\(\int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^3} \, dx\) [16]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^3} \, dx=-\frac {b c d^2}{2 x}+\frac {1}{2} b c^2 d^2 \text {arctanh}(c x)-\frac {d^2 (a+b \text {arctanh}(c x))}{2 x^2}-\frac {2 c d^2 (a+b \text {arctanh}(c x))}{x}+a c^2 d^2 \log (x)+2 b c^2 d^2 \log (x)-b c^2 d^2 \log \left (1-c^2 x^2\right )-\frac {1}{2} b c^2 d^2 \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c^2 d^2 \operatorname {PolyLog}(2,c x) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^3} \, dx=\frac {d^2 \left (-2 a-8 a c x-2 b c x-2 b \text {arctanh}(c x)-8 b c x \text {arctanh}(c x)+4 a c^2 x^2 \log (x)+8 b c^2 x^2 \log (c x)-b c^2 x^2 \log (1-c x)+b c^2 x^2 \log (1+c x)-4 b c^2 x^2 \log \left (1-c^2 x^2\right )-2 b c^2 x^2 \operatorname {PolyLog}(2,-c x)+2 b c^2 x^2 \operatorname {PolyLog}(2,c x)\right )}{4 x^2} \] Input:

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

Output:

(d^2*(-2*a - 8*a*c*x - 2*b*c*x - 2*b*ArcTanh[c*x] - 8*b*c*x*ArcTanh[c*x] + 
 4*a*c^2*x^2*Log[x] + 8*b*c^2*x^2*Log[c*x] - b*c^2*x^2*Log[1 - c*x] + b*c^ 
2*x^2*Log[1 + c*x] - 4*b*c^2*x^2*Log[1 - c^2*x^2] - 2*b*c^2*x^2*PolyLog[2, 
 -(c*x)] + 2*b*c^2*x^2*PolyLog[2, c*x]))/(4*x^2)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 137, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(b*c*d^2)/x + (b*c^2*d^2*ArcTanh[c*x])/2 - (d^2*(a + b*ArcTanh[c*x])) 
/(2*x^2) - (2*c*d^2*(a + b*ArcTanh[c*x]))/x + a*c^2*d^2*Log[x] + 2*b*c^2*d 
^2*Log[x] - b*c^2*d^2*Log[1 - c^2*x^2] - (b*c^2*d^2*PolyLog[2, -(c*x)])/2 
+ (b*c^2*d^2*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 = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90

method	result	size
parts	\(d^{2} a \left (c^{2} \ln \left (x \right )-\frac {2 c}{x}-\frac {1}{2 x^{2}}\right )+d^{2} b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{2 c x}+2 \ln \left (c x \right )-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\)	\(123\)
derivativedivides	\(c^{2} \left (d^{2} a \left (-\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )-\frac {2}{c x}\right )+d^{2} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{2 c x}+2 \ln \left (c x \right )-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\right )\)	\(127\)
default	\(c^{2} \left (d^{2} a \left (-\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )-\frac {2}{c x}\right )+d^{2} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{2 c x}+2 \ln \left (c x \right )-\frac {3 \ln \left (c x +1\right )}{4}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\right )\)	\(127\)
risch	\(-\frac {d^{2} a}{2 x^{2}}+c^{2} d^{2} a \ln \left (-c x \right )-\frac {2 c \,d^{2} a}{x}+\frac {5 c^{2} d^{2} b \ln \left (-c x \right )}{4}-\frac {b c \,d^{2}}{2 x}-\frac {5 b \,c^{2} d^{2} \ln \left (-c x +1\right )}{4}+\frac {d^{2} b \ln \left (-c x +1\right )}{4 x^{2}}+\frac {c^{2} d^{2} b \operatorname {dilog}\left (-c x +1\right )}{2}+\frac {c \,d^{2} b \ln \left (-c x +1\right )}{x}+\frac {3 b \,c^{2} d^{2} \ln \left (c x \right )}{4}-\frac {3 b \,c^{2} d^{2} \ln \left (c x +1\right )}{4}-\frac {b \,d^{2} \ln \left (c x +1\right )}{4 x^{2}}-\frac {b \,c^{2} d^{2} \operatorname {dilog}\left (c x +1\right )}{2}-\frac {b c \,d^{2} \ln \left (c x +1\right )}{x}\)	\(196\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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