Integrand size = 20, antiderivative size = 156 \[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=-\frac {b c d^4}{2 x}+4 a c^3 d^4 x+\frac {1}{2} b c^3 d^4 x+4 b c^3 d^4 x \text {arctanh}(c x)-\frac {d^4 (a+b \text {arctanh}(c x))}{2 x^2}-\frac {4 c d^4 (a+b \text {arctanh}(c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \text {arctanh}(c x))+6 a c^2 d^4 \log (x)+4 b c^2 d^4 \log (x)-3 b c^2 d^4 \operatorname {PolyLog}(2,-c x)+3 b c^2 d^4 \operatorname {PolyLog}(2,c x) \]
-1/2*b*c*d^4/x+4*a*c^3*d^4*x+1/2*b*c^3*d^4*x+4*b*c^3*d^4*x*arctanh(c*x)-1/ 2*d^4*(a+b*arctanh(c*x))/x^2-4*c*d^4*(a+b*arctanh(c*x))/x+1/2*c^4*d^4*x^2* (a+b*arctanh(c*x))+6*a*c^2*d^4*ln(x)+4*b*c^2*d^4*ln(x)-3*b*c^2*d^4*polylog (2,-c*x)+3*b*c^2*d^4*polylog(2,c*x)
Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=\frac {d^4 \left (-a-8 a c x-b c x+8 a c^3 x^3+b c^3 x^3+a c^4 x^4-b \text {arctanh}(c x)-8 b c x \text {arctanh}(c x)+8 b c^3 x^3 \text {arctanh}(c x)+b c^4 x^4 \text {arctanh}(c x)+12 a c^2 x^2 \log (x)+8 b c^2 x^2 \log (c x)-6 b c^2 x^2 \operatorname {PolyLog}(2,-c x)+6 b c^2 x^2 \operatorname {PolyLog}(2,c x)\right )}{2 x^2} \]
(d^4*(-a - 8*a*c*x - b*c*x + 8*a*c^3*x^3 + b*c^3*x^3 + a*c^4*x^4 - b*ArcTa nh[c*x] - 8*b*c*x*ArcTanh[c*x] + 8*b*c^3*x^3*ArcTanh[c*x] + b*c^4*x^4*ArcT anh[c*x] + 12*a*c^2*x^2*Log[x] + 8*b*c^2*x^2*Log[c*x] - 6*b*c^2*x^2*PolyLo g[2, -(c*x)] + 6*b*c^2*x^2*PolyLog[2, c*x]))/(2*x^2)
Time = 0.40 (sec) , antiderivative size = 156, 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.
| \(\displaystyle \int \frac {(c d x+d)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (c^4 d^4 x (a+b \text {arctanh}(c x))+4 c^3 d^4 (a+b \text {arctanh}(c x))+\frac {6 c^2 d^4 (a+b \text {arctanh}(c x))}{x}+\frac {d^4 (a+b \text {arctanh}(c x))}{x^3}+\frac {4 c d^4 (a+b \text {arctanh}(c x))}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} c^4 d^4 x^2 (a+b \text {arctanh}(c x))-\frac {d^4 (a+b \text {arctanh}(c x))}{2 x^2}-\frac {4 c d^4 (a+b \text {arctanh}(c x))}{x}+4 a c^3 d^4 x+6 a c^2 d^4 \log (x)+4 b c^3 d^4 x \text {arctanh}(c x)+\frac {1}{2} b c^3 d^4 x-3 b c^2 d^4 \operatorname {PolyLog}(2,-c x)+3 b c^2 d^4 \operatorname {PolyLog}(2,c x)+4 b c^2 d^4 \log (x)-\frac {b c d^4}{2 x}\) |
-1/2*(b*c*d^4)/x + 4*a*c^3*d^4*x + (b*c^3*d^4*x)/2 + 4*b*c^3*d^4*x*ArcTanh [c*x] - (d^4*(a + b*ArcTanh[c*x]))/(2*x^2) - (4*c*d^4*(a + b*ArcTanh[c*x]) )/x + (c^4*d^4*x^2*(a + b*ArcTanh[c*x]))/2 + 6*a*c^2*d^4*Log[x] + 4*b*c^2* d^4*Log[x] - 3*b*c^2*d^4*PolyLog[2, -(c*x)] + 3*b*c^2*d^4*PolyLog[2, c*x]
3.1.37.3.1 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 = 147, normalized size of antiderivative = 0.94
| method | result | size |
| parts | \(d^{4} a \left (\frac {c^{4} x^{2}}{2}+4 c^{3} x -\frac {4 c}{x}-\frac {1}{2 x^{2}}+6 c^{2} \ln \left (x \right )\right )+b \,d^{4} c^{2} \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+4 c x \,\operatorname {arctanh}\left (c x \right )+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )+\frac {c x}{2}+4 \ln \left (c x \right )-\frac {1}{2 c x}\right )\) | \(147\) |
| derivativedivides | \(c^{2} \left (d^{4} a \left (\frac {c^{2} x^{2}}{2}+4 c x +6 \ln \left (c x \right )-\frac {4}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b \,d^{4} \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+4 c x \,\operatorname {arctanh}\left (c x \right )+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )+\frac {c x}{2}+4 \ln \left (c x \right )-\frac {1}{2 c x}\right )\right )\) | \(150\) |
| default | \(c^{2} \left (d^{4} a \left (\frac {c^{2} x^{2}}{2}+4 c x +6 \ln \left (c x \right )-\frac {4}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b \,d^{4} \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}+4 c x \,\operatorname {arctanh}\left (c x \right )+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )+\frac {c x}{2}+4 \ln \left (c x \right )-\frac {1}{2 c x}\right )\right )\) | \(150\) |
