\(\int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^2} \, dx\) [57]

Optimal result

Integrand size = 20, antiderivative size = 212 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^2} \, dx=-\frac {b c}{2 d^2 x}+\frac {b c^2}{2 d^2 (1+c x)}-\frac {a+b \text {arctanh}(c x)}{2 d^2 x^2}+\frac {2 c (a+b \text {arctanh}(c x))}{d^2 x}+\frac {c^2 (a+b \text {arctanh}(c x))}{d^2 (1+c x)}+\frac {3 a c^2 \log (x)}{d^2}-\frac {2 b c^2 \log (x)}{d^2}+\frac {3 c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{d^2}-\frac {3 b c^2 \operatorname {PolyLog}(2,-c x)}{2 d^2}+\frac {3 b c^2 \operatorname {PolyLog}(2,c x)}{2 d^2}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^2} \, dx=\frac {-\frac {2 a}{x^2}+\frac {8 a c}{x}-\frac {2 b c}{x}+\frac {4 a c^2}{1+c x}+b c^2 \cosh (2 \text {arctanh}(c x))+12 a c^2 \log (x)-12 a c^2 \log (1+c x)-8 b c^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-6 b c^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-b c^2 \sinh (2 \text {arctanh}(c x))+2 b \text {arctanh}(c x) \left (c^2-\frac {1}{x^2}+\frac {4 c}{x}+c^2 \cosh (2 \text {arctanh}(c x))+6 c^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-c^2 \sinh (2 \text {arctanh}(c x))\right )}{4 d^2} \] Input:

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

Output:

((-2*a)/x^2 + (8*a*c)/x - (2*b*c)/x + (4*a*c^2)/(1 + c*x) + b*c^2*Cosh[2*A 
rcTanh[c*x]] + 12*a*c^2*Log[x] - 12*a*c^2*Log[1 + c*x] - 8*b*c^2*Log[(c*x) 
/Sqrt[1 - c^2*x^2]] - 6*b*c^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - b*c^2*Sinh 
[2*ArcTanh[c*x]] + 2*b*ArcTanh[c*x]*(c^2 - x^(-2) + (4*c)/x + c^2*Cosh[2*A 
rcTanh[c*x]] + 6*c^2*Log[1 - E^(-2*ArcTanh[c*x])] - c^2*Sinh[2*ArcTanh[c*x 
]]))/(4*d^2)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 212, 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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (-\frac {3 c^3 (a+b \text {arctanh}(c x))}{d^2 (c x+1)}-\frac {c^3 (a+b \text {arctanh}(c x))}{d^2 (c x+1)^2}+\frac {3 c^2 (a+b \text {arctanh}(c x))}{d^2 x}+\frac {a+b \text {arctanh}(c x)}{d^2 x^3}-\frac {2 c (a+b \text {arctanh}(c x))}{d^2 x^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(b*c)/(d^2*x) + (b*c^2)/(2*d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])/(2*d 
^2*x^2) + (2*c*(a + b*ArcTanh[c*x]))/(d^2*x) + (c^2*(a + b*ArcTanh[c*x]))/ 
(d^2*(1 + c*x)) + (3*a*c^2*Log[x])/d^2 - (2*b*c^2*Log[x])/d^2 + (3*c^2*(a 
+ b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 + (b*c^2*Log[1 - c^2*x^2])/d^2 - ( 
3*b*c^2*PolyLog[2, -(c*x)])/(2*d^2) + (3*b*c^2*PolyLog[2, c*x])/(2*d^2) - 
(3*b*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^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 = 0.49 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+\frac {2}{c x}+3 \ln \left (c x \right )+\frac {1}{c x +1}-3 \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {3 \ln \left (c x +1\right )^{2}}{4}-\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\ln \left (c x -1\right )-\frac {1}{2 c x}-2 \ln \left (c x \right )+\frac {1}{2 c x +2}+\ln \left (c x +1\right )\right )}{d^{2}}\right )\)	\(218\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+\frac {2}{c x}+3 \ln \left (c x \right )+\frac {1}{c x +1}-3 \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {3 \ln \left (c x +1\right )^{2}}{4}-\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\ln \left (c x -1\right )-\frac {1}{2 c x}-2 \ln \left (c x \right )+\frac {1}{2 c x +2}+\ln \left (c x +1\right )\right )}{d^{2}}\right )\)	\(218\)
parts	\(\frac {a \left (\frac {c^{2}}{c x +1}-3 c^{2} \ln \left (c x +1\right )-\frac {1}{2 x^{2}}+\frac {2 c}{x}+3 c^{2} \ln \left (x \right )\right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \operatorname {dilog}\left (c x \right )}{2}-\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}-\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {3 \ln \left (c x +1\right )^{2}}{4}-\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\ln \left (c x -1\right )-\frac {1}{2 c x}-2 \ln \left (c x \right )+\frac {1}{2 c x +2}+\ln \left (c x +1\right )\right )}{d^{2}}\)	\(220\)
risch	\(-\frac {c^{3} b \ln \left (-c x +1\right ) x}{4 d^{2} \left (-c x -1\right )}+\frac {5 c^{2} b \ln \left (c x +1\right )}{4 d^{2}}+\frac {c^{2} b \ln \left (-c x +1\right )}{4 d^{2} \left (-c x -1\right )}+\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{2}}-\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}-\frac {c b \ln \left (-c x +1\right )}{d^{2} x}+\frac {c^{2} b \ln \left (c x +1\right )}{2 d^{2} \left (c x +1\right )}+\frac {c b \ln \left (c x +1\right )}{d^{2} x}-\frac {a}{2 d^{2} x^{2}}+\frac {2 c a}{d^{2} x}-\frac {c^{2} a}{d^{2} \left (-c x -1\right )}+\frac {3 c^{2} a \ln \left (-c x \right )}{d^{2}}-\frac {3 c^{2} a \ln \left (-c x -1\right )}{d^{2}}-\frac {3 c^{2} b \ln \left (-c x \right )}{4 d^{2}}+\frac {3 c^{2} b \ln \left (-c x +1\right )}{4 d^{2}}+\frac {b \ln \left (-c x +1\right )}{4 d^{2} x^{2}}-\frac {c^{2} b \ln \left (-c x -1\right )}{4 d^{2}}+\frac {3 c^{2} b \operatorname {dilog}\left (-c x +1\right )}{2 d^{2}}-\frac {3 c^{2} b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{2}}-\frac {3 c^{2} b \ln \left (c x +1\right )^{2}}{4 d^{2}}-\frac {5 c^{2} b \ln \left (c x \right )}{4 d^{2}}-\frac {b \ln \left (c x +1\right )}{4 d^{2} x^{2}}-\frac {3 c^{2} b \operatorname {dilog}\left (c x +1\right )}{2 d^{2}}-\frac {b c}{2 d^{2} x}+\frac {b \,c^{2}}{2 d^{2} \left (c x +1\right )}\)	\(412\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arctanh(c*x) + a)/(c^2*d^2*x^5 + 2*c*d^2*x^4 + d^2*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^2} \, dx=\frac {\int \frac {a}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a/(c**2*x**5 + 2*c*x**4 + x**3), x) + Integral(b*atanh(c*x)/(c** 
2*x**5 + 2*c*x**4 + x**3), x))/d**2
 

