\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^2} \, dx\) [25]

Optimal result

Integrand size = 20, antiderivative size = 150 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^2} \, dx=3 a c^2 d^3 x+\frac {1}{2} b c^2 d^3 x-\frac {1}{2} b c d^3 \text {arctanh}(c x)+3 b c^2 d^3 x \text {arctanh}(c x)-\frac {d^3 (a+b \text {arctanh}(c x))}{x}+\frac {1}{2} c^3 d^3 x^2 (a+b \text {arctanh}(c x))+3 a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1-c^2 x^2\right )-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,c x) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^2} \, dx=\frac {d^3 \left (-4 a+12 a c^2 x^2+2 b c^2 x^2+2 a c^3 x^3-4 b \text {arctanh}(c x)+12 b c^2 x^2 \text {arctanh}(c x)+2 b c^3 x^3 \text {arctanh}(c x)+12 a c x \log (x)+4 b c x \log (c x)+b c x \log (1-c x)-b c x \log (1+c x)+4 b c x \log \left (1-c^2 x^2\right )-6 b c x \operatorname {PolyLog}(2,-c x)+6 b c x \operatorname {PolyLog}(2,c x)\right )}{4 x} \] Input:

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 150, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^2,x]
 

Output:

3*a*c^2*d^3*x + (b*c^2*d^3*x)/2 - (b*c*d^3*ArcTanh[c*x])/2 + 3*b*c^2*d^3*x 
*ArcTanh[c*x] - (d^3*(a + b*ArcTanh[c*x]))/x + (c^3*d^3*x^2*(a + b*ArcTanh 
[c*x]))/2 + 3*a*c*d^3*Log[x] + b*c*d^3*Log[x] + b*c*d^3*Log[1 - c^2*x^2] - 
 (3*b*c*d^3*PolyLog[2, -(c*x)])/2 + (3*b*c*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 = 0.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87

method	result	size
parts	\(d^{3} a \left (\frac {c^{3} x^{2}}{2}+3 c^{2} x +3 c \ln \left (x \right )-\frac {1}{x}\right )+d^{3} b c \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )\)	\(131\)
derivativedivides	\(c \left (d^{3} a \left (\frac {c^{2} x^{2}}{2}+3 c x +3 \ln \left (c x \right )-\frac {1}{c x}\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )\right )\)	\(134\)
default	\(c \left (d^{3} a \left (\frac {c^{2} x^{2}}{2}+3 c x +3 \ln \left (c x \right )-\frac {1}{c x}\right )+d^{3} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+3 \,\operatorname {arctanh}\left (c x \right ) c x +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )\right )\)	\(134\)
risch	\(-\frac {c^{3} d^{3} b \ln \left (-c x +1\right ) x^{2}}{4}-\frac {3 c^{2} d^{3} b \ln \left (-c x +1\right ) x}{2}+\frac {5 c \,d^{3} b \ln \left (-c x +1\right )}{4}+\frac {b \,c^{2} d^{3} x}{2}-3 b c \,d^{3}+\frac {3 c \,d^{3} b \operatorname {dilog}\left (-c x +1\right )}{2}+\frac {c \,d^{3} b \ln \left (-c x \right )}{2}+\frac {d^{3} b \ln \left (-c x +1\right )}{2 x}+\frac {c^{3} d^{3} a \,x^{2}}{2}+3 a \,c^{2} d^{3} x -\frac {7 c \,d^{3} a}{2}+3 c \,d^{3} a \ln \left (-c x \right )-\frac {d^{3} a}{x}+\frac {b \,c^{3} d^{3} \ln \left (c x +1\right ) x^{2}}{4}+\frac {3 b \,c^{2} d^{3} \ln \left (c x +1\right ) x}{2}+\frac {3 b c \,d^{3} \ln \left (c x +1\right )}{4}-\frac {3 b c \,d^{3} \operatorname {dilog}\left (c x +1\right )}{2}+\frac {b c \,d^{3} \ln \left (c x \right )}{2}-\frac {b \,d^{3} \ln \left (c x +1\right )}{2 x}\)	\(247\)

Input:

int((c*d*x+d)^3*(a+b*arctanh(c*x))/x^2,x,method=_RETURNVERBOSE)
 

Output:

d^3*a*(1/2*c^3*x^2+3*c^2*x+3*c*ln(x)-1/x)+d^3*b*c*(1/2*arctanh(c*x)*c^2*x^ 
2+3*arctanh(c*x)*c*x+3*arctanh(c*x)*ln(c*x)-arctanh(c*x)/c/x-3/2*dilog(c*x 
)-3/2*dilog(c*x+1)-3/2*ln(c*x)*ln(c*x+1)+1/2*c*x+5/4*ln(c*x-1)+ln(c*x)+3/4 
*ln(c*x+1))
 

Fricas [F]

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

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

Sympy [F]

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

integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.53 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^2} \, dx=\frac {1}{4} \, b c^{3} d^{3} x^{2} \log \left (c x + 1\right ) - \frac {1}{4} \, b c^{3} d^{3} x^{2} \log \left (-c x + 1\right ) + \frac {1}{2} \, a c^{3} d^{3} x^{2} + 3 \, a c^{2} d^{3} x + \frac {1}{2} \, b c^{2} d^{3} x + \frac {3}{2} \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{3} - \frac {3}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b c d^{3} + \frac {3}{2} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b c d^{3} - \frac {1}{4} \, b c d^{3} \log \left (c x + 1\right ) + \frac {1}{4} \, b c d^{3} \log \left (c x - 1\right ) + 3 \, a c d^{3} \log \left (x\right ) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^2,x, algorithm="maxima")
 

Output:

1/4*b*c^3*d^3*x^2*log(c*x + 1) - 1/4*b*c^3*d^3*x^2*log(-c*x + 1) + 1/2*a*c 
^3*d^3*x^2 + 3*a*c^2*d^3*x + 1/2*b*c^2*d^3*x + 3/2*(2*c*x*arctanh(c*x) + l 
og(-c^2*x^2 + 1))*b*c*d^3 - 3/2*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1)) 
*b*c*d^3 + 3/2*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b*c*d^3 - 1/4*b*c 
*d^3*log(c*x + 1) + 1/4*b*c*d^3*log(c*x - 1) + 3*a*c*d^3*log(x) - 1/2*(c*( 
log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*d^3 - a*d^3/x
 

Giac [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x^2,x)
 

Output:

int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x^2, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^2} \, dx=\frac {d^{3} \left (\mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}+6 \mathit {atanh} \left (c x \right ) b \,c^{2} x^{2}+3 \mathit {atanh} \left (c x \right ) b c x -2 \mathit {atanh} \left (c x \right ) b +6 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) b c x +4 \,\mathrm {log}\left (c^{2} x -c \right ) b c x +6 \,\mathrm {log}\left (x \right ) a c x +2 \,\mathrm {log}\left (x \right ) b c x +a \,c^{3} x^{3}+6 a \,c^{2} x^{2}-2 a +b \,c^{2} x^{2}\right )}{2 x} \] Input:

int((c*d*x+d)^3*(a+b*atanh(c*x))/x^2,x)
 

Output:

(d**3*(atanh(c*x)*b*c**3*x**3 + 6*atanh(c*x)*b*c**2*x**2 + 3*atanh(c*x)*b* 
c*x - 2*atanh(c*x)*b + 6*int(atanh(c*x)/x,x)*b*c*x + 4*log(c**2*x - c)*b*c 
*x + 6*log(x)*a*c*x + 2*log(x)*b*c*x + a*c**3*x**3 + 6*a*c**2*x**2 - 2*a + 
 b*c**2*x**2))/(2*x)