\(\int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx\) [6]

Optimal result

Integrand size = 18, antiderivative size = 70 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=-\frac {d (a+b \text {arctanh}(c x))}{x}+a c d \log (x)+b c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )-\frac {1}{2} b c d \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c d \operatorname {PolyLog}(2,c x) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=-\frac {a d}{x}+a c d \log (x)+b c d \left (-\frac {\text {arctanh}(c x)}{c x}+\log (c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )+\frac {1}{2} b c d (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x)) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d*(a + b*ArcTanh[c*x]))/x) + a*c*d*Log[x] + b*c*d*Log[x] - (b*c*d*Log[1 
 - c^2*x^2])/2 - (b*c*d*PolyLog[2, -(c*x)])/2 + (b*c*d*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 = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24

method	result	size
parts	\(a d \left (c \ln \left (x \right )-\frac {1}{x}\right )+d b c \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\)	\(87\)
derivativedivides	\(c \left (a d \left (\ln \left (c x \right )-\frac {1}{c x}\right )+d b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\right )\)	\(91\)
default	\(c \left (a d \left (\ln \left (c x \right )-\frac {1}{c x}\right )+d b \left (\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )\right )\)	\(91\)
risch	\(\frac {c d b \operatorname {dilog}\left (-c x +1\right )}{2}+\frac {c d b \ln \left (-c x \right )}{2}-\frac {\ln \left (-c x +1\right ) b c d}{2}+\frac {d b \ln \left (-c x +1\right )}{2 x}+c d a \ln \left (-c x \right )-\frac {a d}{x}-\frac {b c d \operatorname {dilog}\left (c x +1\right )}{2}+\frac {b c d \ln \left (c x \right )}{2}-\frac {\ln \left (c x +1\right ) b c d}{2}-\frac {b d \ln \left (c x +1\right )}{2 x}\)	\(110\)

Input:

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

Output:

a*d*(c*ln(x)-1/x)+d*b*c*(arctanh(c*x)*ln(c*x)-arctanh(c*x)/c/x-1/2*ln(c*x- 
1)+ln(c*x)-1/2*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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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