Integrand size = 19, antiderivative size = 157 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {b^2}{16 c d^3 (1+c x)^2}-\frac {3 b^2}{16 c d^3 (1+c x)}+\frac {3 b^2 \text {arctanh}(c x)}{16 c d^3}-\frac {b (a+b \text {arctanh}(c x))}{4 c d^3 (1+c x)^2}-\frac {b (a+b \text {arctanh}(c x))}{4 c d^3 (1+c x)}+\frac {(a+b \text {arctanh}(c x))^2}{8 c d^3}-\frac {(a+b \text {arctanh}(c x))^2}{2 c d^3 (1+c x)^2} \]
-1/16*b^2/c/d^3/(c*x+1)^2-3/16*b^2/c/d^3/(c*x+1)+3/16*b^2*arctanh(c*x)/c/d ^3-1/4*b*(a+b*arctanh(c*x))/c/d^3/(c*x+1)^2-1/4*b*(a+b*arctanh(c*x))/c/d^3 /(c*x+1)+1/8*(a+b*arctanh(c*x))^2/c/d^3-1/2*(a+b*arctanh(c*x))^2/c/d^3/(c* x+1)^2
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {-8 a^2-4 a b-b^2}{16 c d^3 (1+c x)^2}-\frac {b (4 a+3 b)}{16 c d^3 (1+c x)}-\frac {b (4 a+2 b+b c x) \text {arctanh}(c x)}{4 c d^3 (1+c x)^2}+\frac {b^2 \left (-3+2 c x+c^2 x^2\right ) \text {arctanh}(c x)^2}{8 c d^3 (1+c x)^2}+\frac {\left (-4 a b-3 b^2\right ) \log (1-c x)}{32 c d^3}+\frac {\left (4 a b+3 b^2\right ) \log (1+c x)}{32 c d^3} \]
(-8*a^2 - 4*a*b - b^2)/(16*c*d^3*(1 + c*x)^2) - (b*(4*a + 3*b))/(16*c*d^3* (1 + c*x)) - (b*(4*a + 2*b + b*c*x)*ArcTanh[c*x])/(4*c*d^3*(1 + c*x)^2) + (b^2*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^2)/(8*c*d^3*(1 + c*x)^2) + ((-4*a *b - 3*b^2)*Log[1 - c*x])/(32*c*d^3) + ((4*a*b + 3*b^2)*Log[1 + c*x])/(32* c*d^3)
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6480, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{(c d x+d)^3} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {b \int \left (\frac {a+b \text {arctanh}(c x)}{4 d^2 \left (1-c^2 x^2\right )}+\frac {a+b \text {arctanh}(c x)}{4 d^2 (c x+1)^2}+\frac {a+b \text {arctanh}(c x)}{2 d^2 (c x+1)^3}\right )dx}{d}-\frac {(a+b \text {arctanh}(c x))^2}{2 c d^3 (c x+1)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {(a+b \text {arctanh}(c x))^2}{8 b c d^2}-\frac {a+b \text {arctanh}(c x)}{4 c d^2 (c x+1)}-\frac {a+b \text {arctanh}(c x)}{4 c d^2 (c x+1)^2}+\frac {3 b \text {arctanh}(c x)}{16 c d^2}-\frac {3 b}{16 c d^2 (c x+1)}-\frac {b}{16 c d^2 (c x+1)^2}\right )}{d}-\frac {(a+b \text {arctanh}(c x))^2}{2 c d^3 (c x+1)^2}\) |
-1/2*(a + b*ArcTanh[c*x])^2/(c*d^3*(1 + c*x)^2) + (b*(-1/16*b/(c*d^2*(1 + c*x)^2) - (3*b)/(16*c*d^2*(1 + c*x)) + (3*b*ArcTanh[c*x])/(16*c*d^2) - (a + b*ArcTanh[c*x])/(4*c*d^2*(1 + c*x)^2) - (a + b*ArcTanh[c*x])/(4*c*d^2*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(8*b*c*d^2)))/d
3.2.15.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Time = 1.01 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {16 c x \,a^{2}+8 c^{2} x^{2} a^{2}+2 b^{2} \operatorname {arctanh}\left (c x \right ) x c +4 x^{2} \operatorname {arctanh}\left (c x \right ) a b \,c^{2}+2 b^{2} \operatorname {arctanh}\left (c x \right )^{2} x^{2} c^{2}+12 a b c x +3 x^{2} \operatorname {arctanh}\left (c x \right ) b^{2} c^{2}+4 b^{2} c^{2} x^{2}+8 c x a b \,\operatorname {arctanh}\left (c x \right )+4 b^{2} c x \operatorname {arctanh}\left (c x \right )^{2}-6 b^{2} \operatorname {arctanh}\left (c x \right )^{2}-5 \,\operatorname {arctanh}\left (c x \right ) b^{2}-12 \,\operatorname {arctanh}\left (c x \right ) a b +8 a b \,c^{2} x^{2}+5 b^{2} c x}{16 d^{3} \left (c x +1\right )^{2} c}\) | \(177\) |
| derivativedivides | \(\frac {-\frac {a^{2}}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x -1\right )^{2}}{32}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x +1\right )^{2}}{32}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {3}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{32}-\frac {3 \ln \left (c x -1\right )}{32}\right )}{d^{3}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}-\frac {\ln \left (c x -1\right )}{16}\right )}{d^{3}}}{c}\) | \(243\) |
