\(\int \frac {a+b \log (c (d x^m)^n)}{e+f x^2} \, dx\) [255]

Optimal result

Integrand size = 24, antiderivative size = 110 \[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt {e} \sqrt {f}}-\frac {b m n \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{2 \sqrt {-e} \sqrt {f}}+\frac {b m n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{2 \sqrt {-e} \sqrt {f}} \] Output:

arctan(f^(1/2)*x/e^(1/2))*(a+b*ln(c*(d*x^m)^n))/e^(1/2)/f^(1/2)-1/2*b*m*n* 
polylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(1/2)/f^(1/2)+1/2*b*m*n*polylog(2,f^ 
(1/2)*x/(-e)^(1/2))/(-e)^(1/2)/f^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\frac {-\left (\left (a+b \log \left (c \left (d x^m\right )^n\right )\right ) \left (\log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )-\log \left (1+\frac {e \sqrt {f} x}{(-e)^{3/2}}\right )\right )\right )+b m n \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )-b m n \operatorname {PolyLog}\left (2,\frac {e \sqrt {f} x}{(-e)^{3/2}}\right )}{2 \sqrt {-e} \sqrt {f}} \] Input:

Integrate[(a + b*Log[c*(d*x^m)^n])/(e + f*x^2),x]
 

Output:

(-((a + b*Log[c*(d*x^m)^n])*(Log[1 + (Sqrt[f]*x)/Sqrt[-e]] - Log[1 + (e*Sq 
rt[f]*x)/(-e)^(3/2)])) + b*m*n*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]] - b*m*n*Po 
lyLog[2, (e*Sqrt[f]*x)/(-e)^(3/2)])/(2*Sqrt[-e]*Sqrt[f])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2895, 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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2}dx\)

\(\Big \downarrow \) 2761

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt {e} \sqrt {f}}-b m n \int \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt {e} \sqrt {f}}-\frac {b m n \int \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x}dx}{\sqrt {e} \sqrt {f}}\)

\(\Big \downarrow \) 5355

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt {e} \sqrt {f}}-\frac {b m n \left (\frac {1}{2} i \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x}dx-\frac {1}{2} i \int \frac {\log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )}{x}dx\right )}{\sqrt {e} \sqrt {f}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{\sqrt {e} \sqrt {f}}-\frac {b m n \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )\right )}{\sqrt {e} \sqrt {f}}\)

Input:

Int[(a + b*Log[c*(d*x^m)^n])/(e + f*x^2),x]
 

Output:

(ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*(d*x^m)^n]))/(Sqrt[e]*Sqrt[f]) - 
 (b*m*n*((I/2)*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - (I/2)*PolyLog[2, (I* 
Sqrt[f]*x)/Sqrt[e]]))/(Sqrt[e]*Sqrt[f])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2761

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]
 

rule 2838

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]
 

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

rule 5355

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]
 

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )}{f \,x^{2}+e}d x\]

Input:

int((a+b*ln(c*(d*x^m)^n))/(f*x^2+e),x)
 

Output:

int((a+b*ln(c*(d*x^m)^n))/(f*x^2+e),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\int { \frac {b \log \left (\left (d x^{m}\right )^{n} c\right ) + a}{f x^{2} + e} \,d x } \] Input:

integrate((a+b*log(c*(d*x^m)^n))/(f*x^2+e),x, algorithm="fricas")
 

Output:

integral((b*log((d*x^m)^n*c) + a)/(f*x^2 + e), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\int \frac {a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}}{e + f x^{2}}\, dx \] Input:

integrate((a+b*ln(c*(d*x**m)**n))/(f*x**2+e),x)
 

Output:

Integral((a + b*log(c*(d*x**m)**n))/(e + f*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*(d*x^m)^n))/(f*x^2+e),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
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\int { \frac {b \log \left (\left (d x^{m}\right )^{n} c\right ) + a}{f x^{2} + e} \,d x } \] Input:

integrate((a+b*log(c*(d*x^m)^n))/(f*x^2+e),x, algorithm="giac")
 

Output:

integrate((b*log((d*x^m)^n*c) + a)/(f*x^2 + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )}{f\,x^2+e} \,d x \] Input:

int((a + b*log(c*(d*x^m)^n))/(e + f*x^2),x)
 

Output:

int((a + b*log(c*(d*x^m)^n))/(e + f*x^2), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d x^m\right )^n\right )}{e+f x^2} \, dx=\frac {\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a +\left (\int \frac {\mathrm {log}\left (x^{m n} d^{n} c \right )}{f \,x^{2}+e}d x \right ) b e f}{e f} \] Input:

int((a+b*log(c*(d*x^m)^n))/(f*x^2+e),x)
 

Output:

(sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a + int(log(x**(m*n)*d**n*c 
)/(e + f*x**2),x)*b*e*f)/(e*f)