Integrand size = 20, antiderivative size = 268 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=-\frac {b c}{2 d^3 x}+\frac {b c^2}{8 d^3 (1+c x)^2}+\frac {13 b c^2}{8 d^3 (1+c x)}-\frac {9 b c^2 \text {arctanh}(c x)}{8 d^3}-\frac {a+b \text {arctanh}(c x)}{2 d^3 x^2}+\frac {3 c (a+b \text {arctanh}(c x))}{d^3 x}+\frac {c^2 (a+b \text {arctanh}(c x))}{2 d^3 (1+c x)^2}+\frac {3 c^2 (a+b \text {arctanh}(c x))}{d^3 (1+c x)}+\frac {6 a c^2 \log (x)}{d^3}-\frac {3 b c^2 \log (x)}{d^3}+\frac {6 c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c^2 \log \left (1-c^2 x^2\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}(2,-c x)}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}(2,c x)}{d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^3} \]
-1/2*b*c/d^3/x+1/8*b*c^2/d^3/(c*x+1)^2+13/8*b*c^2/d^3/(c*x+1)-9/8*b*c^2*ar ctanh(c*x)/d^3+1/2*(-a-b*arctanh(c*x))/d^3/x^2+3*c*(a+b*arctanh(c*x))/d^3/ x+1/2*c^2*(a+b*arctanh(c*x))/d^3/(c*x+1)^2+3*c^2*(a+b*arctanh(c*x))/d^3/(c *x+1)+6*a*c^2*ln(x)/d^3-3*b*c^2*ln(x)/d^3+6*c^2*(a+b*arctanh(c*x))*ln(2/(c *x+1))/d^3+3/2*b*c^2*ln(-c^2*x^2+1)/d^3-3*b*c^2*polylog(2,-c*x)/d^3+3*b*c^ 2*polylog(2,c*x)/d^3-3*b*c^2*polylog(2,1-2/(c*x+1))/d^3
Time = 1.20 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\frac {-\frac {16 a}{x^2}+\frac {96 a c}{x}+\frac {16 a c^2}{(1+c x)^2}+\frac {96 a c^2}{1+c x}+192 a c^2 \log (x)-192 a c^2 \log (1+c x)+b c^2 \left (-\frac {16}{c x}+28 \cosh (2 \text {arctanh}(c x))+\cosh (4 \text {arctanh}(c x))-96 \log (c x)+48 \log \left (1-c^2 x^2\right )-96 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-28 \sinh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (4-\frac {4}{c^2 x^2}+\frac {24}{c x}+14 \cosh (2 \text {arctanh}(c x))+\cosh (4 \text {arctanh}(c x))+48 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-14 \sinh (2 \text {arctanh}(c x))-\sinh (4 \text {arctanh}(c x))\right )-\sinh (4 \text {arctanh}(c x))\right )}{32 d^3} \]
((-16*a)/x^2 + (96*a*c)/x + (16*a*c^2)/(1 + c*x)^2 + (96*a*c^2)/(1 + c*x) + 192*a*c^2*Log[x] - 192*a*c^2*Log[1 + c*x] + b*c^2*(-16/(c*x) + 28*Cosh[2 *ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 96*Log[c*x] + 48*Log[1 - c^2*x^2] - 96*PolyLog[2, E^(-2*ArcTanh[c*x])] - 28*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh [c*x]*(4 - 4/(c^2*x^2) + 24/(c*x) + 14*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTa nh[c*x]] + 48*Log[1 - E^(-2*ArcTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] - Sin h[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c*x]]))/(32*d^3)
Time = 0.58 (sec) , antiderivative size = 268, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^3 (c d x+d)^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {6 c^3 (a+b \text {arctanh}(c x))}{d^3 (c x+1)}-\frac {3 c^3 (a+b \text {arctanh}(c x))}{d^3 (c x+1)^2}-\frac {c^3 (a+b \text {arctanh}(c x))}{d^3 (c x+1)^3}+\frac {6 c^2 (a+b \text {arctanh}(c x))}{d^3 x}+\frac {a+b \text {arctanh}(c x)}{d^3 x^3}-\frac {3 c (a+b \text {arctanh}(c x))}{d^3 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c^2 (a+b \text {arctanh}(c x))}{d^3 (c x+1)}+\frac {c^2 (a+b \text {arctanh}(c x))}{2 d^3 (c x+1)^2}+\frac {6 c^2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {a+b \text {arctanh}(c x)}{2 d^3 x^2}+\frac {3 c (a+b \text {arctanh}(c x))}{d^3 x}+\frac {6 a c^2 \log (x)}{d^3}-\frac {9 b c^2 \text {arctanh}(c x)}{8 d^3}-\frac {3 b c^2 \operatorname {PolyLog}(2,-c x)}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}(2,c x)}{d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{d^3}+\frac {3 b c^2 \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {13 b c^2}{8 d^3 (c x+1)}+\frac {b c^2}{8 d^3 (c x+1)^2}-\frac {3 b c^2 \log (x)}{d^3}-\frac {b c}{2 d^3 x}\) |
-1/2*(b*c)/(d^3*x) + (b*c^2)/(8*d^3*(1 + c*x)^2) + (13*b*c^2)/(8*d^3*(1 + c*x)) - (9*b*c^2*ArcTanh[c*x])/(8*d^3) - (a + b*ArcTanh[c*x])/(2*d^3*x^2) + (3*c*(a + b*ArcTanh[c*x]))/(d^3*x) + (c^2*(a + b*ArcTanh[c*x]))/(2*d^3*( 1 + c*x)^2) + (3*c^2*(a + b*ArcTanh[c*x]))/(d^3*(1 + c*x)) + (6*a*c^2*Log[ x])/d^3 - (3*b*c^2*Log[x])/d^3 + (6*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c* x)])/d^3 + (3*b*c^2*Log[1 - c^2*x^2])/(2*d^3) - (3*b*c^2*PolyLog[2, -(c*x) ])/d^3 + (3*b*c^2*PolyLog[2, c*x])/d^3 - (3*b*c^2*PolyLog[2, 1 - 2/(1 + c* x)])/d^3
3.1.65.3.1 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 = 1.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a \left (\frac {1}{2 \left (c x +1\right )^{2}}+\frac {3}{c x +1}-6 \ln \left (c x +1\right )-\frac {1}{2 c^{2} x^{2}}+\frac {3}{c x}+6 \ln \left (c x \right )\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )-3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x +1\right )^{2}}{2}+\frac {1}{8 \left (c x +1\right )^{2}}+\frac {13}{8 \left (c x +1\right )}+\frac {15 \ln \left (c x +1\right )}{16}+\frac {33 \ln \left (c x -1\right )}{16}-\frac {1}{2 c x}-3 \ln \left (c x \right )\right )}{d^{3}}\right )\) | \(256\) |
| default | \(c^{2} \left (\frac {a \left (\frac {1}{2 \left (c x +1\right )^{2}}+\frac {3}{c x +1}-6 \ln \left (c x +1\right )-\frac {1}{2 c^{2} x^{2}}+\frac {3}{c x}+6 \ln \left (c x \right )\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )-3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x +1\right )^{2}}{2}+\frac {1}{8 \left (c x +1\right )^{2}}+\frac {13}{8 \left (c x +1\right )}+\frac {15 \ln \left (c x +1\right )}{16}+\frac {33 \ln \left (c x -1\right )}{16}-\frac {1}{2 c x}-3 \ln \left (c x \right )\right )}{d^{3}}\right )\) | \(256\) |
| parts | \(\frac {a \left (\frac {c^{2}}{2 \left (c x +1\right )^{2}}+\frac {3 c^{2}}{c x +1}-6 c^{2} \ln \left (c x +1\right )-\frac {1}{2 x^{2}}+\frac {3 c}{x}+6 c^{2} \ln \left (x \right )\right )}{d^{3}}+\frac {b \,c^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\frac {3 \,\operatorname {arctanh}\left (c x \right )}{c x}+6 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-3 \operatorname {dilog}\left (c x +1\right )-3 \ln \left (c x \right ) \ln \left (c x +1\right )-3 \operatorname {dilog}\left (c x \right )-3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {3 \ln \left (c x +1\right )^{2}}{2}+\frac {1}{8 \left (c x +1\right )^{2}}+\frac {13}{8 \left (c x +1\right )}+\frac {15 \ln \left (c x +1\right )}{16}+\frac {33 \ln \left (c x -1\right )}{16}-\frac {1}{2 c x}-3 \ln \left (c x \right )\right )}{d^{3}}\) | \(260\) |
