3.3.19 \(\int \frac {a+b \log (c x^n)}{x^2 (d+e x^2)} \, dx\) [219]

3.3.19.1 Optimal result

Integrand size = 23, antiderivative size = 134 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {i b \sqrt {e} n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2}}-\frac {i b \sqrt {e} n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2}} \]

output

-b*n/d/x+(-a-b*ln(c*x^n))/d/x-arctan(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*e^ 
(1/2)/d^(3/2)+1/2*I*b*n*polylog(2,-I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(3/2)-1/ 
2*I*b*n*polylog(2,I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(3/2)
 

3.3.19.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx=\frac {d \left (-2 b (-d)^{3/2} n+2 \sqrt {-d} d \left (a+b \log \left (c x^n\right )\right )-d \sqrt {e} x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )+d \sqrt {e} x \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )+b d \sqrt {e} n x \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )-b d \sqrt {e} n x \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )\right )}{2 (-d)^{7/2} x} \]

input

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

output

(d*(-2*b*(-d)^(3/2)*n + 2*Sqrt[-d]*d*(a + b*Log[c*x^n]) - d*Sqrt[e]*x*(a + 
 b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]] + d*Sqrt[e]*x*(a + b*Log[c*x^ 
n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)] + b*d*Sqrt[e]*n*x*PolyLog[2, (Sqrt[e 
]*x)/Sqrt[-d]] - b*d*Sqrt[e]*n*x*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)]))/(2 
*(-d)^(7/2)*x)
 

3.3.19.3 Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2780, 2741, 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.

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

\(\Big \downarrow \) 2780

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

\(\Big \downarrow \) 2741

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

\(\Big \downarrow \) 2761

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5355

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

\(\Big \downarrow \) 2838

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

input

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

output

(-((b*n)/x) - (a + b*Log[c*x^n])/x)/d - (e*((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] 
)))/d
 

3.3.19.3.1 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 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

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 2780

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* 
(x_)^(r_.)), x_Symbol] :> Simp[1/d   Int[x^m*(a + b*Log[c*x^n])^p, x], x] - 
 Simp[e/d   Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
 

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 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]
 

3.3.19.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.46 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.43

method	result	size
risch	\(\frac {b e \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{d \sqrt {d e}}-\frac {b e \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{d \sqrt {d e}}-\frac {b \ln \left (x^{n}\right )}{d x}-\frac {b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d \sqrt {-d e}}+\frac {b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d \sqrt {-d e}}-\frac {b n e \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d \sqrt {-d e}}+\frac {b n e \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d \sqrt {-d e}}-\frac {b n}{d x}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d \sqrt {d e}}-\frac {1}{d x}\right )\)	\(325\)

input

int((a+b*ln(c*x^n))/x^2/(e*x^2+d),x,method=_RETURNVERBOSE)
 

output

b*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)-b*e/d/(d*e)^(1/2)*arctan 
(x*e/(d*e)^(1/2))*ln(x^n)-b*ln(x^n)/d/x-1/2*b*n*e/d*ln(x)/(-d*e)^(1/2)*ln( 
(-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*e/d*ln(x)/(-d*e)^(1/2)*ln((e*x+( 
-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n*e/d/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/ 
2))/(-d*e)^(1/2))+1/2*b*n*e/d/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e) 
^(1/2))-b*n/d/x+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*P 
i*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)*(-e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1 
/d/x)
 

3.3.19.5 Fricas [F]

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

input

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

output

integral((b*log(c*x^n) + a)/(e*x^4 + d*x^2), x)
 

3.3.19.6 Sympy [F]

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

input

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

output

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

3.3.19.7 Maxima [F(-2)]

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

input

integrate((a+b*log(c*x^n))/x^2/(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
 

3.3.19.8 Giac [F]

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

input

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

output

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

3.3.19.9 Mupad [F(-1)]

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

input

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

output

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