Maxima [F]

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

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

Output:

-1/2*a*(6*c^2*log(c*x + 1)/d^2 - 6*c^2*log(x)/d^2 - (6*c^2*x^2 + 3*c*x - 1 
)/(c*d^2*x^3 + d^2*x^2)) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/ 
(c^2*d^2*x^5 + 2*c*d^2*x^4 + d^2*x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^2} \, dx=\frac {-6 \mathit {atanh} \left (c x \right )^{2} b \,c^{3} x^{3}-6 \mathit {atanh} \left (c x \right )^{2} b \,c^{2} x^{2}-12 \mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}+12 \mathit {atanh} \left (c x \right ) b c x +4 \mathit {atanh} \left (c x \right ) b -24 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{6}+c^{2} x^{5}-c \,x^{4}-x^{3}}d x \right ) b c \,x^{3}-24 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{6}+c^{2} x^{5}-c \,x^{4}-x^{3}}d x \right ) b \,x^{2}+\mathrm {log}\left (c x -1\right ) b \,c^{3} x^{3}+\mathrm {log}\left (c x -1\right ) b \,c^{2} x^{2}-48 \,\mathrm {log}\left (c x +1\right ) a \,c^{3} x^{3}-48 \,\mathrm {log}\left (c x +1\right ) a \,c^{2} x^{2}+7 \,\mathrm {log}\left (c x +1\right ) b \,c^{3} x^{3}+7 \,\mathrm {log}\left (c x +1\right ) b \,c^{2} x^{2}+48 \,\mathrm {log}\left (x \right ) a \,c^{3} x^{3}+48 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-8 \,\mathrm {log}\left (x \right ) b \,c^{3} x^{3}-8 \,\mathrm {log}\left (x \right ) b \,c^{2} x^{2}-48 a \,c^{3} x^{3}+24 a c x -8 a -6 b \,c^{3} x^{3}+4 b c x}{16 d^{2} x^{2} \left (c x +1\right )} \] Input:

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

Output:

( - 6*atanh(c*x)**2*b*c**3*x**3 - 6*atanh(c*x)**2*b*c**2*x**2 - 12*atanh(c 
*x)*b*c**3*x**3 + 12*atanh(c*x)*b*c*x + 4*atanh(c*x)*b - 24*int(atanh(c*x) 
/(c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b*c*x**3 - 24*int(atanh(c*x)/( 
c**3*x**6 + c**2*x**5 - c*x**4 - x**3),x)*b*x**2 + log(c*x - 1)*b*c**3*x** 
3 + log(c*x - 1)*b*c**2*x**2 - 48*log(c*x + 1)*a*c**3*x**3 - 48*log(c*x + 
1)*a*c**2*x**2 + 7*log(c*x + 1)*b*c**3*x**3 + 7*log(c*x + 1)*b*c**2*x**2 + 
 48*log(x)*a*c**3*x**3 + 48*log(x)*a*c**2*x**2 - 8*log(x)*b*c**3*x**3 - 8* 
log(x)*b*c**2*x**2 - 48*a*c**3*x**3 + 24*a*c*x - 8*a - 6*b*c**3*x**3 + 4*b 
*c*x)/(16*d**2*x**2*(c*x + 1))