Integrand size = 20, antiderivative size = 176 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=-\frac {b c d^3}{6 x^2}-\frac {3 b c^2 d^3}{2 x}+\frac {3}{2} b c^3 d^3 \text {arctanh}(c x)-\frac {d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{2 x^2}-\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{x}+a c^3 d^3 \log (x)+\frac {10}{3} b c^3 d^3 \log (x)-\frac {5}{3} b c^3 d^3 \log \left (1-c^2 x^2\right )-\frac {1}{2} b c^3 d^3 \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c^3 d^3 \operatorname {PolyLog}(2,c x) \]
-1/6*b*c*d^3/x^2-3/2*b*c^2*d^3/x+3/2*b*c^3*d^3*arctanh(c*x)-1/3*d^3*(a+b*a rctanh(c*x))/x^3-3/2*c*d^3*(a+b*arctanh(c*x))/x^2-3*c^2*d^3*(a+b*arctanh(c *x))/x+a*c^3*d^3*ln(x)+10/3*b*c^3*d^3*ln(x)-5/3*b*c^3*d^3*ln(-c^2*x^2+1)-1 /2*b*c^3*d^3*polylog(2,-c*x)+1/2*b*c^3*d^3*polylog(2,c*x)
Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=\frac {d^3 \left (-4 a-18 a c x-2 b c x-36 a c^2 x^2-18 b c^2 x^2-4 b \text {arctanh}(c x)-18 b c x \text {arctanh}(c x)-36 b c^2 x^2 \text {arctanh}(c x)+12 a c^3 x^3 \log (x)+40 b c^3 x^3 \log (c x)-9 b c^3 x^3 \log (1-c x)+9 b c^3 x^3 \log (1+c x)-20 b c^3 x^3 \log \left (1-c^2 x^2\right )-6 b c^3 x^3 \operatorname {PolyLog}(2,-c x)+6 b c^3 x^3 \operatorname {PolyLog}(2,c x)\right )}{12 x^3} \]
(d^3*(-4*a - 18*a*c*x - 2*b*c*x - 36*a*c^2*x^2 - 18*b*c^2*x^2 - 4*b*ArcTan h[c*x] - 18*b*c*x*ArcTanh[c*x] - 36*b*c^2*x^2*ArcTanh[c*x] + 12*a*c^3*x^3* Log[x] + 40*b*c^3*x^3*Log[c*x] - 9*b*c^3*x^3*Log[1 - c*x] + 9*b*c^3*x^3*Lo g[1 + c*x] - 20*b*c^3*x^3*Log[1 - c^2*x^2] - 6*b*c^3*x^3*PolyLog[2, -(c*x) ] + 6*b*c^3*x^3*PolyLog[2, c*x]))/(12*x^3)
Time = 0.42 (sec) , antiderivative size = 176, 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.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {c^3 d^3 (a+b \text {arctanh}(c x))}{x}+\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{x^2}+\frac {d^3 (a+b \text {arctanh}(c x))}{x^4}+\frac {3 c d^3 (a+b \text {arctanh}(c x))}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{x}-\frac {d^3 (a+b \text {arctanh}(c x))}{3 x^3}-\frac {3 c d^3 (a+b \text {arctanh}(c x))}{2 x^2}+a c^3 d^3 \log (x)+\frac {3}{2} b c^3 d^3 \text {arctanh}(c x)-\frac {1}{2} b c^3 d^3 \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c^3 d^3 \operatorname {PolyLog}(2,c x)+\frac {10}{3} b c^3 d^3 \log (x)-\frac {3 b c^2 d^3}{2 x}-\frac {5}{3} b c^3 d^3 \log \left (1-c^2 x^2\right )-\frac {b c d^3}{6 x^2}\) |
-1/6*(b*c*d^3)/x^2 - (3*b*c^2*d^3)/(2*x) + (3*b*c^3*d^3*ArcTanh[c*x])/2 - (d^3*(a + b*ArcTanh[c*x]))/(3*x^3) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(2*x^2 ) - (3*c^2*d^3*(a + b*ArcTanh[c*x]))/x + a*c^3*d^3*Log[x] + (10*b*c^3*d^3* Log[x])/3 - (5*b*c^3*d^3*Log[1 - c^2*x^2])/3 - (b*c^3*d^3*PolyLog[2, -(c*x )])/2 + (b*c^3*d^3*PolyLog[2, c*x])/2
3.1.27.3.1 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 = 1.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86
| method | result | size |
| parts | \(d^{3} a \left (-\frac {3 c^{2}}{x}-\frac {1}{3 x^{3}}-\frac {3 c}{2 x^{2}}+c^{3} \ln \left (x \right )\right )+d^{3} b \,c^{3} \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {11 \ln \left (c x +1\right )}{12}-\frac {29 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {3}{2 c x}+\frac {10 \ln \left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\) | \(151\) |
