Integrand size = 18, antiderivative size = 176 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=-\frac {b^2}{54 c (1+c x)^3}-\frac {5 b^2}{144 c (1+c x)^2}-\frac {11 b^2}{144 c (1+c x)}+\frac {11 b^2 \text {arctanh}(c x)}{144 c}-\frac {b (a+b \text {arctanh}(c x))}{9 c (1+c x)^3}-\frac {b (a+b \text {arctanh}(c x))}{12 c (1+c x)^2}-\frac {b (a+b \text {arctanh}(c x))}{12 c (1+c x)}+\frac {(a+b \text {arctanh}(c x))^2}{24 c}-\frac {(a+b \text {arctanh}(c x))^2}{3 c (1+c x)^3} \] Output:
-1/54*b^2/c/(c*x+1)^3-5/144*b^2/c/(c*x+1)^2-11/144*b^2/c/(c*x+1)+11/144*b^ 2*arctanh(c*x)/c-1/9*b*(a+b*arctanh(c*x))/c/(c*x+1)^3-1/12*b*(a+b*arctanh( c*x))/c/(c*x+1)^2-1/12*b*(a+b*arctanh(c*x))/c/(c*x+1)+1/24*(a+b*arctanh(c* x))^2/c-1/3*(a+b*arctanh(c*x))^2/c/(c*x+1)^3
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=-\frac {16 \left (18 a^2+6 a b+b^2\right )+6 b (12 a+5 b) (1+c x)+6 b (12 a+11 b) (1+c x)^2+24 b \left (24 a+b \left (10+9 c x+3 c^2 x^2\right )\right ) \text {arctanh}(c x)-36 b^2 \left (-7+3 c x+3 c^2 x^2+c^3 x^3\right ) \text {arctanh}(c x)^2+3 b (12 a+11 b) (1+c x)^3 \log (1-c x)-3 b (12 a+11 b) (1+c x)^3 \log (1+c x)}{864 c (1+c x)^3} \] Input:
Integrate[(a + b*ArcTanh[c*x])^2/(1 + c*x)^4,x]
Output:
-1/864*(16*(18*a^2 + 6*a*b + b^2) + 6*b*(12*a + 5*b)*(1 + c*x) + 6*b*(12*a + 11*b)*(1 + c*x)^2 + 24*b*(24*a + b*(10 + 9*c*x + 3*c^2*x^2))*ArcTanh[c* x] - 36*b^2*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x^3)*ArcTanh[c*x]^2 + 3*b*(12*a + 11*b)*(1 + c*x)^3*Log[1 - c*x] - 3*b*(12*a + 11*b)*(1 + c*x)^3*Log[1 + c *x])/(c*(1 + c*x)^3)
Time = 0.47 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{(c x+1)^4} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {2}{3} b \int \left (\frac {a+b \text {arctanh}(c x)}{8 \left (1-c^2 x^2\right )}+\frac {a+b \text {arctanh}(c x)}{8 (c x+1)^2}+\frac {a+b \text {arctanh}(c x)}{4 (c x+1)^3}+\frac {a+b \text {arctanh}(c x)}{2 (c x+1)^4}\right )dx-\frac {(a+b \text {arctanh}(c x))^2}{3 c (c x+1)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} b \left (\frac {(a+b \text {arctanh}(c x))^2}{16 b c}-\frac {a+b \text {arctanh}(c x)}{8 c (c x+1)}-\frac {a+b \text {arctanh}(c x)}{8 c (c x+1)^2}-\frac {a+b \text {arctanh}(c x)}{6 c (c x+1)^3}+\frac {11 b \text {arctanh}(c x)}{96 c}-\frac {11 b}{96 c (c x+1)}-\frac {5 b}{96 c (c x+1)^2}-\frac {b}{36 c (c x+1)^3}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 c (c x+1)^3}\) |
Input:
Int[(a + b*ArcTanh[c*x])^2/(1 + c*x)^4,x]
Output:
-1/3*(a + b*ArcTanh[c*x])^2/(c*(1 + c*x)^3) + (2*b*(-1/36*b/(c*(1 + c*x)^3 ) - (5*b)/(96*c*(1 + c*x)^2) - (11*b)/(96*c*(1 + c*x)) + (11*b*ArcTanh[c*x ])/(96*c) - (a + b*ArcTanh[c*x])/(6*c*(1 + c*x)^3) - (a + b*ArcTanh[c*x])/ (8*c*(1 + c*x)^2) - (a + b*ArcTanh[c*x])/(8*c*(1 + c*x)) + (a + b*ArcTanh[ c*x])^2/(16*b*c)))/3
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 = 0.51 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.43
| method | result | size |
| parallelrisch | \(-\frac {-54 b^{2} c x \operatorname {arctanh}\left (c x \right )^{2}-432 a^{2} c x -144 a^{2} c^{3} x^{3}-324 a b \,c^{2} x^{2}-252 a b c x -432 a^{2} c^{2} x^{2}-135 b^{2} x^{2} c^{2}-56 b^{2} x^{3} c^{3}-87 b^{2} c x -18 b^{2} c^{3} x^{3} \operatorname {arctanh}\left (c x \right )^{2}-54 b^{2} c^{2} x^{2} \operatorname {arctanh}\left (c x \right )^{2}-33 \,\operatorname {arctanh}\left (c x \right ) b^{2} c^{3} x^{3}+9 c \,b^{2} \operatorname {arctanh}\left (c x \right ) x +87 b^{2} \operatorname {arctanh}\left (c x \right )-63 b^{2} \operatorname {arctanh}\left (c x \right ) c^{2} x^{2}-36 \,\operatorname {arctanh}\left (c x \right ) a b \,c^{3} x^{3}-108 \,\operatorname {arctanh}\left (c x \right ) a b \,c^{2} x^{2}-108 \,\operatorname {arctanh}\left (c x \right ) a b c x -120 a b \,c^{3} x^{3}+126 b^{2} \operatorname {arctanh}\left (c x \right )^{2}+252 \,\operatorname {arctanh}\left (c x \right ) a b}{432 \left (c x +1\right )^{3} c}\) | \(252\) |
