Integrand size = 20, antiderivative size = 149 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\frac {a x}{c^2 d^2}-\frac {b}{2 c^3 d^2 (1+c x)}+\frac {b \text {arctanh}(c x)}{2 c^3 d^2}+\frac {b x \text {arctanh}(c x)}{c^2 d^2}-\frac {a+b \text {arctanh}(c x)}{c^3 d^2 (1+c x)}+\frac {2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^3 d^2} \]
a*x/c^2/d^2-1/2*b/c^3/d^2/(c*x+1)+1/2*b*arctanh(c*x)/c^3/d^2+b*x*arctanh(c *x)/c^2/d^2+(-a-b*arctanh(c*x))/c^3/d^2/(c*x+1)+2*(a+b*arctanh(c*x))*ln(2/ (c*x+1))/c^3/d^2+1/2*b*ln(-c^2*x^2+1)/c^3/d^2-b*polylog(2,1-2/(c*x+1))/c^3 /d^2
Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\frac {4 a c x-\frac {4 a}{1+c x}-8 a \log (1+c x)+b \left (-\cosh (2 \text {arctanh}(c x))+2 \log \left (1-c^2 x^2\right )-4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \left (2 c x-\cosh (2 \text {arctanh}(c x))+4 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))\right )\right )}{4 c^3 d^2} \]
(4*a*c*x - (4*a)/(1 + c*x) - 8*a*Log[1 + c*x] + b*(-Cosh[2*ArcTanh[c*x]] + 2*Log[1 - c^2*x^2] - 4*PolyLog[2, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[ c*x]] + 2*ArcTanh[c*x]*(2*c*x - Cosh[2*ArcTanh[c*x]] + 4*Log[1 + E^(-2*Arc Tanh[c*x])] + Sinh[2*ArcTanh[c*x]])))/(4*c^3*d^2)
Time = 0.41 (sec) , antiderivative size = 149, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arctanh}(c x))}{(c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {2 (a+b \text {arctanh}(c x))}{c^2 d^2 (c x+1)}+\frac {a+b \text {arctanh}(c x)}{c^2 d^2}+\frac {a+b \text {arctanh}(c x)}{c^2 d^2 (c x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a+b \text {arctanh}(c x)}{c^3 d^2 (c x+1)}+\frac {2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^3 d^2}+\frac {a x}{c^2 d^2}+\frac {b \text {arctanh}(c x)}{2 c^3 d^2}+\frac {b x \text {arctanh}(c x)}{c^2 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{c^3 d^2}-\frac {b}{2 c^3 d^2 (c x+1)}+\frac {b \log \left (1-c^2 x^2\right )}{2 c^3 d^2}\) |
(a*x)/(c^2*d^2) - b/(2*c^3*d^2*(1 + c*x)) + (b*ArcTanh[c*x])/(2*c^3*d^2) + (b*x*ArcTanh[c*x])/(c^2*d^2) - (a + b*ArcTanh[c*x])/(c^3*d^2*(1 + c*x)) + (2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^3*d^2) + (b*Log[1 - c^2*x^2] )/(2*c^3*d^2) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(c^3*d^2)
3.1.52.3.1 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 = 1.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {\frac {a \left (c x -2 \ln \left (c x +1\right )-\frac {1}{c x +1}\right )}{d^{2}}+\frac {b \left (c x \,\operatorname {arctanh}\left (c x \right )-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}-\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {\ln \left (c x +1\right )^{2}}{2}\right )}{d^{2}}}{c^{3}}\) | \(137\) |
| default | \(\frac {\frac {a \left (c x -2 \ln \left (c x +1\right )-\frac {1}{c x +1}\right )}{d^{2}}+\frac {b \left (c x \,\operatorname {arctanh}\left (c x \right )-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}-\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {\ln \left (c x +1\right )^{2}}{2}\right )}{d^{2}}}{c^{3}}\) | \(137\) |