| default | \(\frac {-\frac {a^{2}}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x -1\right )^{2}}{32}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x +1\right )^{2}}{32}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {3}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{32}-\frac {3 \ln \left (c x -1\right )}{32}\right )}{d^{3}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}-\frac {\ln \left (c x -1\right )}{16}\right )}{d^{3}}}{c}\) | \(243\) |
| parts | \(-\frac {a^{2}}{2 d^{3} c \left (c x +1\right )^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{8}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{8}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x -1\right )^{2}}{32}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{16}-\frac {\ln \left (c x +1\right )^{2}}{32}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {3}{16 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{32}-\frac {3 \ln \left (c x -1\right )}{32}\right )}{d^{3} c}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {1}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}-\frac {\ln \left (c x -1\right )}{16}\right )}{d^{3} c}\) | \(248\) |
| risch | \(\frac {b^{2} \left (c^{2} x^{2}+2 c x -3\right ) \ln \left (c x +1\right )^{2}}{32 d^{3} \left (c x +1\right )^{2} c}-\frac {b \left (b \,x^{2} \ln \left (-c x +1\right ) c^{2}+2 b c x \ln \left (-c x +1\right )+2 b c x -3 b \ln \left (-c x +1\right )+8 a +4 b \right ) \ln \left (c x +1\right )}{16 d^{3} \left (c x +1\right )^{2} c}+\frac {b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+4 b \,c^{2} \ln \left (-c x -1\right ) x^{2} a +3 b^{2} c^{2} \ln \left (-c x -1\right ) x^{2}-4 a b \,c^{2} \ln \left (c x -1\right ) x^{2}-3 \ln \left (c x -1\right ) x^{2} b^{2} c^{2}+2 \ln \left (-c x +1\right )^{2} b^{2} c x +8 \ln \left (-c x -1\right ) a b c x +6 \ln \left (-c x -1\right ) b^{2} c x -8 a b c \ln \left (c x -1\right ) x -6 \ln \left (c x -1\right ) b^{2} c x +4 b^{2} c x \ln \left (-c x +1\right )-8 a b c x -6 b^{2} c x -3 b^{2} \ln \left (-c x +1\right )^{2}+4 \ln \left (-c x -1\right ) a b +3 b^{2} \ln \left (-c x -1\right )-4 a b \ln \left (c x -1\right )-3 \ln \left (c x -1\right ) b^{2}+16 b \ln \left (-c x +1\right ) a +8 b^{2} \ln \left (-c x +1\right )-16 a^{2}-16 a b -8 b^{2}}{32 d^{3} \left (c x +1\right )^{2} c}\) | \(405\) |
1/16*(16*c*x*a^2+8*c^2*x^2*a^2+2*b^2*arctanh(c*x)*x*c+4*x^2*arctanh(c*x)*a *b*c^2+2*b^2*arctanh(c*x)^2*x^2*c^2+12*a*b*c*x+3*x^2*arctanh(c*x)*b^2*c^2+ 4*b^2*c^2*x^2+8*c*x*a*b*arctanh(c*x)+4*b^2*c*x*arctanh(c*x)^2-6*b^2*arctan h(c*x)^2-5*arctanh(c*x)*b^2-12*arctanh(c*x)*a*b+8*a*b*c^2*x^2+5*b^2*c*x)/d ^3/(c*x+1)^2/c
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {2 \, {\left (4 \, a b + 3 \, b^{2}\right )} c x - {\left (b^{2} c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 16 \, a^{2} + 16 \, a b + 8 \, b^{2} - {\left ({\left (4 \, a b + 3 \, b^{2}\right )} c^{2} x^{2} + 2 \, {\left (4 \, a b + b^{2}\right )} c x - 12 \, a b - 5 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{32 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} \]
-1/32*(2*(4*a*b + 3*b^2)*c*x - (b^2*c^2*x^2 + 2*b^2*c*x - 3*b^2)*log(-(c*x + 1)/(c*x - 1))^2 + 16*a^2 + 16*a*b + 8*b^2 - ((4*a*b + 3*b^2)*c^2*x^2 + 2*(4*a*b + b^2)*c*x - 12*a*b - 5*b^2)*log(-(c*x + 1)/(c*x - 1)))/(c^3*d^3* x^2 + 2*c^2*d^3*x + c*d^3)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {\int \frac {a^{2}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \]
(Integral(a**2/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b**2*a tanh(c*x)**2/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(2*a*b*at anh(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. 399 vs. \(2 (143) = 286\).
