\(\int \frac {(a+b \cot ^{-1}(c x^n)) (d+e \log (f x^m))}{x} \, dx\) [69]

Optimal result

Integrand size = 24, antiderivative size = 187 \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right )}{2 n}-\frac {i b e \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right )}{2 n}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right )}{2 n}+\frac {i b e \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right )}{2 n}-\frac {i b e m \operatorname {PolyLog}\left (3,-\frac {i x^{-n}}{c}\right )}{2 n^2}+\frac {i b e m \operatorname {PolyLog}\left (3,\frac {i x^{-n}}{c}\right )}{2 n^2} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right )}{n^2}-\frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right ) \left (d+e \log \left (f x^m\right )\right )}{n}-\frac {1}{2} \left (a+b \cot ^{-1}\left (c x^n\right )+b \arctan \left (c x^n\right )\right ) \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \] Input:

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

Output:

(b*c*e*m*x^n*HypergeometricPFQ[{1/2, 1/2, 1/2, 1}, {3/2, 3/2, 3/2}, -(c^2* 
x^(2*n))])/n^2 - (b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, -(c 
^2*x^(2*n))]*(d + e*Log[f*x^m]))/n - ((a + b*ArcCot[c*x^n] + b*ArcTan[c*x^ 
n])*Log[x]*(e*m*Log[x] - 2*(d + e*Log[f*x^m])))/2
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right )}{2 n}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right )}{2 n}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}+\frac {i b e \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}-\frac {i b e m \operatorname {PolyLog}\left (3,-\frac {i x^{-n}}{c}\right )}{2 n^2}+\frac {i b e m \operatorname {PolyLog}\left (3,\frac {i x^{-n}}{c}\right )}{2 n^2}\)

Input:

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

Output:

a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) - ((I/2)*b*d*PolyLog[2, (-I)/(c*x^n) 
])/n - ((I/2)*b*e*Log[f*x^m]*PolyLog[2, (-I)/(c*x^n)])/n + ((I/2)*b*d*Poly 
Log[2, I/(c*x^n)])/n + ((I/2)*b*e*Log[f*x^m]*PolyLog[2, I/(c*x^n)])/n - (( 
I/2)*b*e*m*PolyLog[3, (-I)/(c*x^n)])/n^2 + ((I/2)*b*e*m*PolyLog[3, I/(c*x^ 
n)])/n^2
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (warning: unable to verify)

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

Time = 196.37 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.03

method	result	size
risch	\(\frac {\left (-\frac {i e \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{4}+\frac {i e \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{4}+\frac {i e \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{4}-\frac {i e \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{4}+\frac {e \ln \left (f \right )}{2}+\frac {d}{2}\right ) \left (\left (\pi b +2 a \right ) \ln \left (x^{n}\right )-i b \operatorname {dilog}\left (1+i c \,x^{n}\right )+i b \operatorname {dilog}\left (1-i c \,x^{n}\right )\right )}{n}+\frac {e \ln \left (x^{m}\right )^{2} \pi b}{4 m}+\frac {e \ln \left (x^{m}\right )^{2} a}{2 m}+\frac {i e b m \ln \left (x \right ) \operatorname {polylog}\left (2, i c \,x^{n}\right )}{2 n}-\frac {i e b \ln \left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x \right )^{2} m}{2}+\frac {i e b \ln \left (1-i c \,x^{n}\right ) \ln \left (x \right )^{2} m}{2}+\frac {i e b \ln \left (x \right ) \ln \left (1+i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2}-\frac {i e b m \operatorname {polylog}\left (3, i c \,x^{n}\right )}{2 n^{2}}-\frac {i e b \operatorname {dilog}\left (-i c \,x^{n}\right ) m \ln \left (x \right )}{2 n}+\frac {i e b \operatorname {dilog}\left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x^{m}\right )}{2 n}-\frac {i e b m \ln \left (x \right ) \operatorname {polylog}\left (2, -i c \,x^{n}\right )}{2 n}-\frac {i e b \ln \left (1-i c \,x^{n}\right ) \ln \left (x \right ) \ln \left (x^{m}\right )}{2}-\frac {i e b \ln \left (x \right )^{2} \ln \left (1+i c \,x^{n}\right ) m}{2}-\frac {i e b \operatorname {dilog}\left (-i \left (c \,x^{n}+i\right )\right ) m \ln \left (x \right )}{2 n}+\frac {i e b \ln \left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x \right ) \ln \left (x^{m}\right )}{2}+\frac {i e b m \operatorname {polylog}\left (3, -i c \,x^{n}\right )}{2 n^{2}}-\frac {i e b \ln \left (-i \left (-c \,x^{n}+i\right )\right ) \ln \left (-i c \,x^{n}\right ) m \ln \left (x \right )}{2 n}+\frac {i e b \ln \left (x \right )^{2} \ln \left (-i \left (-c \,x^{n}+i\right )\right ) m}{2}-\frac {i e b \ln \left (x \right ) \ln \left (-i \left (-c \,x^{n}+i\right )\right ) \ln \left (x^{m}\right )}{2}+\frac {i e b \ln \left (-i \left (-c \,x^{n}+i\right )\right ) \ln \left (-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}+\frac {i e b \operatorname {dilog}\left (-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}\)	\(566\)

Input:

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

Output:

