e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n/(-d*e)^(1/2)/d*dilog((e*x+(-d*e)^ (1/2))/(-d*e)^(1/2))+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2* I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/ 2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(1/2*x/d/(e*x^2+d)+1/2/d/(d*e)^(1/2)*a rctan(x*e/(d*e)^(1/2)))
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]