| derivativedivides | \(\frac {-\frac {a^{2}}{3 \left (c x +1\right )^{3}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{24}-\frac {\operatorname {arctanh}\left (c x \right )}{9 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{24}-\frac {\ln \left (c x -1\right )^{2}}{96}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (c x +1\right )^{2}}{96}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {11 \ln \left (c x -1\right )}{288}-\frac {1}{54 \left (c x +1\right )^{3}}-\frac {5}{144 \left (c x +1\right )^{2}}-\frac {11}{144 \left (c x +1\right )}+\frac {11 \ln \left (c x +1\right )}{288}\right )+2 b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(265\) |
| default | \(\frac {-\frac {a^{2}}{3 \left (c x +1\right )^{3}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{24}-\frac {\operatorname {arctanh}\left (c x \right )}{9 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{24}-\frac {\ln \left (c x -1\right )^{2}}{96}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (c x +1\right )^{2}}{96}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {11 \ln \left (c x -1\right )}{288}-\frac {1}{54 \left (c x +1\right )^{3}}-\frac {5}{144 \left (c x +1\right )^{2}}-\frac {11}{144 \left (c x +1\right )}+\frac {11 \ln \left (c x +1\right )}{288}\right )+2 b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(265\) |
| parts | \(-\frac {a^{2}}{3 \left (c x +1\right )^{3} c}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{24}-\frac {\operatorname {arctanh}\left (c x \right )}{9 \left (c x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{12 \left (c x +1\right )}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{24}-\frac {\ln \left (c x -1\right )^{2}}{96}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (c x +1\right )^{2}}{96}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{48}-\frac {11 \ln \left (c x -1\right )}{288}-\frac {1}{54 \left (c x +1\right )^{3}}-\frac {5}{144 \left (c x +1\right )^{2}}-\frac {11}{144 \left (c x +1\right )}+\frac {11 \ln \left (c x +1\right )}{288}\right )}{c}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(270\) |
| orering | \(\frac {\left (135 c^{5} x^{5}-9 c^{4} x^{4}-324 x^{3} c^{3}+98 c^{2} x^{2}+337 c x -237\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2}}{216 \left (c x +1\right )^{3} c}+\frac {\left (c x +1\right )^{2} \left (c x -1\right ) \left (270 c^{4} x^{4}+231 x^{3} c^{3}-447 c^{2} x^{2}-281 c x +319\right ) \left (\frac {2 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b c}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )}-\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c}{\left (c x +1\right )^{5}}\right )}{864 c^{2}}+\frac {\left (27 x^{3} c^{3}+48 c^{2} x^{2}-29\right ) \left (c x +1\right )^{3} \left (c x -1\right )^{2} \left (\frac {2 b^{2} c^{2}}{\left (-c^{2} x^{2}+1\right )^{2} \left (c x +1\right )^{4}}-\frac {16 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{2}}{\left (c x +1\right )^{5} \left (-c^{2} x^{2}+1\right )}+\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{3} x}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {20 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c^{2}}{\left (c x +1\right )^{6}}\right )}{864 c^{3}}\) | \(315\) |
