\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx\) [24]

Optimal result

Integrand size = 20, antiderivative size = 152 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=3 a c d^3 x+\frac {3}{2} b c d^3 x+\frac {1}{6} b c^2 d^3 x^2-\frac {3}{2} b d^3 \text {arctanh}(c x)+3 b c d^3 x \text {arctanh}(c x)+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))+\frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))+a d^3 \log (x)+\frac {5}{3} b d^3 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^3 \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b d^3 \operatorname {PolyLog}(2,c x) \] Output:

3*a*c*d^3*x+3/2*b*c*d^3*x+1/6*b*c^2*d^3*x^2-3/2*b*d^3*arctanh(c*x)+3*b*c*d 
^3*x*arctanh(c*x)+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))+1/3*c^3*d^3*x^3*(a+b* 
arctanh(c*x))+a*d^3*ln(x)+5/3*b*d^3*ln(-c^2*x^2+1)-1/2*b*d^3*polylog(2,-c* 
x)+1/2*b*d^3*polylog(2,c*x)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\frac {1}{12} d^3 \left (36 a c x+18 b c x+18 a c^2 x^2+2 b c^2 x^2+4 a c^3 x^3+36 b c x \text {arctanh}(c x)+18 b c^2 x^2 \text {arctanh}(c x)+4 b c^3 x^3 \text {arctanh}(c x)+12 a \log (x)+9 b \log (1-c x)-9 b \log (1+c x)+18 b \log \left (1-c^2 x^2\right )+2 b \log \left (-1+c^2 x^2\right )-6 b \operatorname {PolyLog}(2,-c x)+6 b \operatorname {PolyLog}(2,c x)\right ) \] Input:

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x,x]
 

Output:

(d^3*(36*a*c*x + 18*b*c*x + 18*a*c^2*x^2 + 2*b*c^2*x^2 + 4*a*c^3*x^3 + 36* 
b*c*x*ArcTanh[c*x] + 18*b*c^2*x^2*ArcTanh[c*x] + 4*b*c^3*x^3*ArcTanh[c*x] 
+ 12*a*Log[x] + 9*b*Log[1 - c*x] - 9*b*Log[1 + c*x] + 18*b*Log[1 - c^2*x^2 
] + 2*b*Log[-1 + c^2*x^2] - 6*b*PolyLog[2, -(c*x)] + 6*b*PolyLog[2, c*x])) 
/12
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6502, 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.

\(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))}{x} \, dx\)

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (c^3 d^3 x^2 (a+b \text {arctanh}(c x))+3 c^2 d^3 x (a+b \text {arctanh}(c x))+3 c d^3 (a+b \text {arctanh}(c x))+\frac {d^3 (a+b \text {arctanh}(c x))}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} c^3 d^3 x^3 (a+b \text {arctanh}(c x))+\frac {3}{2} c^2 d^3 x^2 (a+b \text {arctanh}(c x))+3 a c d^3 x+a d^3 \log (x)-\frac {3}{2} b d^3 \text {arctanh}(c x)+3 b c d^3 x \text {arctanh}(c x)+\frac {1}{6} b c^2 d^3 x^2+\frac {5}{3} b d^3 \log \left (1-c^2 x^2\right )-\frac {1}{2} b d^3 \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b d^3 \operatorname {PolyLog}(2,c x)+\frac {3}{2} b c d^3 x\)

Input:

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x,x]
 

Output:

3*a*c*d^3*x + (3*b*c*d^3*x)/2 + (b*c^2*d^3*x^2)/6 - (3*b*d^3*ArcTanh[c*x]) 
/2 + 3*b*c*d^3*x*ArcTanh[c*x] + (3*c^2*d^3*x^2*(a + b*ArcTanh[c*x]))/2 + ( 
c^3*d^3*x^3*(a + b*ArcTanh[c*x]))/3 + a*d^3*Log[x] + (5*b*d^3*Log[1 - c^2* 
x^2])/3 - (b*d^3*PolyLog[2, -(c*x)])/2 + (b*d^3*PolyLog[2, c*x])/2
 

Defintions of rubi rules used

rule 2009

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

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.86

method	result	size
parts	\(d^{3} a \left (\frac {x^{3} c^{3}}{3}+\frac {3 c^{2} x^{2}}{2}+3 c x +\ln \left (x \right )\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c^{2} x^{2}}{6}+\frac {3 c x}{2}+\frac {29 \ln \left (c x -1\right )}{12}+\frac {11 \ln \left (c x +1\right )}{12}\right )\)	\(131\)
derivativedivides	\(d^{3} a \left (\frac {x^{3} c^{3}}{3}+\frac {3 c^{2} x^{2}}{2}+3 c x +\ln \left (c x \right )\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c^{2} x^{2}}{6}+\frac {3 c x}{2}+\frac {29 \ln \left (c x -1\right )}{12}+\frac {11 \ln \left (c x +1\right )}{12}\right )\)	\(133\)
default	\(d^{3} a \left (\frac {x^{3} c^{3}}{3}+\frac {3 c^{2} x^{2}}{2}+3 c x +\ln \left (c x \right )\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c^{2} x^{2}}{6}+\frac {3 c x}{2}+\frac {29 \ln \left (c x -1\right )}{12}+\frac {11 \ln \left (c x +1\right )}{12}\right )\)	\(133\)
risch	\(-\frac {d^{3} b \ln \left (-c x +1\right ) c^{3} x^{3}}{6}-\frac {3 d^{3} b \ln \left (-c x +1\right ) c^{2} x^{2}}{4}-\frac {3 d^{3} b \ln \left (-c x +1\right ) c x}{2}+\frac {29 d^{3} b \ln \left (-c x +1\right )}{12}+\frac {b \,c^{2} d^{3} x^{2}}{6}+\frac {3 b c \,d^{3} x}{2}-\frac {65 d^{3} b}{18}+\frac {d^{3} b \operatorname {dilog}\left (-c x +1\right )}{2}+\frac {a \,c^{3} d^{3} x^{3}}{3}+\frac {3 a \,c^{2} d^{3} x^{2}}{2}+3 d^{3} a c x -\frac {29 d^{3} a}{6}+d^{3} a \ln \left (-c x \right )+\frac {d^{3} b \ln \left (c x +1\right ) c^{3} x^{3}}{6}+\frac {3 d^{3} b \ln \left (c x +1\right ) c^{2} x^{2}}{4}+\frac {3 d^{3} b \ln \left (c x +1\right ) c x}{2}+\frac {11 d^{3} b \ln \left (c x +1\right )}{12}-\frac {d^{3} b \operatorname {dilog}\left (c x +1\right )}{2}\)	\(229\)

