Integrand size = 18, antiderivative size = 180 \[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=-\frac {5 e n x^2}{36 a}-d n x \coth ^{-1}(a x)-\frac {1}{9} e n x^3 \coth ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1-a^2 x^2\right )}{2 a}-\frac {e n \log \left (1-a^2 x^2\right )}{18 a^3}+\frac {\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d+e\right ) n \operatorname {PolyLog}\left (2,a^2 x^2\right )}{12 a^3} \] Output:
-5/36*e*n*x^2/a-d*n*x*arccoth(a*x)-1/9*e*n*x^3*arccoth(a*x)+1/6*e*x^2*ln(c *x^n)/a+d*x*arccoth(a*x)*ln(c*x^n)+1/3*e*x^3*arccoth(a*x)*ln(c*x^n)-1/2*d* n*ln(-a^2*x^2+1)/a-1/18*e*n*ln(-a^2*x^2+1)/a^3+1/6*(3*a^2*d+e)*ln(c*x^n)*l n(-a^2*x^2+1)/a^3+1/12*(3*a^2*d+e)*n*polylog(2,a^2*x^2)/a^3
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\frac {-5 a^2 e n x^2+36 a^2 d n \log \left (\frac {1}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\right )+6 a^2 e x^2 \log \left (c x^n\right )-4 a^3 x \coth ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+18 a^2 d \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )+6 e \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )-2 e n \log \left (-1+a^2 x^2\right )+3 \left (3 a^2 d+e\right ) n \operatorname {PolyLog}\left (2,a^2 x^2\right )}{36 a^3} \] Input:
Integrate[(d + e*x^2)*ArcCoth[a*x]*Log[c*x^n],x]
Output:
(-5*a^2*e*n*x^2 + 36*a^2*d*n*Log[1/(a*Sqrt[1 - 1/(a^2*x^2)]*x)] + 6*a^2*e* x^2*Log[c*x^n] - 4*a^3*x*ArcCoth[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*L og[c*x^n]) + 18*a^2*d*Log[c*x^n]*Log[1 - a^2*x^2] + 6*e*Log[c*x^n]*Log[1 - a^2*x^2] - 2*e*n*Log[-1 + a^2*x^2] + 3*(3*a^2*d + e)*n*PolyLog[2, a^2*x^2 ])/(36*a^3)
Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2835, 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.
| \(\displaystyle \int \coth ^{-1}(a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2835 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \coth ^{-1}(a x) x^2+\frac {e x}{6 a}+d \coth ^{-1}(a x)+\frac {\left (3 d a^2+e\right ) \log \left (1-a^2 x^2\right )}{6 a^3 x}\right )dx+\frac {\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}-n \left (\frac {d \log \left (1-a^2 x^2\right )}{2 a}-\frac {\left (3 a^2 d+e\right ) \operatorname {PolyLog}\left (2,a^2 x^2\right )}{12 a^3}+\frac {e \log \left (1-a^2 x^2\right )}{18 a^3}+d x \coth ^{-1}(a x)+\frac {1}{9} e x^3 \coth ^{-1}(a x)+\frac {5 e x^2}{36 a}\right )+d x \coth ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \coth ^{-1}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\) |
Input:
Int[(d + e*x^2)*ArcCoth[a*x]*Log[c*x^n],x]
Output:
(e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCoth[a*x]*Log[c*x^n] + (e*x^3*ArcCoth[a* x]*Log[c*x^n])/3 + ((3*a^2*d + e)*Log[c*x^n]*Log[1 - a^2*x^2])/(6*a^3) - n *((5*e*x^2)/(36*a) + d*x*ArcCoth[a*x] + (e*x^3*ArcCoth[a*x])/9 + (d*Log[1 - a^2*x^2])/(2*a) + (e*Log[1 - a^2*x^2])/(18*a^3) - ((3*a^2*d + e)*PolyLog [2, a^2*x^2])/(12*a^3))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)* (x_))], x_Symbol] :> With[{u = IntHide[Px*F[d*(e + f*x)], x]}, Simp[(a + b* Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, Ar cCoth}, F]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 61.01 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.97