| risch | \(\frac {b^{2} \left (x^{3} c^{3}+3 c^{2} x^{2}+3 c x -7\right ) \ln \left (c x +1\right )^{2}}{96 \left (c x +1\right )^{3} c}-\frac {b \left (3 b \,x^{3} \ln \left (-c x +1\right ) c^{3}+9 b \,c^{2} x^{2} \ln \left (-c x +1\right )+6 b \,c^{2} x^{2}+9 b c x \ln \left (-c x +1\right )+18 b c x -21 b \ln \left (-c x +1\right )+48 a +20 b \right ) \ln \left (c x +1\right )}{144 \left (c x +1\right )^{3} c}-\frac {36 a b \ln \left (c x -1\right )+288 a^{2}+72 a b \,c^{2} x^{2}+240 b a +33 b^{2} \ln \left (c x -1\right )-33 b^{2} \ln \left (-c x -1\right )+216 a b c x +112 b^{2}+99 \ln \left (c x -1\right ) b^{2} c^{2} x^{2}-99 \ln \left (-c x -1\right ) b^{2} c^{2} x^{2}-36 \ln \left (-c x -1\right ) a b +99 \ln \left (c x -1\right ) b^{2} c x -99 \ln \left (-c x -1\right ) b^{2} c x +33 \ln \left (c x -1\right ) b^{2} c^{3} x^{3}-33 \ln \left (-c x -1\right ) b^{2} c^{3} x^{3}-27 b^{2} c x \ln \left (-c x +1\right )^{2}-120 b^{2} \ln \left (-c x +1\right )-36 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )-9 b^{2} c^{3} x^{3} \ln \left (-c x +1\right )^{2}-108 \ln \left (-c x -1\right ) a b c x -36 \ln \left (-c x -1\right ) a b \,c^{3} x^{3}+66 b^{2} x^{2} c^{2}+162 b^{2} c x -108 b^{2} c x \ln \left (-c x +1\right )-27 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}-108 \ln \left (-c x -1\right ) a b \,c^{2} x^{2}+108 a b c \ln \left (c x -1\right ) x +108 a b \,c^{2} \ln \left (c x -1\right ) x^{2}-288 \ln \left (-c x +1\right ) a b +36 a b \ln \left (c x -1\right ) x^{3} c^{3}+63 \ln \left (-c x +1\right )^{2} b^{2}}{864 \left (c x +1\right )^{3} c}\) | \(558\) |
Input:
int((a+b*arctanh(c*x))^2/(c*x+1)^4,x,method=_RETURNVERBOSE)
Output:
-1/432*(-54*b^2*c*x*arctanh(c*x)^2-432*a^2*c*x-144*a^2*c^3*x^3-324*a*b*c^2 *x^2-252*a*b*c*x-432*a^2*c^2*x^2-135*b^2*x^2*c^2-56*b^2*x^3*c^3-87*b^2*c*x -18*b^2*c^3*x^3*arctanh(c*x)^2-54*b^2*c^2*x^2*arctanh(c*x)^2-33*arctanh(c* x)*b^2*c^3*x^3+9*c*b^2*arctanh(c*x)*x+87*b^2*arctanh(c*x)-63*b^2*arctanh(c *x)*c^2*x^2-36*arctanh(c*x)*a*b*c^3*x^3-108*arctanh(c*x)*a*b*c^2*x^2-108*a rctanh(c*x)*a*b*c*x-120*a*b*c^3*x^3+126*b^2*arctanh(c*x)^2+252*arctanh(c*x )*a*b)/(c*x+1)^3/c
Time = 0.10 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=-\frac {6 \, {\left (12 \, a b + 11 \, b^{2}\right )} c^{2} x^{2} + 54 \, {\left (4 \, a b + 3 \, b^{2}\right )} c x - 9 \, {\left (b^{2} c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x - 7 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 288 \, a^{2} + 240 \, a b + 112 \, b^{2} - 3 \, {\left ({\left (12 \, a b + 11 \, b^{2}\right )} c^{3} x^{3} + 3 \, {\left (12 \, a b + 7 \, b^{2}\right )} c^{2} x^{2} + 3 \, {\left (12 \, a b - b^{2}\right )} c x - 84 \, a b - 29 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{864 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*x+1)^4,x, algorithm="fricas")
Output:
-1/864*(6*(12*a*b + 11*b^2)*c^2*x^2 + 54*(4*a*b + 3*b^2)*c*x - 9*(b^2*c^3* x^3 + 3*b^2*c^2*x^2 + 3*b^2*c*x - 7*b^2)*log(-(c*x + 1)/(c*x - 1))^2 + 288 *a^2 + 240*a*b + 112*b^2 - 3*((12*a*b + 11*b^2)*c^3*x^3 + 3*(12*a*b + 7*b^ 2)*c^2*x^2 + 3*(12*a*b - b^2)*c*x - 84*a*b - 29*b^2)*log(-(c*x + 1)/(c*x - 1)))/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{\left (c x + 1\right )^{4}}\, dx \] Input:
integrate((a+b*atanh(c*x))**2/(c*x+1)**4,x)
Output:
Integral((a + b*atanh(c*x))**2/(c*x + 1)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (158) = 316\).
