\(\int \frac {a+b \log (c x^n)}{(d+e \log (c x^n))^2} \, dx\) [182]

Optimal result

Integrand size = 23, antiderivative size = 89 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {d}{e n}} (-b d+a e+b e n) x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {(b d-a e) x}{e^2 n \left (d+e \log \left (c x^n\right )\right )} \] Output:

(b*e*n+a*e-b*d)*x*Ei((d+e*ln(c*x^n))/e/n)/e^3/exp(d/e/n)/n^2/((c*x^n)^(1/n 
))+(-a*e+b*d)*x/e^2/n/(d+e*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {d}{e n}} (-b d+a e+b e n) x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )-\frac {e (-b d+a e) n x}{d+e \log \left (c x^n\right )}}{e^3 n^2} \] Input:

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

Output:

(((-(b*d) + a*e + b*e*n)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/ 
(e*n))*(c*x^n)^n^(-1)) - (e*(-(b*d) + a*e)*n*x)/(d + e*Log[c*x^n]))/(e^3*n 
^2)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2807, 2009}

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.

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

\(\Big \downarrow \) 2807

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} (b d-a e) \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^3 n^2}+\frac {x (b d-a e)}{e^2 n \left (e \log \left (c x^n\right )+d\right )}+\frac {b x \left (c x^n\right )^{-1/n} e^{-\frac {d}{e n}} \operatorname {ExpIntegralEi}\left (\frac {d+e \log \left (c x^n\right )}{e n}\right )}{e^2 n}\)

Input:

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

Output:

-(((b*d - a*e)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(e^3*E^(d/(e*n)) 
*n^2*(c*x^n)^n^(-1))) + (b*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(e^2 
*E^(d/(e*n))*n*(c*x^n)^n^(-1)) + ((b*d - a*e)*x)/(e^2*n*(d + e*Log[c*x^n]) 
)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2807

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]*(e_. 
) + (d_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*x^n])^p*(d + 
 e*Log[c*x^n])^q, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p] && 
IntegerQ[q]
 

Maple [C] (warning: unable to verify)

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

Time = 0.87 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.16

method	result	size
risch	\(-\frac {2 x \left (e a -b d \right )}{e^{2} n \left (2 d +2 e \ln \left (c \right )+2 e \ln \left (x^{n}\right )+i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )\right )}-\frac {\left (b e n +e a -b d \right ) x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 d}{2 e n}} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )-\frac {i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i e \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i e \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 e \ln \left (c \right )+2 e \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 d}{2 e n}\right )}{e^{3} n^{2}}\)	\(370\)

Input:

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

Output:

-2/e^2/n*x*(a*e-b*d)/(2*d+2*e*ln(c)+2*e*ln(x^n)+I*e*Pi*csgn(I*x^n)*csgn(I* 
c*x^n)^2-I*e*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*e*Pi*csgn(I*c*x^n)^3 
+I*e*Pi*csgn(I*c*x^n)^2*csgn(I*c))-(b*e*n+a*e-b*d)/e^3/n^2*x*c^(-1/n)*(x^n 
)^(-1/n)*exp(-1/2*(I*e*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e*Pi*csgn(I*x^n)*c 
sgn(I*c*x^n)*csgn(I*c)-I*e*Pi*csgn(I*c*x^n)^3+I*e*Pi*csgn(I*c*x^n)^2*csgn( 
I*c)+2*d)/e/n)*Ei(1,-ln(x)-1/2*(I*e*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*e*Pi* 
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*e*Pi*csgn(I*c*x^n)^3+I*e*Pi*csgn(I*c 
*x^n)^2*csgn(I*c)+2*e*ln(c)+2*e*(ln(x^n)-n*ln(x))+2*d)/e/n)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx=\frac {{\left ({\left (b d e - a e^{2}\right )} n x e^{\left (\frac {e \log \left (c\right ) + d}{e n}\right )} + {\left (b d e n - b d^{2} + a d e + {\left (b e^{2} n - b d e + a e^{2}\right )} \log \left (c\right ) + {\left (b e^{2} n^{2} - {\left (b d e - a e^{2}\right )} n\right )} \log \left (x\right )\right )} \operatorname {log\_integral}\left (x e^{\left (\frac {e \log \left (c\right ) + d}{e n}\right )}\right )\right )} e^{\left (-\frac {e \log \left (c\right ) + d}{e n}\right )}}{e^{4} n^{3} \log \left (x\right ) + e^{4} n^{2} \log \left (c\right ) + d e^{3} n^{2}} \] Input:

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

Output:

((b*d*e - a*e^2)*n*x*e^((e*log(c) + d)/(e*n)) + (b*d*e*n - b*d^2 + a*d*e + 
 (b*e^2*n - b*d*e + a*e^2)*log(c) + (b*e^2*n^2 - (b*d*e - a*e^2)*n)*log(x) 
)*log_integral(x*e^((e*log(c) + d)/(e*n))))*e^(-(e*log(c) + d)/(e*n))/(e^4 
*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (88) = 176\).

Time = 0.14 (sec) , antiderivative size = 712, normalized size of antiderivative = 8.00 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e \log \left (c x^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

b*e^2*n^2*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x)/((e^4*n^3*lo 
g(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) + b*e^2*n*Ei(log(c)/n + d/(e*n 
) + log(x))*e^(-d/(e*n))*log(c)/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3* 
n^2)*c^(1/n)) - b*d*e*n*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x 
)/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) + a*e^2*n*Ei(log 
(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(x)/((e^4*n^3*log(x) + e^4*n^2*l 
og(c) + d*e^3*n^2)*c^(1/n)) + b*d*e*n*x/(e^4*n^3*log(x) + e^4*n^2*log(c) + 
 d*e^3*n^2) - a*e^2*n*x/(e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2) + b* 
d*e*n*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) + e^4* 
n^2*log(c) + d*e^3*n^2)*c^(1/n)) - b*d*e*Ei(log(c)/n + d/(e*n) + log(x))*e 
^(-d/(e*n))*log(c)/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) 
 + a*e^2*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))*log(c)/((e^4*n^3*log 
(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n)) - b*d^2*Ei(log(c)/n + d/(e*n) + 
 log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) + e^4*n^2*log(c) + d*e^3*n^2)*c^(1/ 
n)) + a*d*e*Ei(log(c)/n + d/(e*n) + log(x))*e^(-d/(e*n))/((e^4*n^3*log(x) 
+ e^4*n^2*log(c) + d*e^3*n^2)*c^(1/n))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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