Integrand size = 20, antiderivative size = 157 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {b^2}{16 c^2 d^3 (1+c x)^2}-\frac {5 b^2}{16 c^2 d^3 (1+c x)}+\frac {5 b^2 \text {arctanh}(c x)}{16 c^2 d^3}+\frac {b (a+b \text {arctanh}(c x))}{4 c^2 d^3 (1+c x)^2}-\frac {3 b (a+b \text {arctanh}(c x))}{4 c^2 d^3 (1+c x)}-\frac {(a+b \text {arctanh}(c x))^2}{8 c^2 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 d^3 (1+c x)^2} \] Output:
1/16*b^2/c^2/d^3/(c*x+1)^2-5/16*b^2/c^2/d^3/(c*x+1)+5/16*b^2*arctanh(c*x)/ c^2/d^3+1/4*b*(a+b*arctanh(c*x))/c^2/d^3/(c*x+1)^2-3/4*b*(a+b*arctanh(c*x) )/c^2/d^3/(c*x+1)-1/8*(a+b*arctanh(c*x))^2/c^2/d^3+1/2*x^2*(a+b*arctanh(c* x))^2/d^3/(c*x+1)^2
Time = 0.33 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {2 \left (8 a^2+4 a b+b^2\right )-2 \left (16 a^2+12 a b+5 b^2\right ) (1+c x)-8 b (b (2+3 c x)+a (4+8 c x)) \text {arctanh}(c x)+4 b^2 \left (-1-2 c x+3 c^2 x^2\right ) \text {arctanh}(c x)^2-b (12 a+5 b) (1+c x)^2 \log (1-c x)+b (12 a+5 b) (1+c x)^2 \log (1+c x)}{32 c^2 d^3 (1+c x)^2} \] Input:
Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]
Output:
(2*(8*a^2 + 4*a*b + b^2) - 2*(16*a^2 + 12*a*b + 5*b^2)*(1 + c*x) - 8*b*(b* (2 + 3*c*x) + a*(4 + 8*c*x))*ArcTanh[c*x] + 4*b^2*(-1 - 2*c*x + 3*c^2*x^2) *ArcTanh[c*x]^2 - b*(12*a + 5*b)*(1 + c*x)^2*Log[1 - c*x] + b*(12*a + 5*b) *(1 + c*x)^2*Log[1 + c*x])/(32*c^2*d^3*(1 + c*x)^2)
Time = 0.44 (sec) , antiderivative size = 157, 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 = {6500, 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.
| \(\displaystyle \int \frac {x (a+b \text {arctanh}(c x))^2}{(c d x+d)^3} \, dx\) |
| \(\Big \downarrow \) 6500 |
| \(\displaystyle \frac {x^2 (a+b \text {arctanh}(c x))^2}{2 d^3 (c x+1)^2}-2 b c \int \left (\frac {a+b \text {arctanh}(c x)}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arctanh}(c x))}{8 c^2 d^3 (c x+1)^2}+\frac {a+b \text {arctanh}(c x)}{4 c^2 d^3 (c x+1)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 (a+b \text {arctanh}(c x))^2}{2 d^3 (c x+1)^2}-2 b c \left (\frac {(a+b \text {arctanh}(c x))^2}{16 b c^3 d^3}+\frac {3 (a+b \text {arctanh}(c x))}{8 c^3 d^3 (c x+1)}-\frac {a+b \text {arctanh}(c x)}{8 c^3 d^3 (c x+1)^2}-\frac {5 b \text {arctanh}(c x)}{32 c^3 d^3}+\frac {5 b}{32 c^3 d^3 (c x+1)}-\frac {b}{32 c^3 d^3 (c x+1)^2}\right )\) |
Input:
Int[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x)^3,x]
Output:
(x^2*(a + b*ArcTanh[c*x])^2)/(2*d^3*(1 + c*x)^2) - 2*b*c*(-1/32*b/(c^3*d^3 *(1 + c*x)^2) + (5*b)/(32*c^3*d^3*(1 + c*x)) - (5*b*ArcTanh[c*x])/(32*c^3* d^3) - (a + b*ArcTanh[c*x])/(8*c^3*d^3*(1 + c*x)^2) + (3*(a + b*ArcTanh[c* x]))/(8*c^3*d^3*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(16*b*c^3*d^3))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e _.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Si mp[(a + b*ArcTanh[c*x])^p u, x] - Simp[b*c*p Int[ExpandIntegrand[(a + b *ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && N eQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.08
| method | result | size |
