Integrand size = 18, antiderivative size = 208 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=-\frac {3 b^3}{64 c (1+c x)^2}-\frac {21 b^3}{64 c (1+c x)}+\frac {21 b^3 \text {arctanh}(c x)}{64 c}-\frac {3 b^2 (a+b \text {arctanh}(c x))}{16 c (1+c x)^2}-\frac {9 b^2 (a+b \text {arctanh}(c x))}{16 c (1+c x)}+\frac {9 b (a+b \text {arctanh}(c x))^2}{32 c}-\frac {3 b (a+b \text {arctanh}(c x))^2}{8 c (1+c x)^2}-\frac {3 b (a+b \text {arctanh}(c x))^2}{8 c (1+c x)}+\frac {(a+b \text {arctanh}(c x))^3}{8 c}-\frac {(a+b \text {arctanh}(c x))^3}{2 c (1+c x)^2} \] Output:
-3/64*b^3/c/(c*x+1)^2-21/64*b^3/c/(c*x+1)+21/64*b^3*arctanh(c*x)/c-3/16*b^ 2*(a+b*arctanh(c*x))/c/(c*x+1)^2-9/16*b^2*(a+b*arctanh(c*x))/c/(c*x+1)+9/3 2*b*(a+b*arctanh(c*x))^2/c-3/8*b*(a+b*arctanh(c*x))^2/c/(c*x+1)^2-3/8*b*(a +b*arctanh(c*x))^2/c/(c*x+1)+1/8*(a+b*arctanh(c*x))^3/c-1/2*(a+b*arctanh(c *x))^3/c/(c*x+1)^2
Time = 0.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=\frac {-2 \left (32 a^3+24 a^2 b+12 a b^2+3 b^3\right )-6 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)-24 b \left (8 a^2+4 a b (2+c x)+b^2 (4+3 c x)\right ) \text {arctanh}(c x)+12 b^2 (-1+c x) (4 a (3+c x)+b (5+3 c x)) \text {arctanh}(c x)^2+16 b^3 \left (-3+2 c x+c^2 x^2\right ) \text {arctanh}(c x)^3-3 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)^2 \log (1-c x)+3 b \left (8 a^2+12 a b+7 b^2\right ) (1+c x)^2 \log (1+c x)}{128 c (1+c x)^2} \] Input:
Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^3,x]
Output:
(-2*(32*a^3 + 24*a^2*b + 12*a*b^2 + 3*b^3) - 6*b*(8*a^2 + 12*a*b + 7*b^2)* (1 + c*x) - 24*b*(8*a^2 + 4*a*b*(2 + c*x) + b^2*(4 + 3*c*x))*ArcTanh[c*x] + 12*b^2*(-1 + c*x)*(4*a*(3 + c*x) + b*(5 + 3*c*x))*ArcTanh[c*x]^2 + 16*b^ 3*(-3 + 2*c*x + c^2*x^2)*ArcTanh[c*x]^3 - 3*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 - c*x] + 3*b*(8*a^2 + 12*a*b + 7*b^2)*(1 + c*x)^2*Log[1 + c *x])/(128*c*(1 + c*x)^2)
Time = 0.62 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{(c x+1)^3} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {3}{2} b \int \left (\frac {(a+b \text {arctanh}(c x))^2}{4 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arctanh}(c x))^2}{4 (c x+1)^2}+\frac {(a+b \text {arctanh}(c x))^2}{2 (c x+1)^3}\right )dx-\frac {(a+b \text {arctanh}(c x))^3}{2 c (c x+1)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} b \left (\frac {(a+b \text {arctanh}(c x))^3}{12 b c}+\frac {3 (a+b \text {arctanh}(c x))^2}{16 c}-\frac {(a+b \text {arctanh}(c x))^2}{4 c (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{4 c (c x+1)^2}-\frac {3 b (a+b \text {arctanh}(c x))}{8 c (c x+1)}-\frac {b (a+b \text {arctanh}(c x))}{8 c (c x+1)^2}+\frac {7 b^2 \text {arctanh}(c x)}{32 c}-\frac {7 b^2}{32 c (c x+1)}-\frac {b^2}{32 c (c x+1)^2}\right )-\frac {(a+b \text {arctanh}(c x))^3}{2 c (c x+1)^2}\) |
Input:
Int[(a + b*ArcTanh[c*x])^3/(1 + c*x)^3,x]
Output:
-1/2*(a + b*ArcTanh[c*x])^3/(c*(1 + c*x)^2) + (3*b*(-1/32*b^2/(c*(1 + c*x) ^2) - (7*b^2)/(32*c*(1 + c*x)) + (7*b^2*ArcTanh[c*x])/(32*c) - (b*(a + b*A rcTanh[c*x]))/(8*c*(1 + c*x)^2) - (3*b*(a + b*ArcTanh[c*x]))/(8*c*(1 + c*x )) + (3*(a + b*ArcTanh[c*x])^2)/(16*c) - (a + b*ArcTanh[c*x])^2/(4*c*(1 + c*x)^2) - (a + b*ArcTanh[c*x])^2/(4*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^3/ (12*b*c)))/2