Input:

int((c*d*x+d)^3*(a+b*arctanh(c*x))/x,x,method=_RETURNVERBOSE)
 

Output:

d^3*a*(1/3*x^3*c^3+3/2*c^2*x^2+3*c*x+ln(x))+d^3*b*(1/3*arctanh(c*x)*c^3*x^ 
3+3/2*arctanh(c*x)*c^2*x^2+3*arctanh(c*x)*c*x+arctanh(c*x)*ln(c*x)-1/2*dil 
og(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)+1/6*c^2*x^2+3/2*c*x+29/12*l 
n(c*x-1)+11/12*ln(c*x+1))
 

Fricas [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x,x, algorithm="fricas")
 

Output:

integral((a*c^3*d^3*x^3 + 3*a*c^2*d^3*x^2 + 3*a*c*d^3*x + a*d^3 + (b*c^3*d 
^3*x^3 + 3*b*c^2*d^3*x^2 + 3*b*c*d^3*x + b*d^3)*arctanh(c*x))/x, x)
 

Sympy [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=d^{3} \left (\int 3 a c\, dx + \int \frac {a}{x}\, dx + \int 3 a c^{2} x\, dx + \int a c^{3} x^{2}\, dx + \int 3 b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx + \int b c^{3} x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \] Input:

integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x,x)
 

Output:

d**3*(Integral(3*a*c, x) + Integral(a/x, x) + Integral(3*a*c**2*x, x) + In 
tegral(a*c**3*x**2, x) + Integral(3*b*c*atanh(c*x), x) + Integral(b*atanh( 
c*x)/x, x) + Integral(3*b*c**2*x*atanh(c*x), x) + Integral(b*c**3*x**2*ata 
nh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.50 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\frac {1}{3} \, a c^{3} d^{3} x^{3} + \frac {3}{2} \, a c^{2} d^{3} x^{2} + \frac {1}{6} \, b c^{2} d^{3} x^{2} + 3 \, a c d^{3} x + \frac {3}{2} \, b c d^{3} x + \frac {3}{2} \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{3} - \frac {1}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b d^{3} + \frac {1}{2} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b d^{3} - \frac {7}{12} \, b d^{3} \log \left (c x + 1\right ) + \frac {11}{12} \, b d^{3} \log \left (c x - 1\right ) + a d^{3} \log \left (x\right ) + \frac {1}{12} \, {\left (2 \, b c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2}\right )} \log \left (c x + 1\right ) - \frac {1}{12} \, {\left (2 \, b c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2}\right )} \log \left (-c x + 1\right ) \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x,x, algorithm="maxima")
 

Output:

1/3*a*c^3*d^3*x^3 + 3/2*a*c^2*d^3*x^2 + 1/6*b*c^2*d^3*x^2 + 3*a*c*d^3*x + 
3/2*b*c*d^3*x + 3/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*d^3 - 1/2*( 
log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b*d^3 + 1/2*(log(c*x + 1)*log(-c 
*x) + dilog(c*x + 1))*b*d^3 - 7/12*b*d^3*log(c*x + 1) + 11/12*b*d^3*log(c* 
x - 1) + a*d^3*log(x) + 1/12*(2*b*c^3*d^3*x^3 + 9*b*c^2*d^3*x^2)*log(c*x + 
 1) - 1/12*(2*b*c^3*d^3*x^3 + 9*b*c^2*d^3*x^2)*log(-c*x + 1)
 

Giac [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x,x, algorithm="giac")
 

Output:

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^3}{x} \,d x \] Input:

int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x,x)
 

Output:

int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x} \, dx=\frac {d^{3} \left (2 \mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}+9 \mathit {atanh} \left (c x \right ) b \,c^{2} x^{2}+18 \mathit {atanh} \left (c x \right ) b c x +11 \mathit {atanh} \left (c x \right ) b +6 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) b +20 \,\mathrm {log}\left (c^{2} x -c \right ) b +6 \,\mathrm {log}\left (x \right ) a +2 a \,c^{3} x^{3}+9 a \,c^{2} x^{2}+18 a c x +b \,c^{2} x^{2}+9 b c x \right )}{6} \] Input:

int((c*d*x+d)^3*(a+b*atanh(c*x))/x,x)
 

Output:

(d**3*(2*atanh(c*x)*b*c**3*x**3 + 9*atanh(c*x)*b*c**2*x**2 + 18*atanh(c*x) 
*b*c*x + 11*atanh(c*x)*b + 6*int(atanh(c*x)/x,x)*b + 20*log(c**2*x - c)*b 
+ 6*log(x)*a + 2*a*c**3*x**3 + 9*a*c**2*x**2 + 18*a*c*x + b*c**2*x**2 + 9* 
b*c*x))/6