Integrand size = 20, antiderivative size = 104 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Output:
arctan(e^(1/2)*x/d^(1/2))*(a+b*ln(c*x^n))/d^(1/2)/e^(1/2)-1/2*b*n*polylog( 2,-e^(1/2)*x/(-d)^(1/2))/(-d)^(1/2)/e^(1/2)+1/2*b*n*polylog(2,e^(1/2)*x/(- d)^(1/2))/(-d)^(1/2)/e^(1/2)
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\frac {-\left (\left (a+b \log \left (c x^n\right )\right ) \left (\log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )-\log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Input:
Integrate[(a + b*Log[c*x^n])/(d + e*x^2),x]
Output:
(-((a + b*Log[c*x^n])*(Log[1 + (Sqrt[e]*x)/Sqrt[-d]] - Log[1 + (d*Sqrt[e]* x)/(-d)^(3/2)])) + b*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]] - b*n*PolyLog[2, ( d*Sqrt[e]*x)/(-d)^(3/2)])/(2*Sqrt[-d]*Sqrt[e])
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx\) |
\(\Big \downarrow \) 2761 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}dx}{\sqrt {d} \sqrt {e}}\) |
\(\Big \downarrow \) 5355 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{x}dx\right )}{\sqrt {d} \sqrt {e}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d} \sqrt {e}}-\frac {b n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d} \sqrt {e}}\) |
Input:
Int[(a + b*Log[c*x^n])/(d + e*x^2),x]
Output:
(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])
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_)^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 = 0.45 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.53
method | result | size |
risch | \(-\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{\sqrt {d e}}+\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{\sqrt {d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 \sqrt {-d e}}+\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{\sqrt {d e}}\) | \(263\) |
Input:
int((a+b*ln(c*x^n))/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
-b/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+b/(d*e)^(1/2)*arctan(x*e/(d *e)^(1/2))*ln(x^n)+1/2*b*n*ln(x)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e )^(1/2))-1/2*b*n*ln(x)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/ 2*b*n/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/(-d*e)^ (1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+(1/2*I*Pi*b*csgn(I*x^n)*csgn( I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*Pi*b*csgn( I*c*x^n)^3+1/2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b*ln(c)+a)/(d*e)^(1/2)*arc tan(x*e/(d*e)^(1/2))
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e x^{2} + d} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(e*x^2+d),x, algorithm="fricas")
Output:
integral((b*log(c*x^n) + a)/(e*x^2 + d), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{d + e x^{2}}\, dx \] Input:
integrate((a+b*ln(c*x**n))/(e*x**2+d),x)
Output:
Integral((a + b*log(c*x**n))/(d + e*x**2), x)
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*x^n))/(e*x^2+d),x, algorithm="maxima")
Output:
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
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{e x^{2} + d} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(e*x^2+d),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/(e*x^2 + d), x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{e\,x^2+d} \,d x \] Input:
int((a + b*log(c*x^n))/(d + e*x^2),x)
Output:
int((a + b*log(c*x^n))/(d + e*x^2), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a +\left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e \,x^{2}+d}d x \right ) b d e}{d e} \] Input:
int((a+b*log(c*x^n))/(e*x^2+d),x)
Output:
(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a + int(log(x**n*c)/(d + e* x**2),x)*b*d*e)/(d*e)