| derivativedivides | \(c^{3} \left (d^{3} a \left (\ln \left (c x \right )-\frac {3}{c x}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{3} b \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {11 \ln \left (c x +1\right )}{12}-\frac {29 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {3}{2 c x}+\frac {10 \ln \left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\right )\) | \(155\) |
| default | \(c^{3} \left (d^{3} a \left (\ln \left (c x \right )-\frac {3}{c x}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{3} b \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {11 \ln \left (c x +1\right )}{12}-\frac {29 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {3}{2 c x}+\frac {10 \ln \left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\right )\) | \(155\) |
| risch | \(-\frac {b c \,d^{3}}{6 x^{2}}-\frac {3 b \,c^{2} d^{3}}{2 x}+\frac {29 c^{3} d^{3} b \ln \left (-c x \right )}{12}-\frac {29 \ln \left (-c x +1\right ) b \,c^{3} d^{3}}{12}+\frac {d^{3} b \ln \left (-c x +1\right )}{6 x^{3}}+\frac {3 c \,d^{3} b \ln \left (-c x +1\right )}{4 x^{2}}+\frac {3 c^{2} d^{3} b \ln \left (-c x +1\right )}{2 x}+\frac {c^{3} d^{3} b \operatorname {dilog}\left (-c x +1\right )}{2}-\frac {d^{3} a}{3 x^{3}}-\frac {3 c \,d^{3} a}{2 x^{2}}-\frac {3 c^{2} d^{3} a}{x}+c^{3} d^{3} \ln \left (-c x \right ) a +\frac {11 d^{3} b \,c^{3} \ln \left (c x \right )}{12}-\frac {11 \ln \left (c x +1\right ) b \,c^{3} d^{3}}{12}-\frac {d^{3} b \ln \left (c x +1\right )}{6 x^{3}}-\frac {3 d^{3} b c \ln \left (c x +1\right )}{4 x^{2}}-\frac {3 d^{3} b \,c^{2} \ln \left (c x +1\right )}{2 x}-\frac {d^{3} b \,c^{3} \operatorname {dilog}\left (c x +1\right )}{2}\) | \(258\) |
d^3*a*(-3*c^2/x-1/3/x^3-3/2*c/x^2+c^3*ln(x))+d^3*b*c^3*(ln(c*x)*arctanh(c* x)-3/c/x*arctanh(c*x)-3/2/c^2/x^2*arctanh(c*x)-1/3/c^3/x^3*arctanh(c*x)-11 /12*ln(c*x+1)-29/12*ln(c*x-1)-1/6/c^2/x^2-3/2/c/x+10/3*ln(c*x)-1/2*dilog(c *x+1)-1/2*ln(c*x)*ln(c*x+1)-1/2*dilog(c*x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
integral((a*c^3*d^3*x^3 + 3*a*c^2*d^3*x^2 + 3*a*c*d^3*x + a*d^3 + (b*c^3*d ^3*x^3 + 3*b*c^2*d^3*x^2 + 3*b*c*d^3*x + b*d^3)*arctanh(c*x))/x^4, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=d^{3} \left (\int \frac {a}{x^{4}}\, dx + \int \frac {3 a c}{x^{3}}\, dx + \int \frac {3 a c^{2}}{x^{2}}\, dx + \int \frac {a c^{3}}{x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
d**3*(Integral(a/x**4, x) + Integral(3*a*c/x**3, x) + Integral(3*a*c**2/x* *2, x) + Integral(a*c**3/x, x) + Integral(b*atanh(c*x)/x**4, x) + Integral (3*b*c*atanh(c*x)/x**3, x) + Integral(3*b*c**2*atanh(c*x)/x**2, x) + Integ ral(b*c**3*atanh(c*x)/x, x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
1/2*b*c^3*d^3*integrate((log(c*x + 1) - log(-c*x + 1))/x, x) + a*c^3*d^3*l og(x) - 3/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c^2*d^3 + 3/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b* c*d^3 - 1/6*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c *x)/x^3)*b*d^3 - 3*a*c^2*d^3/x - 3/2*a*c*d^3/x^2 - 1/3*a*d^3/x^3
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^3}{x^4} \,d x \]