Time = 0.05 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=-\frac {1}{72} \, {\left (c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} + \frac {48 \, \operatorname {artanh}\left (c x\right )}{c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c}\right )} a b - \frac {1}{864} \, {\left (12 \, c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left (66 \, c^{2} x^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 9 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 162 \, c x - 3 \, {\left (11 \, c^{3} x^{3} + 33 \, c^{2} x^{2} + 33 \, c x + 6 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 11\right )} \log \left (c x + 1\right ) + 33 \, {\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 112\right )} c^{2}}{c^{6} x^{3} + 3 \, c^{5} x^{2} + 3 \, c^{4} x + c^{3}}\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{3 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} - \frac {a^{2}}{3 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*x+1)^4,x, algorithm="maxima")
Output:
-1/72*(c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2) + 48*arctanh(c*x)/(c^4*x^3 + 3 *c^3*x^2 + 3*c^2*x + c))*a*b - 1/864*(12*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^ 5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c ^2)*arctanh(c*x) + (66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c *x + 1)^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 162*c*x - 3*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1) *log(c*x - 1) + 11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*lo g(c*x - 1) + 112)*c^2/(c^6*x^3 + 3*c^5*x^2 + 3*c^4*x + c^3))*b^2 - 1/3*b^2 *arctanh(c*x)^2/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/3*a^2/(c^4*x^3 + 3 *c^3*x^2 + 3*c^2*x + c)
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (158) = 316\).
Time = 0.13 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.89 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=\frac {1}{1728} \, c {\left (\frac {18 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {3 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {6 \, {\left (\frac {36 \, {\left (c x + 1\right )}^{2} a b}{{\left (c x - 1\right )}^{2}} - \frac {36 \, {\left (c x + 1\right )} a b}{c x - 1} + 12 \, a b + \frac {18 \, {\left (c x + 1\right )}^{2} b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {9 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + 2 \, b^{2}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {{\left (\frac {216 \, {\left (c x + 1\right )}^{2} a^{2}}{{\left (c x - 1\right )}^{2}} - \frac {216 \, {\left (c x + 1\right )} a^{2}}{c x - 1} + 72 \, a^{2} + \frac {216 \, {\left (c x + 1\right )}^{2} a b}{{\left (c x - 1\right )}^{2}} - \frac {108 \, {\left (c x + 1\right )} a b}{c x - 1} + 24 \, a b + \frac {108 \, {\left (c x + 1\right )}^{2} b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {27 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + 4 \, b^{2}\right )} {\left (c x - 1\right )}^{3}}{{\left (c x + 1\right )}^{3} c^{2}}\right )} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*x+1)^4,x, algorithm="giac")