| parallelrisch | \(\frac {-4 b^{2} c x \operatorname {arctanh}\left (c x \right )^{2}+8 a b \,c^{2} x^{2}+4 a b c x +8 a^{2} c^{2} x^{2}+4 b^{2} x^{2} c^{2}+3 b^{2} c x +6 b^{2} c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}-2 c \,b^{2} \operatorname {arctanh}\left (c x \right ) x -3 b^{2} \operatorname {arctanh}\left (c x \right )+5 b^{2} \operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+12 \,\operatorname {arctanh}\left (c x \right ) a b \,c^{2} x^{2}-8 \,\operatorname {arctanh}\left (c x \right ) a b c x -2 b^{2} \operatorname {arctanh}\left (c x \right )^{2}-4 \,\operatorname {arctanh}\left (c x \right ) a b}{16 d^{3} \left (c x +1\right )^{2} c^{2}}\) | \(170\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (\frac {1}{2 \left (c x +1\right )^{2}}-\frac {1}{c x +1}\right )}{d^{3}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {3 \ln \left (c x -1\right )^{2}}{32}+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {3 \ln \left (c x +1\right )^{2}}{32}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {5 \ln \left (c x -1\right )}{32}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{32}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{8 \left (c x +1\right )^{2}}-\frac {3}{8 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}}{c^{2}}\) | \(282\) |
| default | \(\frac {\frac {a^{2} \left (\frac {1}{2 \left (c x +1\right )^{2}}-\frac {1}{c x +1}\right )}{d^{3}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {3 \ln \left (c x -1\right )^{2}}{32}+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {3 \ln \left (c x +1\right )^{2}}{32}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {5 \ln \left (c x -1\right )}{32}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{32}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{8 \left (c x +1\right )^{2}}-\frac {3}{8 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}}{c^{2}}\) | \(282\) |
| parts | \(\frac {a^{2} \left (-\frac {1}{c^{2} \left (c x +1\right )}+\frac {1}{2 c^{2} \left (c x +1\right )^{2}}\right )}{d^{3}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {3 \ln \left (c x -1\right )^{2}}{32}+\frac {3 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {3 \ln \left (c x +1\right )^{2}}{32}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {5 \ln \left (c x -1\right )}{32}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{32}\right )}{d^{3} c^{2}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{8 \left (c x +1\right )^{2}}-\frac {3}{8 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3} c^{2}}\) | \(290\) |
| orering | \(\frac {\left (c x +1\right ) \left (15 c^{6} x^{6}-40 c^{5} x^{5}+47 c^{4} x^{4}-10 x^{3} c^{3}-21 c^{2} x^{2}+12 c x -3\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2}}{32 x^{2} c^{4} \left (c d x +d \right )^{3}}+\frac {\left (c x -1\right ) \left (c x +1\right )^{2} \left (15 c^{4} x^{4}-20 x^{3} c^{3}-2 c^{2} x^{2}+12 c x -3\right ) \left (\frac {\left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2}}{\left (c d x +d \right )^{3}}+\frac {2 x \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b c}{\left (c d x +d \right )^{3} \left (-c^{2} x^{2}+1\right )}-\frac {3 x \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c d}{\left (c d x +d \right )^{4}}\right )}{32 c^{4} x^{2}}+\frac {\left (5 c^{2} x^{2}-3\right ) \left (c x -1\right )^{2} \left (c x +1\right )^{3} \left (\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b c}{\left (c d x +d \right )^{3} \left (-c^{2} x^{2}+1\right )}-\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c d}{\left (c d x +d \right )^{4}}+\frac {2 x \,b^{2} c^{2}}{\left (-c^{2} x^{2}+1\right )^{2} \left (c d x +d \right )^{3}}-\frac {12 x \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{2} d}{\left (c d x +d \right )^{4} \left (-c^{2} x^{2}+1\right )}+\frac {4 x^{2} \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{3}}{\left (c d x +d \right )^{3} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {12 x \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c^{2} d^{2}}{\left (c d x +d \right )^{5}}\right )}{64 c^{4} x}\) | \(421\) |