| risch | \(-\frac {a}{2 d^{3} x^{2}}-\frac {b c}{2 d^{3} x}+\frac {b \,c^{2}}{8 d^{3} \left (c x +1\right )^{2}}+\frac {3 b \,c^{2}}{2 d^{3} \left (c x +1\right )}+\frac {7 b \,c^{2} \ln \left (c x +1\right )}{4 d^{3}}+\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{d^{3}}-\frac {3 c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{3}}-\frac {3 c b \ln \left (-c x +1\right )}{2 d^{3} x}+\frac {3 c^{2} b \ln \left (-c x +1\right )}{4 d^{3} \left (-c x -1\right )}-\frac {3 c^{2} b \ln \left (-c x +1\right )}{16 d^{3} \left (-c x -1\right )^{2}}+\frac {3 b \,c^{2} \ln \left (c x +1\right )}{2 d^{3} \left (c x +1\right )}+\frac {3 b c \ln \left (c x +1\right )}{2 d^{3} x}+\frac {b \,c^{2} \ln \left (c x +1\right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {c^{2} b}{8 d^{3} \left (-c x -1\right )}+\frac {c^{2} a}{2 d^{3} \left (-c x -1\right )^{2}}-\frac {3 c^{2} a}{d^{3} \left (-c x -1\right )}+\frac {3 c a}{d^{3} x}-\frac {5 c^{2} b \ln \left (-c x \right )}{4 d^{3}}+\frac {5 c^{2} b \ln \left (-c x +1\right )}{4 d^{3}}+\frac {b \ln \left (-c x +1\right )}{4 d^{3} x^{2}}-\frac {3 c^{2} b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{3}}-\frac {13 c^{2} b \ln \left (-c x -1\right )}{16 d^{3}}+\frac {3 c^{2} \operatorname {dilog}\left (-c x +1\right ) b}{d^{3}}-\frac {6 c^{2} a \ln \left (-c x -1\right )}{d^{3}}+\frac {6 c^{2} \ln \left (-c x \right ) a}{d^{3}}-\frac {7 b \,c^{2} \ln \left (c x \right )}{4 d^{3}}-\frac {b \ln \left (c x +1\right )}{4 d^{3} x^{2}}-\frac {3 b \,c^{2} \operatorname {dilog}\left (c x +1\right )}{d^{3}}-\frac {3 b \,c^{2} \ln \left (c x +1\right )^{2}}{2 d^{3}}-\frac {3 c^{3} b \ln \left (-c x +1\right ) x}{4 d^{3} \left (-c x -1\right )}+\frac {c^{4} b \ln \left (-c x +1\right ) x^{2}}{16 d^{3} \left (-c x -1\right )^{2}}+\frac {c^{3} b \ln \left (-c x +1\right ) x}{8 d^{3} \left (-c x -1\right )^{2}}\) | \(561\) |
c^2*(a/d^3*(1/2/(c*x+1)^2+3/(c*x+1)-6*ln(c*x+1)-1/2/c^2/x^2+3/c/x+6*ln(c*x ))+b/d^3*(1/2/(c*x+1)^2*arctanh(c*x)+3/(c*x+1)*arctanh(c*x)-6*arctanh(c*x) *ln(c*x+1)-1/2/c^2/x^2*arctanh(c*x)+3/c/x*arctanh(c*x)+6*ln(c*x)*arctanh(c *x)-3*dilog(c*x+1)-3*ln(c*x)*ln(c*x+1)-3*dilog(c*x)-3*(ln(c*x+1)-ln(1/2*c* x+1/2))*ln(-1/2*c*x+1/2)+3*dilog(1/2*c*x+1/2)+3/2*ln(c*x+1)^2+1/8/(c*x+1)^ 2+13/8/(c*x+1)+15/16*ln(c*x+1)+33/16*ln(c*x-1)-1/2/c/x-3*ln(c*x)))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{3}} \,d x } \]
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\frac {\int \frac {a}{c^{3} x^{6} + 3 c^{2} x^{5} + 3 c x^{4} + x^{3}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{6} + 3 c^{2} x^{5} + 3 c x^{4} + x^{3}}\, dx}{d^{3}} \]
(Integral(a/(c**3*x**6 + 3*c**2*x**5 + 3*c*x**4 + x**3), x) + Integral(b*a tanh(c*x)/(c**3*x**6 + 3*c**2*x**5 + 3*c*x**4 + x**3), x))/d**3
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{3}} \,d x } \]
1/2*a*((12*c^3*x^3 + 18*c^2*x^2 + 4*c*x - 1)/(c^2*d^3*x^4 + 2*c*d^3*x^3 + d^3*x^2) - 12*c^2*log(c*x + 1)/d^3 + 12*c^2*log(x)/d^3) + 1/2*b*integrate( (log(c*x + 1) - log(-c*x + 1))/(c^3*d^3*x^6 + 3*c^2*d^3*x^5 + 3*c*d^3*x^4 + d^3*x^3), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)^3} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^3\,{\left (d+c\,d\,x\right )}^3} \,d x \]