\(\int \frac {(a+b \cot ^{-1}(c x^n)) (d+e \log (f x^m))}{x} \, dx\) [217]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2338, 6874, 4945, 4941, 2438, 5128, 5126, 2421, 6724} \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {i b d \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right )}{2 n}+\frac {i b d \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right )}{2 n}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}+\frac {i b e \operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}-\frac {i b e m \operatorname {PolyLog}\left (3,-\frac {i x^{-n}}{c}\right )}{2 n^2}+\frac {i b e m \operatorname {PolyLog}\left (3,\frac {i x^{-n}}{c}\right )}{2 n^2} \]

[In]

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

[Out]

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

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2438

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 4941

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Dist[I*(b/2), Int[Log[1 + I/(c
*x)]/x, x], x] + Dist[I*(b/2), Int[Log[1 - I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4945

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcCot[c*x])^p
/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

Rule 5126

Int[(ArcCot[(c_.)*(x_)^(n_.)]*Log[(d_.)*(x_)^(m_.)])/(x_), x_Symbol] :> Dist[I/2, Int[Log[d*x^m]*(Log[1 - I/(c
*x^n)]/x), x], x] - Dist[I/2, Int[Log[d*x^m]*(Log[1 + I/(c*x^n)]/x), x], x] /; FreeQ[{c, d, m, n}, x]

Rule 5128

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

Rule 6724

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d \left (a+b \cot ^{-1}\left (c x^n\right )\right )}{x}+\frac {e \left (a+b \cot ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx \\ & = d \int \frac {a+b \cot ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x} \, dx \\ & = (a e) \int \frac {\log \left (f x^m\right )}{x} \, dx+(b e) \int \frac {\cot ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac {d \text {Subst}\left (\int \frac {a+b \cot ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n} \\ & = a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {1}{2} (i b e) \int \frac {\log \left (f x^m\right ) \log \left (1-\frac {i x^{-n}}{c}\right )}{x} \, dx-\frac {1}{2} (i b e) \int \frac {\log \left (f x^m\right ) \log \left (1+\frac {i x^{-n}}{c}\right )}{x} \, dx+\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{c x}\right )}{x} \, dx,x,x^n\right )}{2 n}-\frac {(i b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{c x}\right )}{x} \, dx,x,x^n\right )}{2 n} \\ & = 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) \int \frac {\operatorname {PolyLog}\left (2,-\frac {i x^{-n}}{c}\right )}{x} \, dx}{2 n}-\frac {(i b e m) \int \frac {\operatorname {PolyLog}\left (2,\frac {i x^{-n}}{c}\right )}{x} \, dx}{2 n} \\ & = 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} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right )}{n^2}-\frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right ) \left (d+e \log \left (f x^m\right )\right )}{n}-\frac {1}{2} \left (a+b \cot ^{-1}\left (c x^n\right )+b \arctan \left (c x^n\right )\right ) \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \]

[In]

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

[Out]

(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*Hypergeome
tricPFQ[{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

Maple [C] (warning: unable to verify)

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

Time = 224.52 (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 (b \pi +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} b \pi }{4 m}+\frac {e \ln \left (x^{m}\right )^{2} a}{2 m}+\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 \operatorname {dilog}\left (-i c \,x^{n}\right ) m \ln \left (x \right )}{2 n}-\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 \ln \left (x \right )^{2} \ln \left (1-i c \,x^{n}\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 (x \right )^{2} \ln \left (1+i c \,x^{n}\right ) m}{2}+\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 (x \right )^{2} \ln \left (-i \left (-c \,x^{n}+i\right )\right ) m}{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 \left (c \,x^{n}+i\right )\right ) m \ln \left (x \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 (-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 \operatorname {dilog}\left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x^{m}\right )}{2 n}+\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 \ln \left (x \right ) \ln \left (1+i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2}-\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 \operatorname {dilog}\left (-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}\)	\(566\)

[In]

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

[Out]

(-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)*c
sgn(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*di
log(1-I*c*x^n))+1/4*e/m*ln(x^m)^2*b*Pi+1/2*e/m*ln(x^m)^2*a+1/2*I*e*b*ln(x)*ln(-I*(c*x^n+I))*ln(x^m)-1/2*I*e*b/
n*dilog(-I*c*x^n)*m*ln(x)-1/2*I*e*b*ln(x)*ln(1-I*c*x^n)*ln(x^m)-1/2*I*e*b*m/n^2*polylog(3,I*c*x^n)+1/2*I*e*b*l
n(x)^2*ln(1-I*c*x^n)*m-1/2*I*e*b*ln(x)*ln(-I*(-c*x^n+I))*ln(x^m)-1/2*I*e*b*ln(x)^2*ln(1+I*c*x^n)*m+1/2*I*e*b*m
/n*ln(x)*polylog(2,I*c*x^n)+1/2*I*e*b*ln(x)^2*ln(-I*(-c*x^n+I))*m+1/2*I*e*b*m/n^2*polylog(3,-I*c*x^n)-1/2*I*e*
b/n*dilog(-I*(c*x^n+I))*m*ln(x)-1/2*I*e*b*m/n*ln(x)*polylog(2,-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/n*dilog(-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*ln(x)*ln(1+I*c*x^n)*ln(x^m)-1/2*I*e*b*ln(x)^2*ln(-I*(c*x^n+I))*m+1/2*I*e*b/n*dilog(-I*c*x^n)*ln(x^m)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {2 \, a e m n^{2} \log \left (x\right )^{2} - 2 i \, b e m {\rm polylog}\left (3, i \, c x^{n}\right ) + 2 i \, b e m {\rm polylog}\left (3, -i \, c x^{n}\right ) + 2 \, {\left (b e m n^{2} \log \left (x\right )^{2} + 2 \, {\left (b e n^{2} \log \left (f\right ) + b d n^{2}\right )} \log \left (x\right )\right )} \operatorname {arccot}\left (c x^{n}\right ) - 2 \, {\left (-i \, b e m n \log \left (x\right ) - i \, b e n \log \left (f\right ) - i \, b d n\right )} {\rm Li}_2\left (i \, c x^{n}\right ) - 2 \, {\left (i \, b e m n \log \left (x\right ) + i \, b e n \log \left (f\right ) + i \, b d n\right )} {\rm Li}_2\left (-i \, c x^{n}\right ) + {\left (-i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (i \, b e n^{2} \log \left (f\right ) + i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (i \, c x^{n} + 1\right ) + {\left (i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (-i \, b e n^{2} \log \left (f\right ) - i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (-i \, c x^{n} + 1\right ) + 4 \, {\left (a e n^{2} \log \left (f\right ) + a d n^{2}\right )} \log \left (x\right )}{4 \, n^{2}} \]

[In]

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

[Out]

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*polylog(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)*log(x))*arccot(c*x^n) - 2*(-I*b*e*m*n*log(x) - I*b*e*n*log(f) - I*b*d*n)*d
ilog(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))*log(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. \[ \int \frac {\left (a+b \cot ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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)*log(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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