| risch | \(\frac {b^{2} \left (3 c^{2} x^{2}-2 c x -1\right ) \ln \left (c x +1\right )^{2}}{32 c^{2} d^{3} \left (c x +1\right )^{2}}-\frac {b \left (3 b \,c^{2} x^{2} \ln \left (-c x +1\right )-2 b c x \ln \left (-c x +1\right )+16 a c x +6 b c x -b \ln \left (-c x +1\right )+8 a +4 b \right ) \ln \left (c x +1\right )}{16 c^{2} d^{3} \left (c x +1\right )^{2}}-\frac {12 a b \ln \left (c x -1\right )+32 a^{2} c x +16 a^{2}+16 b a +5 b^{2} \ln \left (c x -1\right )-5 b^{2} \ln \left (-c x -1\right )+24 a b c x +8 b^{2}+5 \ln \left (c x -1\right ) b^{2} c^{2} x^{2}-5 \ln \left (-c x -1\right ) b^{2} c^{2} x^{2}-12 \ln \left (-c x -1\right ) a b +10 \ln \left (c x -1\right ) b^{2} c x -10 \ln \left (-c x -1\right ) b^{2} c x +2 b^{2} c x \ln \left (-c x +1\right )^{2}-8 b^{2} \ln \left (-c x +1\right )-24 \ln \left (-c x -1\right ) a b c x +10 b^{2} c x -12 b^{2} c x \ln \left (-c x +1\right )-3 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}-32 a b c x \ln \left (-c x +1\right )-12 \ln \left (-c x -1\right ) a b \,c^{2} x^{2}+24 a b c \ln \left (c x -1\right ) x +12 a b \,c^{2} \ln \left (c x -1\right ) x^{2}-16 \ln \left (-c x +1\right ) a b +\ln \left (-c x +1\right )^{2} b^{2}}{32 c^{2} d^{3} \left (c x +1\right )^{2}}\) | \(432\) |
Input:
int(x*(a+b*arctanh(c*x))^2/(c*d*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/16*(-4*b^2*c*x*arctanh(c*x)^2+8*a*b*c^2*x^2+4*a*b*c*x+8*a^2*c^2*x^2+4*b^ 2*x^2*c^2+3*b^2*c*x+6*b^2*c^2*x^2*arctanh(c*x)^2-2*c*b^2*arctanh(c*x)*x-3* b^2*arctanh(c*x)+5*b^2*arctanh(c*x)*c^2*x^2+12*arctanh(c*x)*a*b*c^2*x^2-8* arctanh(c*x)*a*b*c*x-2*b^2*arctanh(c*x)^2-4*arctanh(c*x)*a*b)/d^3/(c*x+1)^ 2/c^2
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {2 \, {\left (16 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} c x - {\left (3 \, b^{2} c^{2} x^{2} - 2 \, b^{2} c x - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 16 \, a^{2} + 16 \, a b + 8 \, b^{2} - {\left ({\left (12 \, a b + 5 \, b^{2}\right )} c^{2} x^{2} - 2 \, {\left (4 \, a b + b^{2}\right )} c x - 4 \, a b - 3 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{32 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d)^3,x, algorithm="fricas")
Output:
-1/32*(2*(16*a^2 + 12*a*b + 5*b^2)*c*x - (3*b^2*c^2*x^2 - 2*b^2*c*x - b^2) *log(-(c*x + 1)/(c*x - 1))^2 + 16*a^2 + 16*a*b + 8*b^2 - ((12*a*b + 5*b^2) *c^2*x^2 - 2*(4*a*b + b^2)*c*x - 4*a*b - 3*b^2)*log(-(c*x + 1)/(c*x - 1))) /(c^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {\int \frac {a^{2} x}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {2 a b x \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \] Input:
integrate(x*(a+b*atanh(c*x))**2/(c*d*x+d)**3,x)
Output:
(Integral(a**2*x/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b**2 *x*atanh(c*x)**2/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(2*a* b*x*atanh(c*x)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x))/d**3
Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (143) = 286\).