| parts | \(\frac {a \left (\frac {x}{c^{2}}-\frac {2 \ln \left (c x +1\right )}{c^{3}}-\frac {1}{c^{3} \left (c x +1\right )}\right )}{d^{2}}+\frac {b \left (c x \,\operatorname {arctanh}\left (c x \right )-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}-\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )+\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {\ln \left (c x +1\right )^{2}}{2}\right )}{d^{2} c^{3}}\) | \(144\) |
| risch | \(-\frac {b \ln \left (c x +1\right )^{2}}{2 c^{3} d^{2}}+\left (\frac {b x}{2 c^{2} d^{2}}-\frac {b}{2 c^{3} d^{2} \left (c x +1\right )}\right ) \ln \left (c x +1\right )+\frac {b \ln \left (c x +1\right )}{2 c^{3} d^{2}}-\frac {b}{2 c^{3} d^{2} \left (c x +1\right )}+\frac {a x}{c^{2} d^{2}}-\frac {a}{c^{3} d^{2}}-\frac {2 a \ln \left (-c x -1\right )}{c^{3} d^{2}}+\frac {a}{c^{3} d^{2} \left (-c x -1\right )}-\frac {b x \ln \left (-c x +1\right )}{2 c^{2} d^{2}}+\frac {b \ln \left (-c x +1\right )}{2 c^{3} d^{2}}-\frac {b}{2 c^{3} d^{2}}+\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{c^{3} d^{2}}-\frac {b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{c^{3} d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{c^{3} d^{2}}+\frac {b \ln \left (-c x -1\right )}{4 c^{3} d^{2}}+\frac {b \ln \left (-c x +1\right ) x}{4 c^{2} d^{2} \left (-c x -1\right )}-\frac {b \ln \left (-c x +1\right )}{4 c^{3} d^{2} \left (-c x -1\right )}\) | \(302\) |
1/c^3*(a/d^2*(c*x-2*ln(c*x+1)-1/(c*x+1))+b/d^2*(c*x*arctanh(c*x)-2*arctanh (c*x)*ln(c*x+1)-1/(c*x+1)*arctanh(c*x)+1/4*ln(c*x-1)-1/2/(c*x+1)+3/4*ln(c* x+1)-(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+dilog(1/2*c*x+1/2)+1/2*l n(c*x+1)^2))
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\frac {\int \frac {a x^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b x^{2} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \]
(Integral(a*x**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(b*x**2*atanh(c*x)/ (c**2*x**2 + 2*c*x + 1), x))/d**2
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
-1/8*(c^3*(2/(c^7*d^2*x + c^6*d^2) - 4*x/(c^5*d^2) + 5*log(c*x + 1)/(c^6*d ^2) - log(c*x - 1)/(c^6*d^2)) - 4*c^3*integrate(x^3*log(c*x + 1)/(c^5*d^2* x^3 + c^4*d^2*x^2 - c^3*d^2*x - c^2*d^2), x) - 2*c^2*(2/(c^6*d^2*x + c^5*d ^2) + 3*log(c*x + 1)/(c^5*d^2) + log(c*x - 1)/(c^5*d^2)) + 12*c^2*integrat e(x^2*log(c*x + 1)/(c^5*d^2*x^3 + c^4*d^2*x^2 - c^3*d^2*x - c^2*d^2), x) + 16*c*integrate(x*log(c*x + 1)/(c^5*d^2*x^3 + c^4*d^2*x^2 - c^3*d^2*x - c^ 2*d^2), x) + 4*(c^2*x^2 + c*x - 2*(c*x + 1)*log(c*x + 1) - 1)*log(-c*x + 1 )/(c^4*d^2*x + c^3*d^2) + 2/(c^4*d^2*x + c^3*d^2) - log(c*x + 1)/(c^3*d^2) + log(c*x - 1)/(c^3*d^2) + 8*integrate(log(c*x + 1)/(c^5*d^2*x^3 + c^4*d^ 2*x^2 - c^3*d^2*x - c^2*d^2), x))*b - a*(1/(c^4*d^2*x + c^3*d^2) - x/(c^2* d^2) + 2*log(c*x + 1)/(c^3*d^2))
\[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{{\left (c d x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arctanh}(c x))}{(d+c d x)^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^2} \,d x \]