Integrand size = 20, antiderivative size = 151 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {b c d (a+b \text {arctanh}(c x))}{x}+\frac {3}{2} c^2 d (a+b \text {arctanh}(c x))^2-\frac {d (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {c d (a+b \text {arctanh}(c x))^2}{x}+b^2 c^2 d \log (x)-\frac {1}{2} b^2 c^2 d \log \left (1-c^2 x^2\right )+2 b c^2 d (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b^2 c^2 d \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
-b*c*d*(a+b*arctanh(c*x))/x+3/2*c^2*d*(a+b*arctanh(c*x))^2-1/2*d*(a+b*arct anh(c*x))^2/x^2-c*d*(a+b*arctanh(c*x))^2/x+b^2*c^2*d*ln(x)-1/2*b^2*c^2*d*l n(-c^2*x^2+1)+2*b*c^2*d*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b^2*c^2*d*polyl og(2,-1+2/(c*x+1))
Time = 0.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {d \left (a^2+2 a^2 c x+2 a b c x+b^2 \left (1+2 c x-3 c^2 x^2\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a+2 a c x+b c x-2 b c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-4 a b c^2 x^2 \log (c x)+a b c^2 x^2 \log (1-c x)-a b c^2 x^2 \log (1+c x)-2 b^2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+2 a b c^2 x^2 \log \left (1-c^2 x^2\right )+2 b^2 c^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{2 x^2} \]
-1/2*(d*(a^2 + 2*a^2*c*x + 2*a*b*c*x + b^2*(1 + 2*c*x - 3*c^2*x^2)*ArcTanh [c*x]^2 + 2*b*ArcTanh[c*x]*(a + 2*a*c*x + b*c*x - 2*b*c^2*x^2*Log[1 - E^(- 2*ArcTanh[c*x])]) - 4*a*b*c^2*x^2*Log[c*x] + a*b*c^2*x^2*Log[1 - c*x] - a* b*c^2*x^2*Log[1 + c*x] - 2*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 2*a* b*c^2*x^2*Log[1 - c^2*x^2] + 2*b^2*c^2*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])] ))/x^2
Time = 0.57 (sec) , antiderivative size = 151, 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.
| \(\displaystyle \int \frac {(c d x+d) (a+b \text {arctanh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))^2}{x^3}+\frac {c d (a+b \text {arctanh}(c x))^2}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} c^2 d (a+b \text {arctanh}(c x))^2+2 b c^2 d \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {d (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {c d (a+b \text {arctanh}(c x))^2}{x}-\frac {b c d (a+b \text {arctanh}(c x))}{x}-b^2 c^2 d \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )-\frac {1}{2} b^2 c^2 d \log \left (1-c^2 x^2\right )+b^2 c^2 d \log (x)\) |
-((b*c*d*(a + b*ArcTanh[c*x]))/x) + (3*c^2*d*(a + b*ArcTanh[c*x])^2)/2 - ( d*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (c*d*(a + b*ArcTanh[c*x])^2)/x + b^2*c ^2*d*Log[x] - (b^2*c^2*d*Log[1 - c^2*x^2])/2 + 2*b*c^2*d*(a + b*ArcTanh[c* x])*Log[2 - 2/(1 + c*x)] - b^2*c^2*d*PolyLog[2, -1 + 2/(1 + c*x)]
3.1.74.3.1 Defintions of rubi rules used
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])
Time = 1.25 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.83
| method | result | size |
| parts | \(a^{2} d \left (-\frac {c}{x}-\frac {1}{2 x^{2}}\right )+b^{2} d \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}+2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\operatorname {dilog}\left (c x +1\right )-\ln \left (c x \right ) \ln \left (c x +1\right )-\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {3 \ln \left (c x -1\right )^{2}}{8}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c x +1\right )^{2}}{8}\right )+2 a b d \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\ln \left (c x +1\right )}{4}-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 c x}\right )\) | \(276\) |
| derivativedivides | \(c^{2} \left (a^{2} d \left (-\frac {1}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b^{2} d \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}+2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\operatorname {dilog}\left (c x +1\right )-\ln \left (c x \right ) \ln \left (c x +1\right )-\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {3 \ln \left (c x -1\right )^{2}}{8}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c x +1\right )^{2}}{8}\right )+2 a b d \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\ln \left (c x +1\right )}{4}-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 c x}\right )\right )\) | \(279\) |
| default | \(c^{2} \left (a^{2} d \left (-\frac {1}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b^{2} d \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}+2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\operatorname {dilog}\left (c x +1\right )-\ln \left (c x \right ) \ln \left (c x +1\right )-\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {3 \ln \left (c x -1\right )^{2}}{8}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c x +1\right )^{2}}{8}\right )+2 a b d \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\ln \left (c x +1\right )}{4}-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 c x}\right )\right )\) | \(279\) |
a^2*d*(-c/x-1/2/x^2)+b^2*d*c^2*(-1/c/x*arctanh(c*x)^2-1/2/c^2/x^2*arctanh( c*x)^2-1/2*arctanh(c*x)*ln(c*x+1)-3/2*arctanh(c*x)*ln(c*x-1)+2*ln(c*x)*arc tanh(c*x)-1/c/x*arctanh(c*x)-1/2*ln(c*x+1)-1/2*ln(c*x-1)+ln(c*x)-dilog(c*x +1)-ln(c*x)*ln(c*x+1)-dilog(c*x)+dilog(1/2*c*x+1/2)+3/4*ln(c*x-1)*ln(1/2*c *x+1/2)-3/8*ln(c*x-1)^2-1/4*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1 /8*ln(c*x+1)^2)+2*a*b*d*c^2*(-1/c/x*arctanh(c*x)-1/2/c^2/x^2*arctanh(c*x)- 1/4*ln(c*x+1)-3/4*ln(c*x-1)+ln(c*x)-1/2/c/x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral((a^2*c*d*x + a^2*d + (b^2*c*d*x + b^2*d)*arctanh(c*x)^2 + 2*(a*b* c*d*x + a*b*d)*arctanh(c*x))/x^3, x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=d \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {a^{2} c}{x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
d*(Integral(a**2/x**3, x) + Integral(a**2*c/x**2, x) + Integral(b**2*atanh (c*x)**2/x**3, x) + Integral(2*a*b*atanh(c*x)/x**3, x) + Integral(b**2*c*a tanh(c*x)**2/x**2, x) + Integral(2*a*b*c*atanh(c*x)/x**2, x))
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*c*d - 1/4*b^2*c* d*(log(-c*x + 1)^2/x + integrate(-((c*x - 1)*log(c*x + 1)^2 + 2*(c*x - (c* x - 1)*log(c*x + 1))*log(-c*x + 1))/(c*x^3 - x^2), x)) + 1/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*d + 1/8*((2*(log(c *x - 1) - 2)*log(c*x + 1) - log(c*x + 1)^2 - log(c*x - 1)^2 - 4*log(c*x - 1) + 8*log(x))*c^2 + 4*(c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c*arctanh(c *x))*b^2*d - a^2*c*d/x - 1/2*b^2*d*arctanh(c*x)^2/x^2 - 1/2*a^2*d/x^2
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right )}{x^3} \,d x \]