Integrand size = 18, antiderivative size = 180 \[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=-\frac {5 e n x^2}{36 a}-d n x \text {arctanh}(a x)-\frac {1}{9} e n x^3 \text {arctanh}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \text {arctanh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arctanh}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1-a^2 x^2\right )}{2 a}-\frac {e n \log \left (1-a^2 x^2\right )}{18 a^3}+\frac {\left (3 a^2 d+e\right ) \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d+e\right ) n \operatorname {PolyLog}\left (2,a^2 x^2\right )}{12 a^3} \] Output:
-5/36*e*n*x^2/a-d*n*x*arctanh(a*x)-1/9*e*n*x^3*arctanh(a*x)+1/6*e*x^2*ln(c *x^n)/a+d*x*arctanh(a*x)*ln(c*x^n)+1/3*e*x^3*arctanh(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 = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.93 \[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\frac {-5 a^2 e n x^2+6 a^2 e x^2 \log \left (c x^n\right )-4 a^3 x \text {arctanh}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )-18 a^2 d n \log \left (1-a^2 x^2\right )+18 a^2 d \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )+6 e \log \left (c x^n\right ) \log \left (1-a^2 x^2\right )-2 e n \log \left (-1+a^2 x^2\right )+3 \left (3 a^2 d+e\right ) n \operatorname {PolyLog}\left (2,a^2 x^2\right )}{36 a^3} \] Input:
Integrate[(d + e*x^2)*ArcTanh[a*x]*Log[c*x^n],x]
Output:
(-5*a^2*e*n*x^2 + 6*a^2*e*x^2*Log[c*x^n] - 4*a^3*x*ArcTanh[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) - 18*a^2*d*n*Log[1 - a^2*x^2] + 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.
| \(\displaystyle \int \text {arctanh}(a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2835 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \text {arctanh}(a x) x^2+\frac {e x}{6 a}+d \text {arctanh}(a x)+\frac {\left (3 d a^2+e\right ) \log \left (1-a^2 x^2\right )}{6 a^3 x}\right )dx+\frac {\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}+d x \text {arctanh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arctanh}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -n \left (\frac {d \log \left (1-a^2 x^2\right )}{2 a}-\frac {\left (3 a^2 d+e\right ) \operatorname {PolyLog}\left (2,a^2 x^2\right )}{12 a^3}+\frac {e \log \left (1-a^2 x^2\right )}{18 a^3}+d x \text {arctanh}(a x)+\frac {1}{9} e x^3 \text {arctanh}(a x)+\frac {5 e x^2}{36 a}\right )+\frac {\left (3 a^2 d+e\right ) \log \left (1-a^2 x^2\right ) \log \left (c x^n\right )}{6 a^3}+d x \text {arctanh}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {arctanh}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}\) |
Input:
Int[(d + e*x^2)*ArcTanh[a*x]*Log[c*x^n],x]
Output:
(e*x^2*Log[c*x^n])/(6*a) + d*x*ArcTanh[a*x]*Log[c*x^n] + (e*x^3*ArcTanh[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*ArcTanh[a*x] + (e*x^3*ArcTanh[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 = 58.35 (sec) , antiderivative size = 726, normalized size of antiderivative = 4.03
| 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 {\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 {d n x \ln \left (-a x +1\right )}{2}-\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 \,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 \right )}{6 a^{3}}+\frac {e n \,x^{2} \ln \left (x \right )}{6 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 {11 e \ln \left (x^{n}\right )}{18 a^{3}}+\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 )+\frac {e n \ln \left (a x +1\right ) \ln \left (x \right )}{6 a^{3}}-\frac {d n x \ln \left (-a x +1\right ) \ln \left (x \right )}{2}+\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 \,x^{3} \ln \left (-a x +1\right ) \ln \left (x \right )}{6}+\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 {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}\) | \(726\) |
| default | \(\text {Expression too large to display}\) | \(88817\) |
Input:
int((e*x^2+d)*arctanh(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*(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)+1/2*d*n*x*ln(-a*x+1)-1/2*d*n*dilog(a*x)/a-1/2*d*n/a*ln(a*x-1 )+1/18*e*n*x^3*ln(-a*x+1)-1/18*e*n/a^3*ln(a*x-1)-1/6*e*n/a^3*dilog(a*x)+1/ 6*e*n/a*x^2*ln(x)-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)-11/18*e/a^3*ln(x^n)+1/6*e*n/a^3*ln(a*x+1)*ln(x)-1/2*d*n*x*ln(-a*x+1)*l n(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*x^3*l n(-a*x+1)*ln(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+1/2*d*n*x*ln(a*x+1)*ln(x)+1/2*d*n*ln(x)*ln(a*x+1)/a+1/6*e *n*x^3*ln(a*x+1)*ln(x)+(-1/4*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I*Pi*csg n(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))