| method | result | size |
| risch | \(\frac {11 e n \ln \left (x \right )}{18 a^{3}}+\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) d \left (\ln \left (a x +1\right ) \left (a x +1\right )-a x -1\right )}{2 a}+\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \ln \left (a x +1\right ) x^{3}}{6}+\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \ln \left (a x +1\right )}{6 a^{3}}+\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \,x^{2}}{6 a}-\frac {d n \operatorname {dilog}\left (a x \right )}{2 a}-\frac {d n \ln \left (a x -1\right )}{2 a}-\frac {e n \ln \left (a x -1\right )}{18 a^{3}}-\frac {e n \operatorname {dilog}\left (a x \right )}{6 a^{3}}+\frac {e n \,x^{3} \ln \left (a x -1\right )}{18}+\frac {e n \,x^{2} \ln \left (x \right )}{6 a}-\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) d \left (\ln \left (a x -1\right ) \left (a x -1\right )-a x +1\right )}{2 a}-\frac {d n x \ln \left (a x +1\right )}{2}+\frac {d n \operatorname {dilog}\left (a x +1\right )}{2 a}-\frac {d n \ln \left (a x +1\right )}{2 a}-\frac {e n \,x^{3} \ln \left (a x +1\right )}{18}-\frac {e n \ln \left (a x +1\right )}{18 a^{3}}+\frac {e n \operatorname {dilog}\left (a x +1\right )}{6 a^{3}}-\frac {d n x \ln \left (a x -1\right ) \ln \left (x \right )}{2}-\frac {11 e \ln \left (x^{n}\right )}{18 a^{3}}-\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \ln \left (a x -1\right ) x^{3}}{6}+\frac {\left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right ) e \ln \left (a x -1\right )}{6 a^{3}}+\frac {d n x \ln \left (a x -1\right )}{2}+\frac {e n \ln \left (a x +1\right ) \ln \left (x \right )}{6 a^{3}}+\frac {d n \ln \left (-a x +1\right ) \ln \left (x \right )}{2 a}-\frac {d n \ln \left (-a x +1\right ) \ln \left (a x \right )}{2 a}+\frac {e n \ln \left (-a x +1\right ) \ln \left (x \right )}{6 a^{3}}-\frac {e n \ln \left (-a x +1\right ) \ln \left (a x \right )}{6 a^{3}}-\frac {5 e n \,x^{2}}{36 a}-\frac {e n \,x^{3} \ln \left (a x -1\right ) \ln \left (x \right )}{6}+\frac {d n x \ln \left (a x +1\right ) \ln \left (x \right )}{2}+\frac {d n \ln \left (x \right ) \ln \left (a x +1\right )}{2 a}+\frac {e n \,x^{3} \ln \left (a x +1\right ) \ln \left (x \right )}{6}+\left (-\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{4}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{4}-\frac {\ln \left (c \right )}{2}\right ) \left (\frac {d \left (\ln \left (a x -1\right ) \left (a x -1\right )-a x +1\right )}{a}+\frac {e \ln \left (a x -1\right ) x^{3}}{3}-\frac {e \ln \left (a x -1\right )}{3 a^{3}}-\frac {e \,x^{2}}{3 a}+\frac {11 e}{9 a^{3}}-\frac {d \left (\ln \left (a x +1\right ) \left (a x +1\right )-a x -1\right )}{a}-\frac {e \ln \left (a x +1\right ) x^{3}}{3}-\frac {e \ln \left (a x +1\right )}{3 a^{3}}\right )\) | \(715\) |