(-1/4*I*e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/4*I*e*Pi*csgn(I*f)*csgn 
(I*f*x^m)^2+1/4*I*e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*I*e*Pi*csgn(I*f*x^m 
)^3+1/2*e*ln(f)+1/2*d)/n*((Pi*b+2*a)*ln(x^n)-I*b*dilog(1+I*c*x^n)+I*b*dilo 
g(1-I*c*x^n))+1/4*e/m*ln(x^m)^2*Pi*b+1/2*e/m*ln(x^m)^2*a+1/2*I*e*b*m/n*ln( 
x)*polylog(2,I*c*x^n)-1/2*I*e*b*ln(-I*(c*x^n+I))*ln(x)^2*m+1/2*I*e*b*ln(1- 
I*c*x^n)*ln(x)^2*m+1/2*I*e*b*ln(x)*ln(1+I*c*x^n)*ln(x^m)-1/2*I*e*b*m/n^2*p 
olylog(3,I*c*x^n)-1/2*I*e*b/n*dilog(-I*c*x^n)*m*ln(x)+1/2*I*e*b/n*dilog(-I 
*(c*x^n+I))*ln(x^m)-1/2*I*e*b*m/n*ln(x)*polylog(2,-I*c*x^n)-1/2*I*e*b*ln(1 
-I*c*x^n)*ln(x)*ln(x^m)-1/2*I*e*b*ln(x)^2*ln(1+I*c*x^n)*m-1/2*I*e*b/n*dilo 
g(-I*(c*x^n+I))*m*ln(x)+1/2*I*e*b*ln(-I*(c*x^n+I))*ln(x)*ln(x^m)+1/2*I*e*b 
*m/n^2*polylog(3,-I*c*x^n)-1/2*I*e*b/n*ln(-I*(-c*x^n+I))*ln(-I*c*x^n)*m*ln 
(x)+1/2*I*e*b*ln(x)^2*ln(-I*(-c*x^n+I))*m-1/2*I*e*b*ln(x)*ln(-I*(-c*x^n+I) 
)*ln(x^m)+1/2*I*e*b/n*ln(-I*(-c*x^n+I))*ln(-I*c*x^n)*ln(x^m)+1/2*I*e*b/n*d 
ilog(-I*c*x^n)*ln(x^m)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {2 \, a e m n^{2} \log \left (x\right )^{2} - 2 i \, b e m {\rm polylog}\left (3, i \, c x^{n}\right ) + 2 i \, b e m {\rm polylog}\left (3, -i \, c x^{n}\right ) + 2 \, {\left (b e m n^{2} \log \left (x\right )^{2} + 2 \, {\left (b e n^{2} \log \left (f\right ) + b d n^{2}\right )} \log \left (x\right )\right )} \operatorname {arccot}\left (c x^{n}\right ) - 2 \, {\left (-i \, b e m n \log \left (x\right ) - i \, b e n \log \left (f\right ) - i \, b d n\right )} {\rm Li}_2\left (i \, c x^{n}\right ) - 2 \, {\left (i \, b e m n \log \left (x\right ) + i \, b e n \log \left (f\right ) + i \, b d n\right )} {\rm Li}_2\left (-i \, c x^{n}\right ) + {\left (-i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (i \, b e n^{2} \log \left (f\right ) + i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (i \, c x^{n} + 1\right ) + {\left (i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (-i \, b e n^{2} \log \left (f\right ) - i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (-i \, c x^{n} + 1\right ) + 4 \, {\left (a e n^{2} \log \left (f\right ) + a d n^{2}\right )} \log \left (x\right )}{4 \, n^{2}} \] Input:

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

Output:

1/4*(2*a*e*m*n^2*log(x)^2 - 2*I*b*e*m*polylog(3, I*c*x^n) + 2*I*b*e*m*poly 
log(3, -I*c*x^n) + 2*(b*e*m*n^2*log(x)^2 + 2*(b*e*n^2*log(f) + b*d*n^2)*lo 
g(x))*arccot(c*x^n) - 2*(-I*b*e*m*n*log(x) - I*b*e*n*log(f) - I*b*d*n)*dil 
og(I*c*x^n) - 2*(I*b*e*m*n*log(x) + I*b*e*n*log(f) + I*b*d*n)*dilog(-I*c*x 
^n) + (-I*b*e*m*n^2*log(x)^2 - 2*(I*b*e*n^2*log(f) + I*b*d*n^2)*log(x))*lo 
g(I*c*x^n + 1) + (I*b*e*m*n^2*log(x)^2 - 2*(-I*b*e*n^2*log(f) - I*b*d*n^2) 
*log(x))*log(-I*c*x^n + 1) + 4*(a*e*n^2*log(f) + a*d*n^2)*log(x))/n^2
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x} \,d x } \] Input:

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

Output:

1/2*a*e*log(f*x^m)^2/m + a*d*log(x) - 1/2*(b*e*m*log(x)^2 - 2*b*e*log(x)*l 
og(x^m) - 2*(b*e*log(f) + b*d)*log(x))*arctan(1/(c*x^n)) + integrate(-1/2* 
(b*c*e*m*n*x^n*log(x)^2 - 2*b*c*e*n*x^n*log(x)*log(x^m) - 2*(b*c*e*log(f) 
+ b*c*d)*n*x^n*log(x))/(c^2*x*x^(2*n) + x), x)
 

Giac [F]

\[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acot}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathit {acot} \left (x^{n} c \right )}{x}d x \right ) b d m +2 \left (\int \frac {\mathit {acot} \left (x^{n} c \right ) \mathrm {log}\left (x^{m} f \right )}{x}d x \right ) b e m +\mathrm {log}\left (x^{m} f \right )^{2} a e +2 \,\mathrm {log}\left (x \right ) a d m}{2 m} \] Input:

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

Output:

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