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 = 1.40 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \(-\frac {-48 a^{2} b \,c^{2} x^{2}-72 a^{2} b c x -12 b^{3} \operatorname {arctanh}\left (c x \right )^{2} c x -18 b^{3} \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}-24 b^{3} c^{2} x^{2}-32 a^{3} c^{2} x^{2}-64 a^{3} c x +72 \,\operatorname {arctanh}\left (c x \right ) a^{2} b -21 \,\operatorname {arctanh}\left (c x \right ) b^{3} c^{2} x^{2}-6 \,\operatorname {arctanh}\left (c x \right ) b^{3} c x -27 b^{3} c x -48 a \,b^{2} c^{2} x^{2}-60 a \,b^{2} c x +24 \operatorname {arctanh}\left (c x \right )^{3} b^{3}+30 b^{3} \operatorname {arctanh}\left (c x \right )^{2}+27 \,\operatorname {arctanh}\left (c x \right ) b^{3}+72 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2}-24 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2} c^{2} x^{2}-24 \,\operatorname {arctanh}\left (c x \right ) a^{2} b \,c^{2} x^{2}-48 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2} c x -48 \,\operatorname {arctanh}\left (c x \right ) a^{2} b c x -36 a \,b^{2} \operatorname {arctanh}\left (c x \right ) c^{2} x^{2}-24 a \,b^{2} \operatorname {arctanh}\left (c x \right ) c x +60 a \,b^{2} \operatorname {arctanh}\left (c x \right )-8 \operatorname {arctanh}\left (c x \right )^{3} b^{3} c^{2} x^{2}-16 \operatorname {arctanh}\left (c x \right )^{3} b^{3} c x}{64 \left (c x +1\right )^{2} c}\) | \(327\) |
| orering | \(-\frac {\left (105 c^{5} x^{5}-231 c^{4} x^{4}+18 x^{3} c^{3}+280 c^{2} x^{2}-253 c x +81\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3}}{64 \left (c x +1\right )^{2} c}-\frac {\left (c x -1\right ) \left (c x +1\right )^{2} \left (105 c^{4} x^{4}-154 x^{3} c^{3}-88 c^{2} x^{2}+208 c x -73\right ) \left (\frac {3 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b c}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )}-\frac {3 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c}{\left (c x +1\right )^{4}}\right )}{64 c^{2}}-\frac {\left (c x +1\right )^{3} \left (c x -1\right )^{2} \left (105 x^{3} c^{3}-77 c^{2} x^{2}-139 c x +99\right ) \left (\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{2}}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )^{2}}-\frac {18 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{2}}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )}+\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3} x}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {12 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c^{2}}{\left (c x +1\right )^{5}}\right )}{256 c^{3}}-\frac {\left (7 c^{2} x^{2}-9\right ) \left (c x +1\right )^{4} \left (c x -1\right )^{3} \left (\frac {6 b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{3} \left (c x +1\right )^{3}}-\frac {54 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{3}}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {36 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{4} x}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )^{3}}+\frac {108 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3}}{\left (c x +1\right )^{5} \left (-c^{2} x^{2}+1\right )}-\frac {54 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{4} x}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {24 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )^{3}}+\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3}}{\left (c x +1\right )^{3} \left (-c^{2} x^{2}+1\right )^{2}}-\frac {60 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c^{3}}{\left (c x +1\right )^{6}}\right )}{256 c^{4}}\) | \(627\) |