Output:
1/1728*c*(18*(3*(c*x + 1)^2*b^2/(c*x - 1)^2 - 3*(c*x + 1)*b^2/(c*x - 1) + b^2)*(c*x - 1)^3*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)^3*c^2) + 6*(36*(c* x + 1)^2*a*b/(c*x - 1)^2 - 36*(c*x + 1)*a*b/(c*x - 1) + 12*a*b + 18*(c*x + 1)^2*b^2/(c*x - 1)^2 - 9*(c*x + 1)*b^2/(c*x - 1) + 2*b^2)*(c*x - 1)^3*log (-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^2) + (216*(c*x + 1)^2*a^2/(c*x - 1)^ 2 - 216*(c*x + 1)*a^2/(c*x - 1) + 72*a^2 + 216*(c*x + 1)^2*a*b/(c*x - 1)^2 - 108*(c*x + 1)*a*b/(c*x - 1) + 24*a*b + 108*(c*x + 1)^2*b^2/(c*x - 1)^2 - 27*(c*x + 1)*b^2/(c*x - 1) + 4*b^2)*(c*x - 1)^3/((c*x + 1)^3*c^2))
Time = 4.92 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=\ln \left (1-c\,x\right )\,\left (\ln \left (c\,x+1\right )\,\left (\frac {b^2}{3\,c\,\left (2\,c^3\,x^3+6\,c^2\,x^2+6\,c\,x+2\right )}-\frac {b^2\,\left (c^3\,x^3+3\,c^2\,x^2+3\,c\,x+1\right )}{24\,c\,\left (2\,c^3\,x^3+6\,c^2\,x^2+6\,c\,x+2\right )}\right )+\frac {b^2}{3\,c\,\left (6\,c^3\,x^3+18\,c^2\,x^2+18\,c\,x+6\right )}+\frac {b\,\left (6\,a-b\right )}{3\,c\,\left (6\,c^3\,x^3+18\,c^2\,x^2+18\,c\,x+6\right )}+\frac {b^2\,\left (11\,c^3\,x^3+45\,c^2\,x^2+69\,c\,x+51\right )}{48\,c\,\left (6\,c^3\,x^3+18\,c^2\,x^2+18\,c\,x+6\right )}\right )-\frac {x\,\left (27\,b^2+36\,a\,b\right )+x^2\,\left (11\,c\,b^2+12\,a\,c\,b\right )+\frac {8\,\left (18\,a^2+15\,a\,b+7\,b^2\right )}{3\,c}}{144\,c^3\,x^3+432\,c^2\,x^2+432\,c\,x+144}+{\ln \left (c\,x+1\right )}^2\,\left (\frac {b^2}{96\,c}-\frac {b^2}{12\,c^2\,\left (3\,x+3\,c\,x^2+\frac {1}{c}+c^2\,x^3\right )}\right )+{\ln \left (1-c\,x\right )}^2\,\left (\frac {b^2}{96\,c}-\frac {b^2}{3\,c\,\left (4\,c^3\,x^3+12\,c^2\,x^2+12\,c\,x+4\right )}\right )-\frac {\ln \left (c\,x+1\right )\,\left (\frac {7\,b^2}{96\,c^2}+\frac {5\,b^2\,x^2}{32}+\frac {23\,b^2\,x}{96\,c}+\frac {11\,b^2\,c\,x^3}{288}+\frac {b\,\left (16\,a+5\,b\right )}{48\,c^2}\right )}{3\,x+3\,c\,x^2+\frac {1}{c}+c^2\,x^3}-\frac {b\,\mathrm {atan}\left (c\,x\,1{}\mathrm {i}\right )\,\left (6\,a+11\,b\right )\,1{}\mathrm {i}}{72\,c} \] Input:
int((a + b*atanh(c*x))^2/(c*x + 1)^4,x)
Output:
log(1 - c*x)*(log(c*x + 1)*(b^2/(3*c*(6*c*x + 6*c^2*x^2 + 2*c^3*x^3 + 2)) - (b^2*(3*c*x + 3*c^2*x^2 + c^3*x^3 + 1))/(24*c*(6*c*x + 6*c^2*x^2 + 2*c^3 *x^3 + 2))) + b^2/(3*c*(18*c*x + 18*c^2*x^2 + 6*c^3*x^3 + 6)) + (b*(6*a - b))/(3*c*(18*c*x + 18*c^2*x^2 + 6*c^3*x^3 + 6)) + (b^2*(69*c*x + 45*c^2*x^ 2 + 11*c^3*x^3 + 51))/(48*c*(18*c*x + 18*c^2*x^2 + 6*c^3*x^3 + 6))) - (x*( 36*a*b + 27*b^2) + x^2*(11*b^2*c + 12*a*b*c) + (8*(15*a*b + 18*a^2 + 7*b^2 ))/(3*c))/(432*c*x + 432*c^2*x^2 + 144*c^3*x^3 + 144) + log(c*x + 1)^2*(b^ 2/(96*c) - b^2/(12*c^2*(3*x + 3*c*x^2 + 1/c + c^2*x^3))) + log(1 - c*x)^2* (b^2/(96*c) - b^2/(3*c*(12*c*x + 12*c^2*x^2 + 4*c^3*x^3 + 4))) - (log(c*x + 1)*((7*b^2)/(96*c^2) + (5*b^2*x^2)/32 + (23*b^2*x)/(96*c) + (11*b^2*c*x^ 3)/288 + (b*(16*a + 5*b))/(48*c^2)))/(3*x + 3*c*x^2 + 1/c + c^2*x^3) - (b* atan(c*x*1i)*(6*a + 11*b)*1i)/(72*c)