Time = 0.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.73 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {{\left (2 \, c x + 1\right )} b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} - \frac {1}{8} \, {\left (c {\left (\frac {2 \, {\left (3 \, c x + 2\right )}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}} - \frac {3 \, \log \left (c x + 1\right )}{c^{3} d^{3}} + \frac {3 \, \log \left (c x - 1\right )}{c^{3} d^{3}}\right )} + \frac {8 \, {\left (2 \, c x + 1\right )} \operatorname {artanh}\left (c x\right )}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}}\right )} a b - \frac {1}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (3 \, c x + 2\right )}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}} - \frac {3 \, \log \left (c x + 1\right )}{c^{3} d^{3}} + \frac {3 \, \log \left (c x - 1\right )}{c^{3} d^{3}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left (3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 10 \, c x - {\left (5 \, c^{2} x^{2} + 10 \, c x + 6 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 5\right )} \log \left (c x + 1\right ) + 5 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}}\right )} b^{2} - \frac {{\left (2 \, c x + 1\right )} a^{2}}{2 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d)^3,x, algorithm="maxima")
Output:
-1/2*(2*c*x + 1)*b^2*arctanh(c*x)^2/(c^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3) - 1/8*(c*(2*(3*c*x + 2)/(c^5*d^3*x^2 + 2*c^4*d^3*x + c^3*d^3) - 3*log(c*x + 1)/(c^3*d^3) + 3*log(c*x - 1)/(c^3*d^3)) + 8*(2*c*x + 1)*arctanh(c*x)/(c ^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3))*a*b - 1/32*(4*c*(2*(3*c*x + 2)/(c^5*d ^3*x^2 + 2*c^4*d^3*x + c^3*d^3) - 3*log(c*x + 1)/(c^3*d^3) + 3*log(c*x - 1 )/(c^3*d^3))*arctanh(c*x) + (3*(c^2*x^2 + 2*c*x + 1)*log(c*x + 1)^2 + 3*(c ^2*x^2 + 2*c*x + 1)*log(c*x - 1)^2 + 10*c*x - (5*c^2*x^2 + 10*c*x + 6*(c^2 *x^2 + 2*c*x + 1)*log(c*x - 1) + 5)*log(c*x + 1) + 5*(c^2*x^2 + 2*c*x + 1) *log(c*x - 1) + 8)*c^2/(c^6*d^3*x^2 + 2*c^5*d^3*x + c^4*d^3))*b^2 - 1/2*(2 *c*x + 1)*a^2/(c^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3)
Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.44 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {1}{64} \, c {\left (\frac {2 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{3} d^{3}} + \frac {2 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b}{c x - 1} + 4 \, a b + \frac {4 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{3} d^{3}} + \frac {{\left (\frac {16 \, {\left (c x + 1\right )} a^{2}}{c x - 1} + 8 \, a^{2} + \frac {16 \, {\left (c x + 1\right )} a b}{c x - 1} + 4 \, a b + \frac {8 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{3} d^{3}}\right )} \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d)^3,x, algorithm="giac")
Output:
1/64*c*(2*(2*(c*x + 1)*b^2/(c*x - 1) + b^2)*(c*x - 1)^2*log(-(c*x + 1)/(c* x - 1))^2/((c*x + 1)^2*c^3*d^3) + 2*(8*(c*x + 1)*a*b/(c*x - 1) + 4*a*b + 4 *(c*x + 1)*b^2/(c*x - 1) + b^2)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))/((c* x + 1)^2*c^3*d^3) + (16*(c*x + 1)*a^2/(c*x - 1) + 8*a^2 + 16*(c*x + 1)*a*b /(c*x - 1) + 4*a*b + 8*(c*x + 1)*b^2/(c*x - 1) + b^2)*(c*x - 1)^2/((c*x + 1)^2*c^3*d^3))