Time = 0.19 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.54 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=-\frac {1}{8} \, {\left (c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {8 \, \operatorname {artanh}\left (c x\right )}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}}\right )} a b - \frac {1}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{3}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}}\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} - \frac {a^{2}}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} \]
-1/8*(c*(2*(c*x + 2)/(c^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3) - log(c*x + 1)/ (c^2*d^3) + log(c*x - 1)/(c^2*d^3)) + 8*arctanh(c*x)/(c^3*d^3*x^2 + 2*c^2* d^3*x + c*d^3))*a*b - 1/32*(4*c*(2*(c*x + 2)/(c^4*d^3*x^2 + 2*c^3*d^3*x + c^2*d^3) - log(c*x + 1)/(c^2*d^3) + log(c*x - 1)/(c^2*d^3))*arctanh(c*x) + ((c^2*x^2 + 2*c*x + 1)*log(c*x + 1)^2 + (c^2*x^2 + 2*c*x + 1)*log(c*x - 1 )^2 + 6*c*x - (3*c^2*x^2 + 6*c*x + 2*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 3)*log(c*x + 1) + 3*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 8)*c^2/(c^5*d^3*x ^2 + 2*c^4*d^3*x + c^3*d^3))*b^2 - 1/2*b^2*arctanh(c*x)^2/(c^3*d^3*x^2 + 2 *c^2*d^3*x + c*d^3) - 1/2*a^2/(c^3*d^3*x^2 + 2*c^2*d^3*x + c*d^3)
Time = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {1}{64} \, c {\left (\frac {2 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{2} d^{3}} + \frac {2 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b}{c x - 1} - 4 \, a b + \frac {4 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2} d^{3}} + \frac {{\left (\frac {16 \, {\left (c x + 1\right )} a^{2}}{c x - 1} - 8 \, a^{2} + \frac {16 \, {\left (c x + 1\right )} a b}{c x - 1} - 4 \, a b + \frac {8 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{2} d^{3}}\right )} \]
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^2*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^2*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^2*d^3))
Time = 5.29 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^3} \, dx=\frac {11\,b^2\,\ln \left (1-c\,x\right )-11\,b^2\,\ln \left (c\,x+1\right )-16\,a\,b-3\,b^2\,{\ln \left (c\,x+1\right )}^2-3\,b^2\,{\ln \left (1-c\,x\right )}^2+12\,b^2\,\mathrm {atanh}\left (c\,x\right )-16\,a^2-8\,b^2-16\,a\,b\,\ln \left (c\,x+1\right )+16\,a\,b\,\ln \left (1-c\,x\right )+6\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+8\,a\,b\,\mathrm {atanh}\left (c\,x\right )-6\,b^2\,c\,x-10\,b^2\,c\,x\,\ln \left (c\,x+1\right )+10\,b^2\,c\,x\,\ln \left (1-c\,x\right )+b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2+12\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+2\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+2\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2+24\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )-3\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )+3\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )-8\,a\,b\,c\,x-4\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+8\,a\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+16\,a\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )-2\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{32\,c\,d^3\,{\left (c\,x+1\right )}^2} \]
(11*b^2*log(1 - c*x) - 11*b^2*log(c*x + 1) - 16*a*b - 3*b^2*log(c*x + 1)^2 - 3*b^2*log(1 - c*x)^2 + 12*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) + 6*b^2*log(c*x + 1)*log(1 - c*x) + 8*a*b*a tanh(c*x) - 6*b^2*c*x - 10*b^2*c*x*log(c*x + 1) + 10*b^2*c*x*log(1 - c*x) + b^2*c^2*x^2*log(c*x + 1)^2 + b^2*c^2*x^2*log(1 - c*x)^2 + 12*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 + 24*b^2 *c*x*atanh(c*x) - 3*b^2*c^2*x^2*log(c*x + 1) + 3*b^2*c^2*x^2*log(1 - c*x) - 8*a*b*c*x - 4*b^2*c*x*log(c*x + 1)*log(1 - c*x) + 8*a*b*c^2*x^2*atanh(c* x) + 16*a*b*c*x*atanh(c*x) - 2*b^2*c^2*x^2*log(c*x + 1)*log(1 - c*x))/(32* c*d^3*(c*x + 1)^2)