| risch | \(4 d^{4} a \,c^{3} x -4 b \,c^{2} d^{4}-\frac {9 a \,c^{2} d^{4}}{2}-\frac {b c \,d^{4}}{2 x}+\frac {b \,c^{3} d^{4} x}{2}+\frac {c^{4} d^{4} a \,x^{2}}{2}-\frac {d^{4} a}{2 x^{2}}-\frac {4 c \,d^{4} a}{x}+6 c^{2} d^{4} \ln \left (-c x \right ) a +\frac {9 c^{2} d^{4} b \ln \left (-c x \right )}{4}+\frac {d^{4} b \ln \left (-c x +1\right )}{4 x^{2}}+3 c^{2} d^{4} \operatorname {dilog}\left (-c x +1\right ) b +\frac {7 b \,c^{2} d^{4} \ln \left (c x \right )}{4}-\frac {b \,d^{4} \ln \left (c x +1\right )}{4 x^{2}}-3 b \,c^{2} d^{4} \operatorname {dilog}\left (c x +1\right )-\frac {c^{4} d^{4} b \,x^{2} \ln \left (-c x +1\right )}{4}-2 c^{3} d^{4} b x \ln \left (-c x +1\right )+\frac {2 c \,d^{4} b \ln \left (-c x +1\right )}{x}+\frac {b \,c^{4} d^{4} \ln \left (c x +1\right ) x^{2}}{4}+2 b \,c^{3} d^{4} \ln \left (c x +1\right ) x -\frac {2 b c \,d^{4} \ln \left (c x +1\right )}{x}\) | \(287\) |
d^4*a*(1/2*c^4*x^2+4*c^3*x-4*c/x-1/2/x^2+6*c^2*ln(x))+b*d^4*c^2*(1/2*c^2*x ^2*arctanh(c*x)+4*c*x*arctanh(c*x)+6*ln(c*x)*arctanh(c*x)-4/c/x*arctanh(c* x)-1/2/c^2/x^2*arctanh(c*x)-3*dilog(c*x+1)-3*ln(c*x)*ln(c*x+1)-3*dilog(c*x )+1/2*c*x+4*ln(c*x)-1/2/c/x)
\[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{4} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
integral((a*c^4*d^4*x^4 + 4*a*c^3*d^4*x^3 + 6*a*c^2*d^4*x^2 + 4*a*c*d^4*x + a*d^4 + (b*c^4*d^4*x^4 + 4*b*c^3*d^4*x^3 + 6*b*c^2*d^4*x^2 + 4*b*c*d^4*x + b*d^4)*arctanh(c*x))/x^3, x)
\[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=d^{4} \left (\int 4 a c^{3}\, dx + \int \frac {a}{x^{3}}\, dx + \int \frac {4 a c}{x^{2}}\, dx + \int \frac {6 a c^{2}}{x}\, dx + \int a c^{4} x\, dx + \int 4 b c^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {4 b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {6 b c^{2} \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**4*(Integral(4*a*c**3, x) + Integral(a/x**3, x) + Integral(4*a*c/x**2, x ) + Integral(6*a*c**2/x, x) + Integral(a*c**4*x, x) + Integral(4*b*c**3*at anh(c*x), x) + Integral(b*atanh(c*x)/x**3, x) + Integral(4*b*c*atanh(c*x)/ x**2, x) + Integral(6*b*c**2*atanh(c*x)/x, x) + Integral(b*c**4*x*atanh(c* x), x))
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (146) = 292\).
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.88 \[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=\frac {1}{4} \, b c^{4} d^{4} x^{2} \log \left (c x + 1\right ) - \frac {1}{4} \, b c^{4} d^{4} x^{2} \log \left (-c x + 1\right ) + \frac {1}{2} \, a c^{4} d^{4} x^{2} + 4 \, a c^{3} d^{4} x + \frac {1}{2} \, b c^{3} d^{4} x + 2 \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c^{2} d^{4} - 3 \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b c^{2} d^{4} + 3 \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b c^{2} d^{4} - \frac {1}{4} \, b c^{2} d^{4} \log \left (c x + 1\right ) + \frac {1}{4} \, b c^{2} d^{4} \log \left (c x - 1\right ) + 6 \, a c^{2} d^{4} \log \left (x\right ) - 2 \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b c d^{4} + \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b d^{4} - \frac {4 \, a c d^{4}}{x} - \frac {a d^{4}}{2 \, x^{2}} \]
1/4*b*c^4*d^4*x^2*log(c*x + 1) - 1/4*b*c^4*d^4*x^2*log(-c*x + 1) + 1/2*a*c ^4*d^4*x^2 + 4*a*c^3*d^4*x + 1/2*b*c^3*d^4*x + 2*(2*c*x*arctanh(c*x) + log (-c^2*x^2 + 1))*b*c^2*d^4 - 3*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b *c^2*d^4 + 3*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))*b*c^2*d^4 - 1/4*b*c ^2*d^4*log(c*x + 1) + 1/4*b*c^2*d^4*log(c*x - 1) + 6*a*c^2*d^4*log(x) - 2* (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c*d^4 + 1/4*((c*log (c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*d^4 - 4*a*c*d^ 4/x - 1/2*a*d^4/x^2
\[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{4} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^4 (a+b \text {arctanh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^4}{x^3} \,d x \]