\[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {artanh}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:
integrate((e*x^2+d)*arctanh(a*x)*log(c*x^n),x, algorithm="fricas")
Output:
integral((e*x^2 + d)*arctanh(a*x)*log(c*x^n), x)
\[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {atanh}{\left (a x \right )}\, dx \] Input:
integrate((e*x**2+d)*atanh(a*x)*ln(c*x**n),x)
Output:
Integral((d + e*x**2)*log(c*x**n)*atanh(a*x), x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.97 \[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=-\frac {1}{36} \, n {\left (\frac {18 \, {\left (i \, \pi d - 2 \, d\right )} \log \left (x\right )}{a} + \frac {6 \, {\left (3 \, a^{2} d + e\right )} {\left (\log \left (a x - 1\right ) \log \left (a x\right ) + {\rm Li}_2\left (-a x + 1\right )\right )}}{a^{3}} + \frac {6 \, {\left (3 \, a^{2} d + e\right )} {\left (\log \left (a x + 1\right ) \log \left (-a x\right ) + {\rm Li}_2\left (a x + 1\right )\right )}}{a^{3}} + \frac {2 \, {\left (9 \, a^{2} d + e\right )} \log \left (a x + 1\right )}{a^{3}} + \frac {-2 i \, \pi a^{3} e x^{3} - 18 i \, \pi a^{3} d x + 5 \, a^{2} e x^{2} + 2 \, {\left (a^{3} e x^{3} + 9 \, a^{3} d x\right )} \log \left (a x + 1\right ) - 2 \, {\left (a^{3} e x^{3} + 9 \, a^{3} d x - 9 \, a^{2} d - e\right )} \log \left (a x - 1\right )}{a^{3}}\right )} + \frac {1}{36} \, {\left ({\left (6 \, x^{3} \log \left (a x + 1\right ) - a {\left (\frac {2 \, a^{2} x^{3} - 3 \, a x^{2} + 6 \, x}{a^{3}} - \frac {6 \, \log \left (a x + 1\right )}{a^{4}}\right )}\right )} e - {\left (6 \, x^{3} \log \left (-a x + 1\right ) - a {\left (\frac {2 \, a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{a^{3}} + \frac {6 \, \log \left (a x - 1\right )}{a^{4}}\right )}\right )} e - \frac {18 \, {\left (a x - {\left (a x + 1\right )} \log \left (a x + 1\right ) + 1\right )} d}{a} + \frac {18 \, {\left (a x - {\left (a x - 1\right )} \log \left (-a x + 1\right ) - 1\right )} d}{a}\right )} \log \left (c x^{n}\right ) \] Input:
integrate((e*x^2+d)*arctanh(a*x)*log(c*x^n),x, algorithm="maxima")
Output:
-1/36*n*(18*(I*pi*d - 2*d)*log(x)/a + 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) + dilo g(a*x + 1))/a^3 + 2*(9*a^2*d + e)*log(a*x + 1)/a^3 + (-2*I*pi*a^3*e*x^3 - 18*I*pi*a^3*d*x + 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) + 1/36*((6*x^3*l og(a*x + 1) - a*((2*a^2*x^3 - 3*a*x^2 + 6*x)/a^3 - 6*log(a*x + 1)/a^4))*e - (6*x^3*log(-a*x + 1) - a*((2*a^2*x^3 + 3*a*x^2 + 6*x)/a^3 + 6*log(a*x - 1)/a^4))*e - 18*(a*x - (a*x + 1)*log(a*x + 1) + 1)*d/a + 18*(a*x - (a*x - 1)*log(-a*x + 1) - 1)*d/a)*log(c*x^n)
\[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \operatorname {artanh}\left (a x\right ) \log \left (c x^{n}\right ) \,d x } \] Input:
integrate((e*x^2+d)*arctanh(a*x)*log(c*x^n),x, algorithm="giac")
Output:
integrate((e*x^2 + d)*arctanh(a*x)*log(c*x^n), x)
Timed out. \[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {atanh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \] Input:
int(log(c*x^n)*atanh(a*x)*(d + e*x^2),x)
Output:
int(log(c*x^n)*atanh(a*x)*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) \text {arctanh}(a x) \log \left (c x^n\right ) \, dx=\frac {36 \mathit {atanh} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} d n x +12 \mathit {atanh} \left (a x \right ) \mathrm {log}\left (x^{n} c \right ) a^{3} e n \,x^{3}-36 \mathit {atanh} \left (a x \right ) a^{3} d \,n^{2} x -4 \mathit {atanh} \left (a x \right ) a^{3} e \,n^{2} x^{3}-36 \mathit {atanh} \left (a x \right ) a^{2} d \,n^{2}-4 \mathit {atanh} \left (a x \right ) e \,n^{2}+36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}-x}d x \right ) a^{2} d n +12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{a^{2} x^{3}-x}d x \right ) e n -36 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} d \,n^{2}-4 \,\mathrm {log}\left (a^{2} x -a \right ) e \,n^{2}+18 \mathrm {log}\left (x^{n} c \right )^{2} a^{2} d +6 \mathrm {log}\left (x^{n} c \right )^{2} e +6 \,\mathrm {log}\left (x^{n} c \right ) a^{2} e n \,x^{2}-5 a^{2} e \,n^{2} x^{2}}{36 a^{3} n} \] Input:
int((e*x^2+d)*atanh(a*x)*log(c*x^n),x)
Output:
(36*atanh(a*x)*log(x**n*c)*a**3*d*n*x + 12*atanh(a*x)*log(x**n*c)*a**3*e*n *x**3 - 36*atanh(a*x)*a**3*d*n**2*x - 4*atanh(a*x)*a**3*e*n**2*x**3 - 36*a tanh(a*x)*a**2*d*n**2 - 4*atanh(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)