Input:
int((e*x^2+d)*arccoth(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)
Output:
11/18*e/a^3*n*ln(x)+1/2*(ln(x^n)-n*ln(x))*d/a*(ln(a*x+1)*(a*x+1)-a*x-1)+1/ 6*(ln(x^n)-n*ln(x))*e*ln(a*x+1)*x^3+1/6*(ln(x^n)-n*ln(x))*e/a^3*ln(a*x+1)+ 1/6*(ln(x^n)-n*ln(x))*e/a*x^2-1/2*d*n*dilog(a*x)/a-1/2*d*n/a*ln(a*x-1)-1/1 8*e*n/a^3*ln(a*x-1)-1/6*e*n/a^3*dilog(a*x)+1/18*e*n*x^3*ln(a*x-1)+1/6*e*n/ a*x^2*ln(x)-1/2*(ln(x^n)-n*ln(x))*d/a*(ln(a*x-1)*(a*x-1)-a*x+1)-1/2*d*n*x* ln(a*x+1)+1/2*d*n*dilog(a*x+1)/a-1/2*d*n*ln(a*x+1)/a-1/18*e*n*x^3*ln(a*x+1 )-1/18*e*n*ln(a*x+1)/a^3+1/6*e*n/a^3*dilog(a*x+1)-1/2*d*n*x*ln(a*x-1)*ln(x )-11/18*e/a^3*ln(x^n)-1/6*(ln(x^n)-n*ln(x))*e*ln(a*x-1)*x^3+1/6*(ln(x^n)-n *ln(x))*e/a^3*ln(a*x-1)+1/2*d*n*x*ln(a*x-1)+1/6*e*n/a^3*ln(a*x+1)*ln(x)+1/ 2*d*n*ln(-a*x+1)/a*ln(x)-1/2*d*n*ln(-a*x+1)/a*ln(a*x)+1/6*e*n/a^3*ln(-a*x+ 1)*ln(x)-1/6*e*n/a^3*ln(-a*x+1)*ln(a*x)-5/36*e*n*x^2/a+(-1/4*I*Pi*csgn(I*x ^n)*csgn(I*c*x^n)^2+1/4*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/4*I*Pi* csgn(I*c*x^n)^3-1/4*I*Pi*csgn(I*c*x^n)^2*csgn(I*c)-1/2*ln(c))*(d/a*(ln(a*x -1)*(a*x-1)-a*x+1)+1/3*e*ln(a*x-1)*x^3-1/3*e/a^3*ln(a*x-1)-1/3*e/a*x^2+11/ 9*e/a^3-d/a*(ln(a*x+1)*(a*x+1)-a*x-1)-1/3*e*ln(a*x+1)*x^3-1/3*e/a^3*ln(a*x +1))-1/6*e*n*x^3*ln(a*x-1)*ln(x)+1/2*d*n*x*ln(a*x+1)*ln(x)+1/2*d*n*ln(x)*l n(a*x+1)/a+1/6*e*n*x^3*ln(a*x+1)*ln(x)
\[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arcoth}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:
integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="fricas")
Output:
integral((e*x^2 + d)*arccoth(a*x)*log(c*x^n), x)
\[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {acoth}{\left (a x \right )}\, dx \] Input:
integrate((e*x**2+d)*acoth(a*x)*ln(c*x**n),x)
Output:
Integral((d + e*x**2)*log(c*x**n)*acoth(a*x), x)
Time = 0.08 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.77 \[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=-\frac {1}{36} \, n {\left (\frac {6 \, {\left (3 \, a^{2} d + e\right )} {\left (\log \left (a x - 1\right ) \log \left (a x\right ) + {\rm Li}_2\left (-a x + 1\right )\right )}}{a^{3}} + \frac {6 \, {\left (3 \, a^{2} d + e\right )} {\left (\log \left (a x + 1\right ) \log \left (-a x\right ) + {\rm Li}_2\left (a x + 1\right )\right )}}{a^{3}} + \frac {2 \, {\left (9 \, a^{2} d + e\right )} \log \left (a x + 1\right )}{a^{3}} + \frac {5 \, a^{2} e x^{2} + 2 \, {\left (a^{3} e x^{3} + 9 \, a^{3} d x\right )} \log \left (a x + 1\right ) - 2 \, {\left (a^{3} e x^{3} + 9 \, a^{3} d x - 9 \, a^{2} d - e\right )} \log \left (a x - 1\right )}{a^{3}}\right )} + \frac {1}{12} \, {\left (6 \, {\left (x \log \left (\frac {1}{a x} + 1\right ) + \frac {\log \left (a x + 1\right )}{a}\right )} d - 6 \, {\left (x \log \left (-\frac {1}{a x} + 1\right ) - \frac {\log \left (a x - 1\right )}{a}\right )} d + {\left (2 \, x^{3} \log \left (\frac {1}{a x} + 1\right ) + \frac {\frac {a x^{2} - 2 \, x}{a} + \frac {2 \, \log \left (a x + 1\right )}{a^{2}}}{a}\right )} e - {\left (2 \, x^{3} \log \left (-\frac {1}{a x} + 1\right ) - \frac {\frac {a x^{2} + 2 \, x}{a} + \frac {2 \, \log \left (a x - 1\right )}{a^{2}}}{a}\right )} e\right )} \log \left (c x^{n}\right ) \] Input:
integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="maxima")
Output:
-1/36*n*(6*(3*a^2*d + e)*(log(a*x - 1)*log(a*x) + dilog(-a*x + 1))/a^3 + 6 *(3*a^2*d + e)*(log(a*x + 1)*log(-a*x) + dilog(a*x + 1))/a^3 + 2*(9*a^2*d + e)*log(a*x + 1)/a^3 + (5*a^2*e*x^2 + 2*(a^3*e*x^3 + 9*a^3*d*x)*log(a*x + 1) - 2*(a^3*e*x^3 + 9*a^3*d*x - 9*a^2*d - e)*log(a*x - 1))/a^3) + 1/12*(6 *(x*log(1/(a*x) + 1) + log(a*x + 1)/a)*d - 6*(x*log(-1/(a*x) + 1) - log(a* x - 1)/a)*d + (2*x^3*log(1/(a*x) + 1) + ((a*x^2 - 2*x)/a + 2*log(a*x + 1)/ a^2)/a)*e - (2*x^3*log(-1/(a*x) + 1) - ((a*x^2 + 2*x)/a + 2*log(a*x - 1)/a ^2)/a)*e)*log(c*x^n)
\[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {arcoth}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:
integrate((e*x^2+d)*arccoth(a*x)*log(c*x^n),x, algorithm="giac")
Output:
integrate((e*x^2 + d)*arccoth(a*x)*log(c*x^n), x)
Timed out. \[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {acoth}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \] Input:
int(log(c*x^n)*acoth(a*x)*(d + e*x^2),x)
Output:
int(log(c*x^n)*acoth(a*x)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \coth ^{-1}(a x) \log \left (c x^n\right ) \, dx=\frac {36 \mathit {acoth} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} d n x +12 \mathit {acoth} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} e n \,x^{3}-36 \mathit {acoth} \left (a x \right ) a^{3} d \,n^{2} x -4 \mathit {acoth} \left (a x \right ) a^{3} e \,n^{2} x^{3}-36 \mathit {acoth} \left (a x \right ) a^{2} d \,n^{2}-4 \mathit {acoth} \left (a x \right ) e \,n^{2}-36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}-x}d x \right ) a^{2} d n -12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}-x}d x \right ) e n +36 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} d \,n^{2}+4 \,\mathrm {log}\left (a^{2} x -a \right ) e \,n^{2}-18 \mathrm {log}\left (x^{n} c \right )^{2} a^{2} d -6 \mathrm {log}\left (x^{n} c \right )^{2} e -6 \,\mathrm {log}\left (x^{n} c \right ) a^{2} e n \,x^{2}+5 a^{2} e \,n^{2} x^{2}}{36 a^{3} n} \] Input:
int((e*x^2+d)*acoth(a*x)*log(c*x^n),x)
Output:
(36*acoth(a*x)*log(x**n*c)*a**3*d*n*x + 12*acoth(a*x)*log(x**n*c)*a**3*e*n *x**3 - 36*acoth(a*x)*a**3*d*n**2*x - 4*acoth(a*x)*a**3*e*n**2*x**3 - 36*a coth(a*x)*a**2*d*n**2 - 4*acoth(a*x)*e*n**2 - 36*int(log(x**n*c)/(a**2*x** 3 - x),x)*a**2*d*n - 12*int(log(x**n*c)/(a**2*x**3 - x),x)*e*n + 36*log(a* *2*x - a)*a**2*d*n**2 + 4*log(a**2*x - a)*e*n**2 - 18*log(x**n*c)**2*a**2* d - 6*log(x**n*c)**2*e - 6*log(x**n*c)*a**2*e*n*x**2 + 5*a**2*e*n**2*x**2) /(36*a**3*n)