Integrand size = 20, antiderivative size = 206 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}+\frac {1}{3} b^2 c^3 d \text {arctanh}(c x)-\frac {b c d (a+b \text {arctanh}(c x))}{3 x^2}-\frac {b c^2 d (a+b \text {arctanh}(c x))}{x}+\frac {5}{6} c^3 d (a+b \text {arctanh}(c x))^2-\frac {d (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {c d (a+b \text {arctanh}(c x))^2}{2 x^2}+b^2 c^3 d \log (x)-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )+\frac {2}{3} b c^3 d (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
-1/3*b^2*c^2*d/x+1/3*b^2*c^3*d*arctanh(c*x)-1/3*b*c*d*(a+b*arctanh(c*x))/x ^2-b*c^2*d*(a+b*arctanh(c*x))/x+5/6*c^3*d*(a+b*arctanh(c*x))^2-1/3*d*(a+b* arctanh(c*x))^2/x^3-1/2*c*d*(a+b*arctanh(c*x))^2/x^2+b^2*c^3*d*ln(x)-1/2*b ^2*c^3*d*ln(-c^2*x^2+1)+2/3*b*c^3*d*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-1/3 *b^2*c^3*d*polylog(2,-1+2/(c*x+1))
Time = 0.36 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.19 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {d \left (2 a^2+3 a^2 c x+2 a b c x+6 a b c^2 x^2+2 b^2 c^2 x^2+b^2 \left (2+3 c x-5 c^3 x^3\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a (2+3 c x)+b c x \left (1+3 c x-c^2 x^2\right )-2 b c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (1+c x)-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+2 a b c^3 x^3 \log \left (1-c^2 x^2\right )+2 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{6 x^3} \]
-1/6*(d*(2*a^2 + 3*a^2*c*x + 2*a*b*c*x + 6*a*b*c^2*x^2 + 2*b^2*c^2*x^2 + b ^2*(2 + 3*c*x - 5*c^3*x^3)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*(2 + 3*c*x ) + b*c*x*(1 + 3*c*x - c^2*x^2) - 2*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])] ) - 4*a*b*c^3*x^3*Log[c*x] + 3*a*b*c^3*x^3*Log[1 - c*x] - 3*a*b*c^3*x^3*Lo g[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 2*a*b*c^3*x^3*Lo g[1 - c^2*x^2] + 2*b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^3
Time = 0.68 (sec) , antiderivative size = 206, 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.
\(\displaystyle \int \frac {(c d x+d) (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))^2}{x^4}+\frac {c d (a+b \text {arctanh}(c x))^2}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{6} c^3 d (a+b \text {arctanh}(c x))^2+\frac {2}{3} b c^3 d \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {b c^2 d (a+b \text {arctanh}(c x))}{x}-\frac {d (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {c d (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {b c d (a+b \text {arctanh}(c x))}{3 x^2}+\frac {1}{3} b^2 c^3 d \text {arctanh}(c x)-\frac {1}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+b^2 c^3 d \log (x)-\frac {b^2 c^2 d}{3 x}-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )\) |
-1/3*(b^2*c^2*d)/x + (b^2*c^3*d*ArcTanh[c*x])/3 - (b*c*d*(a + b*ArcTanh[c* x]))/(3*x^2) - (b*c^2*d*(a + b*ArcTanh[c*x]))/x + (5*c^3*d*(a + b*ArcTanh[ c*x])^2)/6 - (d*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (c*d*(a + b*ArcTanh[c*x] )^2)/(2*x^2) + b^2*c^3*d*Log[x] - (b^2*c^3*d*Log[1 - c^2*x^2])/2 + (2*b*c^ 3*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/3 - (b^2*c^3*d*PolyLog[2, - 1 + 2/(1 + c*x)])/3
3.1.75.3.1 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 = 1.42 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.50
method | result | size |
parts | \(a^{2} d \left (-\frac {1}{3 x^{3}}-\frac {c}{2 x^{2}}\right )+b^{2} d \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\ln \left (c x +1\right )}{3}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )\right )+2 a b d \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\ln \left (c x +1\right )}{12}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}\right )\) | \(308\) |