Time = 5.08 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.58 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {16\,a\,b+17\,b^2\,\ln \left (c\,x+1\right )-17\,b^2\,\ln \left (1-c\,x\right )+b^2\,{\ln \left (c\,x+1\right )}^2+b^2\,{\ln \left (1-c\,x\right )}^2-28\,b^2\,\mathrm {atanh}\left (c\,x\right )+16\,a^2+8\,b^2+16\,a\,b\,\ln \left (c\,x+1\right )-16\,a\,b\,\ln \left (1-c\,x\right )-2\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-24\,a\,b\,\mathrm {atanh}\left (c\,x\right )+32\,a^2\,c\,x+10\,b^2\,c\,x+30\,b^2\,c\,x\,\ln \left (c\,x+1\right )-30\,b^2\,c\,x\,\ln \left (1-c\,x\right )-3\,b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2-3\,b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2-28\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+2\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+2\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2-56\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+9\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )-9\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )+24\,a\,b\,c\,x+32\,a\,b\,c\,x\,\ln \left (c\,x+1\right )-32\,a\,b\,c\,x\,\ln \left (1-c\,x\right )-4\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-24\,a\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )-48\,a\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )+6\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{32\,c^2\,d^3\,{\left (c\,x+1\right )}^2} \] Input:
int((x*(a + b*atanh(c*x))^2)/(d + c*d*x)^3,x)
Output:
-(16*a*b + 17*b^2*log(c*x + 1) - 17*b^2*log(1 - c*x) + b^2*log(c*x + 1)^2 + b^2*log(1 - c*x)^2 - 28*b^2*atanh(c*x) + 16*a^2 + 8*b^2 + 16*a*b*log(c*x + 1) - 16*a*b*log(1 - c*x) - 2*b^2*log(c*x + 1)*log(1 - c*x) - 24*a*b*ata nh(c*x) + 32*a^2*c*x + 10*b^2*c*x + 30*b^2*c*x*log(c*x + 1) - 30*b^2*c*x*l og(1 - c*x) - 3*b^2*c^2*x^2*log(c*x + 1)^2 - 3*b^2*c^2*x^2*log(1 - c*x)^2 - 28*b^2*c^2*x^2*atanh(c*x) + 2*b^2*c*x*log(c*x + 1)^2 + 2*b^2*c*x*log(1 - c*x)^2 - 56*b^2*c*x*atanh(c*x) + 9*b^2*c^2*x^2*log(c*x + 1) - 9*b^2*c^2*x ^2*log(1 - c*x) + 24*a*b*c*x + 32*a*b*c*x*log(c*x + 1) - 32*a*b*c*x*log(1 - c*x) - 4*b^2*c*x*log(c*x + 1)*log(1 - c*x) - 24*a*b*c^2*x^2*atanh(c*x) - 48*a*b*c*x*atanh(c*x) + 6*b^2*c^2*x^2*log(c*x + 1)*log(1 - c*x))/(32*c^2* d^3*(c*x + 1)^2)
Time = 0.17 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.91 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {12 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-8 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -4 \mathit {atanh} \left (c x \right )^{2} b^{2}+32 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}+12 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-4 \mathit {atanh} \left (c x \right ) b^{2}+4 \,\mathrm {log}\left (c x -1\right ) a b \,c^{2} x^{2}+8 \,\mathrm {log}\left (c x -1\right ) a b c x +4 \,\mathrm {log}\left (c x -1\right ) a b +\mathrm {log}\left (c x -1\right ) b^{2} c^{2} x^{2}+2 \,\mathrm {log}\left (c x -1\right ) b^{2} c x +\mathrm {log}\left (c x -1\right ) b^{2}-4 \,\mathrm {log}\left (c x +1\right ) a b \,c^{2} x^{2}-8 \,\mathrm {log}\left (c x +1\right ) a b c x -4 \,\mathrm {log}\left (c x +1\right ) a b -\mathrm {log}\left (c x +1\right ) b^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (c x +1\right ) b^{2} c x -\mathrm {log}\left (c x +1\right ) b^{2}+16 a^{2} c^{2} x^{2}+12 a b \,c^{2} x^{2}-4 a b +5 b^{2} c^{2} x^{2}-3 b^{2}}{32 c^{2} d^{3} \left (c^{2} x^{2}+2 c x +1\right )} \] Input:
int(x*(a+b*atanh(c*x))^2/(c*d*x+d)^3,x)
Output:
(12*atanh(c*x)**2*b**2*c**2*x**2 - 8*atanh(c*x)**2*b**2*c*x - 4*atanh(c*x) **2*b**2 + 32*atanh(c*x)*a*b*c**2*x**2 + 12*atanh(c*x)*b**2*c**2*x**2 - 4* atanh(c*x)*b**2 + 4*log(c*x - 1)*a*b*c**2*x**2 + 8*log(c*x - 1)*a*b*c*x + 4*log(c*x - 1)*a*b + log(c*x - 1)*b**2*c**2*x**2 + 2*log(c*x - 1)*b**2*c*x + log(c*x - 1)*b**2 - 4*log(c*x + 1)*a*b*c**2*x**2 - 8*log(c*x + 1)*a*b*c *x - 4*log(c*x + 1)*a*b - log(c*x + 1)*b**2*c**2*x**2 - 2*log(c*x + 1)*b** 2*c*x - log(c*x + 1)*b**2 + 16*a**2*c**2*x**2 + 12*a*b*c**2*x**2 - 4*a*b + 5*b**2*c**2*x**2 - 3*b**2)/(32*c**2*d**3*(c**2*x**2 + 2*c*x + 1))