\(\int \frac {a+b \log (c x^n)}{x (d+e x^2)^2} \, dx\) [224]

Optimal result

Integrand size = 23, antiderivative size = 82 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\frac {a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^2} \] Output:

1/2*(a+b*ln(c*x^n))/d/(e*x^2+d)-1/4*ln(1+d/e/x^2)*(2*a-b*n+2*b*ln(c*x^n))/ 
d^2+1/4*b*n*polylog(2,-d/e/x^2)/d^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.40 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\frac {a-b n \log (x)+b \log \left (c x^n\right )}{2 d^2+2 d e x^2}+\frac {\log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d^2}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )}{2 d^2}+\frac {b n \left (\frac {\sqrt {e} x \log (x)}{i \sqrt {d}-\sqrt {e} x}-\frac {\sqrt {e} x \log (x)}{i \sqrt {d}+\sqrt {e} x}+2 \log ^2(x)+\log \left (i \sqrt {d}-\sqrt {e} x\right )+\log \left (i \sqrt {d}+\sqrt {e} x\right )-2 \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )-2 \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )\right )}{4 d^2} \] Input:

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

Output:

(a - b*n*Log[x] + b*Log[c*x^n])/(2*d^2 + 2*d*e*x^2) + (Log[x]*(a - b*n*Log 
[x] + b*Log[c*x^n]))/d^2 - ((a - b*n*Log[x] + b*Log[c*x^n])*Log[d + e*x^2] 
)/(2*d^2) + (b*n*((Sqrt[e]*x*Log[x])/(I*Sqrt[d] - Sqrt[e]*x) - (Sqrt[e]*x* 
Log[x])/(I*Sqrt[d] + Sqrt[e]*x) + 2*Log[x]^2 + Log[I*Sqrt[d] - Sqrt[e]*x] 
+ Log[I*Sqrt[d] + Sqrt[e]*x] - 2*(Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] + 
PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]]) - 2*(Log[x]*Log[1 - (I*Sqrt[e]*x)/Sq 
rt[d]] + PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])))/(4*d^2)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2785, 25, 2779, 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.

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

\(\Big \downarrow \) 2785

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2779

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

rule 2785

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^2)^(q_.), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(q + 1)*((a + 
 b*Log[c*x^n])/(2*d*f*(q + 1))), x] + Simp[1/(2*d*(q + 1))   Int[(f*x)^m*(d 
 + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.78 (sec) , antiderivative size = 338, normalized size of antiderivative = 4.12

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right )}{2 d \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}+\frac {b n \ln \left (e \,x^{2}+d \right )}{4 d^{2}}-\frac {b n \ln \left (x \right )}{2 d^{2}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{2}}+\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e \left (-\frac {d}{e \left (e \,x^{2}+d \right )}+\frac {\ln \left (e \,x^{2}+d \right )}{e}\right )}{2 d^{2}}+\frac {\ln \left (x \right )}{d^{2}}\right )\)	\(338\)

Input:

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

Output:

1/2*b*ln(x^n)/d/(e*x^2+d)-1/2*b*ln(x^n)/d^2*ln(e*x^2+d)+b*ln(x^n)/d^2*ln(x 
)+1/4*b*n/d^2*ln(e*x^2+d)-1/2*b*n/d^2*ln(x)-1/2*b*n/d^2*ln(x)^2+1/2*b*n/d^ 
2*ln(x)*ln(e*x^2+d)-1/2*b*n/d^2*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2)) 
-1/2*b*n/d^2*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^2*dilog(( 
-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^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)*c 
sgn(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)*(-1/2*e/d^2*(-d/e/(e*x^2+d)+ln(e*x^2+d)/e)+1/d^2*l 
n(x))
 

Fricas [F]

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

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

Output:

integral((b*log(c*x^n) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*a*(1/(d*e*x^2 + d^2) - log(e*x^2 + d)/d^2 + 2*log(x)/d^2) + b*integrat 
e((log(c) + log(x^n))/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{3}+2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{2} e \,x^{2}-\mathrm {log}\left (e \,x^{2}+d \right ) a d -\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}+2 \,\mathrm {log}\left (x \right ) a d +2 \,\mathrm {log}\left (x \right ) a e \,x^{2}-a e \,x^{2}}{2 d^{2} \left (e \,x^{2}+d \right )} \] Input:

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

Output:

(2*int(log(x**n*c)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**3 + 2*int(log 
(x**n*c)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**2*e*x**2 - log(d + e*x* 
*2)*a*d - log(d + e*x**2)*a*e*x**2 + 2*log(x)*a*d + 2*log(x)*a*e*x**2 - a* 
e*x**2)/(2*d**2*(d + e*x**2))