derivativedivides | \(c^{3} \left (a^{2} d \left (-\frac {1}{2 c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+b^{2} d \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\ln \left (c x +1\right )}{3}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )\right )+2 a b d \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\ln \left (c x +1\right )}{12}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}\right )\right )\) | \(311\) |
default | \(c^{3} \left (a^{2} d \left (-\frac {1}{2 c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+b^{2} d \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\ln \left (c x +1\right )}{3}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )\right )+2 a b d \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\ln \left (c x +1\right )}{12}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}\right )\right )\) | \(311\) |
a^2*d*(-1/3/x^3-1/2*c/x^2)+b^2*d*c^3*(-1/2/c^2/x^2*arctanh(c*x)^2-1/3/c^3/ x^3*arctanh(c*x)^2+1/6*arctanh(c*x)*ln(c*x+1)-5/6*arctanh(c*x)*ln(c*x-1)-1 /3/c^2/x^2*arctanh(c*x)-1/c/x*arctanh(c*x)+2/3*ln(c*x)*arctanh(c*x)+1/12*( ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/3*dilog(1/2*c*x+1/2)-1/24*ln (c*x+1)^2-1/3*dilog(c*x+1)-1/3*ln(c*x)*ln(c*x+1)-1/3*dilog(c*x)+5/12*ln(c* x-1)*ln(1/2*c*x+1/2)-5/24*ln(c*x-1)^2-1/3*ln(c*x+1)-2/3*ln(c*x-1)-1/3/c/x+ ln(c*x))+2*a*b*d*c^3*(-1/2/c^2/x^2*arctanh(c*x)-1/3/c^3/x^3*arctanh(c*x)+1 /12*ln(c*x+1)-5/12*ln(c*x-1)-1/6/c^2/x^2-1/2/c/x+1/3*ln(c*x))
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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^4, x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=d \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {a^{2} c}{x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx\right ) \]
d*(Integral(a**2/x**4, x) + Integral(a**2*c/x**3, x) + Integral(b**2*atanh (c*x)**2/x**4, x) + Integral(2*a*b*atanh(c*x)/x**4, x) + Integral(b**2*c*a tanh(c*x)**2/x**3, x) + Integral(2*a*b*c*atanh(c*x)/x**3, x))
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (187) = 374\).
Time = 0.62 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.02 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d + \frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, b^{2} c^{3} d \log \left (c x + 1\right ) - \frac {2}{3} \, b^{2} c^{3} d \log \left (c x - 1\right ) + b^{2} c^{3} d \log \left (x\right ) + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c d - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b d - \frac {a^{2} c d}{2 \, x^{2}} - \frac {a^{2} d}{3 \, x^{3}} - \frac {8 \, b^{2} c^{2} d x^{2} - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )^{2} - {\left (5 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (3 \, b^{2} c^{2} d x^{2} + b^{2} c d x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{24 \, x^{3}} \]
-1/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^3*d - 1/3*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^3*d + 1/3*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^3*d - 1/3*b^2*c^3*d*log(c*x + 1) - 2 /3*b^2*c^3*d*log(c*x - 1) + b^2*c^3*d*log(x) + 1/2*((c*log(c*x + 1) - c*lo g(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c*d - 1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*a*b*d - 1/2*a^2*c*d/x ^2 - 1/3*a^2*d/x^3 - 1/24*(8*b^2*c^2*d*x^2 - (b^2*c^3*d*x^3 - 3*b^2*c*d*x - 2*b^2*d)*log(c*x + 1)^2 - (5*b^2*c^3*d*x^3 - 3*b^2*c*d*x - 2*b^2*d)*log( -c*x + 1)^2 + 4*(3*b^2*c^2*d*x^2 + b^2*c*d*x)*log(c*x + 1) - 2*(6*b^2*c^2* d*x^2 + 2*b^2*c*d*x - (b^2*c^3*d*x^3 - 3*b^2*c*d*x - 2*b^2*d)*log(c*x + 1) )*log(-c*x + 1))/x^3
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right )}{x^4} \,d x \]