Time = 0.17 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(1+c x)^4} \, dx=\frac {-96 b^{2} c x -576 \mathit {atanh} \left (c x \right ) a b -36 \,\mathrm {log}\left (c x -1\right ) a b +36 \,\mathrm {log}\left (c x +1\right ) a b +24 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-144 \mathit {atanh} \left (c x \right ) b^{2} c x +24 a b \,c^{3} x^{3}-144 a b c x +22 b^{2} c^{3} x^{3}+108 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -288 a^{2}+36 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}+108 \,\mathrm {log}\left (c x +1\right ) a b \,c^{2} x^{2}-108 \,\mathrm {log}\left (c x -1\right ) a b c x +108 \,\mathrm {log}\left (c x +1\right ) a b c x +108 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-21 \,\mathrm {log}\left (c x -1\right ) b^{2}+21 \,\mathrm {log}\left (c x +1\right ) b^{2}-252 \mathit {atanh} \left (c x \right )^{2} b^{2}-216 \mathit {atanh} \left (c x \right ) b^{2}-63 \,\mathrm {log}\left (c x -1\right ) b^{2} c x +63 \,\mathrm {log}\left (c x +1\right ) b^{2} c x -90 b^{2}-216 a b -63 \,\mathrm {log}\left (c x -1\right ) b^{2} c^{2} x^{2}+63 \,\mathrm {log}\left (c x +1\right ) b^{2} c^{2} x^{2}+36 \,\mathrm {log}\left (c x +1\right ) a b \,c^{3} x^{3}-21 \,\mathrm {log}\left (c x -1\right ) b^{2} c^{3} x^{3}+21 \,\mathrm {log}\left (c x +1\right ) b^{2} c^{3} x^{3}-36 \,\mathrm {log}\left (c x -1\right ) a b \,c^{3} x^{3}-108 \,\mathrm {log}\left (c x -1\right ) a b \,c^{2} x^{2}}{864 c \left (c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1\right )} \] Input:
int((a+b*atanh(c*x))^2/(c*x+1)^4,x)
Output:
(36*atanh(c*x)**2*b**2*c**3*x**3 + 108*atanh(c*x)**2*b**2*c**2*x**2 + 108* atanh(c*x)**2*b**2*c*x - 252*atanh(c*x)**2*b**2 - 576*atanh(c*x)*a*b + 24* atanh(c*x)*b**2*c**3*x**3 - 144*atanh(c*x)*b**2*c*x - 216*atanh(c*x)*b**2 - 36*log(c*x - 1)*a*b*c**3*x**3 - 108*log(c*x - 1)*a*b*c**2*x**2 - 108*log (c*x - 1)*a*b*c*x - 36*log(c*x - 1)*a*b - 21*log(c*x - 1)*b**2*c**3*x**3 - 63*log(c*x - 1)*b**2*c**2*x**2 - 63*log(c*x - 1)*b**2*c*x - 21*log(c*x - 1)*b**2 + 36*log(c*x + 1)*a*b*c**3*x**3 + 108*log(c*x + 1)*a*b*c**2*x**2 + 108*log(c*x + 1)*a*b*c*x + 36*log(c*x + 1)*a*b + 21*log(c*x + 1)*b**2*c** 3*x**3 + 63*log(c*x + 1)*b**2*c**2*x**2 + 63*log(c*x + 1)*b**2*c*x + 21*lo g(c*x + 1)*b**2 - 288*a**2 + 24*a*b*c**3*x**3 - 144*a*b*c*x - 216*a*b + 22 *b**2*c**3*x**3 - 96*b**2*c*x - 90*b**2)/(864*c*(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1))