| risch | \(\frac {b^{3} \left (c^{2} x^{2}+2 c x -3\right ) \ln \left (c x +1\right )^{3}}{64 \left (c x +1\right )^{2} c}+\frac {3 b^{2} \left (-2 b \,c^{2} x^{2} \ln \left (-c x +1\right )+4 a \,c^{2} x^{2}+3 b \,c^{2} x^{2}-4 b c x \ln \left (-c x +1\right )+8 a c x +2 b c x +6 b \ln \left (-c x +1\right )-12 a -5 b \right ) \ln \left (c x +1\right )^{2}}{128 \left (c x +1\right )^{2} c}-\frac {3 b \left (-b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+4 a b \,c^{2} \ln \left (-c x +1\right ) x^{2}+3 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )-2 b^{2} c x \ln \left (-c x +1\right )^{2}+8 a b c x \ln \left (-c x +1\right )+2 b^{2} c x \ln \left (-c x +1\right )+8 a b c x +6 b^{2} c x +3 \ln \left (-c x +1\right )^{2} b^{2}-12 \ln \left (-c x +1\right ) a b -5 b^{2} \ln \left (-c x +1\right )+16 a^{2}+16 b a +8 b^{2}\right ) \ln \left (c x +1\right )}{64 \left (c x +1\right )^{2} c}-\frac {96 a \,b^{2}+96 a^{2} b +48 a^{2} b c x -96 a \,b^{2} \ln \left (-c x +1\right )-24 \ln \left (-c x -1\right ) a^{2} b -36 \ln \left (-c x -1\right ) a \,b^{2}+64 a^{3}+36 a \,b^{2} \ln \left (c x -1\right )+24 a^{2} b \ln \left (c x -1\right )+36 \ln \left (-c x +1\right )^{2} a \,b^{2}-96 \ln \left (-c x +1\right ) a^{2} b -42 \ln \left (-c x -1\right ) b^{3} c x +42 \ln \left (c x -1\right ) b^{3} c x +48 b^{3}-6 \ln \left (-c x +1\right )^{3} b^{3}-6 b^{3} c x \ln \left (-c x +1\right )^{2}+42 b^{3} c x +72 a \,b^{2} c x -12 \ln \left (-c x +1\right )^{2} a \,b^{2} c^{2} x^{2}-24 \ln \left (-c x +1\right )^{2} a \,b^{2} c x -48 a \,b^{2} c x \ln \left (-c x +1\right )-21 b^{3} \ln \left (-c x -1\right )+21 b^{3} \ln \left (c x -1\right )-48 b^{3} \ln \left (-c x +1\right )+15 b^{3} \ln \left (-c x +1\right )^{2}-48 \ln \left (-c x -1\right ) a^{2} b c x -72 \ln \left (-c x -1\right ) a \,b^{2} c x +48 \ln \left (c x -1\right ) a^{2} b c x +72 \ln \left (c x -1\right ) a \,b^{2} c x +24 \ln \left (c x -1\right ) a^{2} b \,c^{2} x^{2}+36 \ln \left (c x -1\right ) a \,b^{2} c^{2} x^{2}-24 \ln \left (-c x -1\right ) a^{2} b \,c^{2} x^{2}-36 \ln \left (-c x -1\right ) a \,b^{2} c^{2} x^{2}+2 \ln \left (-c x +1\right )^{3} b^{3} c^{2} x^{2}+4 \ln \left (-c x +1\right )^{3} b^{3} c x +21 \ln \left (c x -1\right ) b^{3} c^{2} x^{2}-21 \ln \left (-c x -1\right ) b^{3} c^{2} x^{2}-9 b^{3} c^{2} x^{2} \ln \left (-c x +1\right )^{2}-36 b^{3} c x \ln \left (-c x +1\right )}{128 \left (c x +1\right )^{2} c}\) | \(861\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(991\) |
| default | \(\text {Expression too large to display}\) | \(991\) |
| parts | \(\text {Expression too large to display}\) | \(999\) |
Input:
int((a+b*arctanh(c*x))^3/(c*x+1)^3,x,method=_RETURNVERBOSE)
Output:
-1/64*(-48*a^2*b*c^2*x^2-72*a^2*b*c*x-12*b^3*arctanh(c*x)^2*c*x-18*b^3*arc tanh(c*x)^2*c^2*x^2-24*b^3*c^2*x^2-32*a^3*c^2*x^2-64*a^3*c*x+72*arctanh(c* x)*a^2*b-21*arctanh(c*x)*b^3*c^2*x^2-6*arctanh(c*x)*b^3*c*x-27*b^3*c*x-48* a*b^2*c^2*x^2-60*a*b^2*c*x+24*arctanh(c*x)^3*b^3+30*b^3*arctanh(c*x)^2+27* arctanh(c*x)*b^3+72*arctanh(c*x)^2*a*b^2-24*arctanh(c*x)^2*a*b^2*c^2*x^2-2 4*arctanh(c*x)*a^2*b*c^2*x^2-48*arctanh(c*x)^2*a*b^2*c*x-48*arctanh(c*x)*a ^2*b*c*x-36*a*b^2*arctanh(c*x)*c^2*x^2-24*a*b^2*arctanh(c*x)*c*x+60*a*b^2* arctanh(c*x)-8*arctanh(c*x)^3*b^3*c^2*x^2-16*arctanh(c*x)^3*b^3*c*x)/(c*x+ 1)^2/c
Time = 0.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=\frac {2 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3} c x - 3 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} - 64 \, a^{3} - 96 \, a^{2} b - 96 \, a b^{2} - 48 \, b^{3} - 6 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x + 3 \, {\left ({\left (4 \, a b^{2} + 3 \, b^{3}\right )} c^{2} x^{2} - 12 \, a b^{2} - 5 \, b^{3} + 2 \, {\left (4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 3 \, {\left ({\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 24 \, a^{2} b - 20 \, a b^{2} - 9 \, b^{3} + 2 \, {\left (8 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{128 \, {\left (c^{3} x^{2} + 2 \, c^{2} x + c\right )}} \] Input:
integrate((a+b*arctanh(c*x))^3/(c*x+1)^3,x, algorithm="fricas")
Output:
1/128*(2*(b^3*c^2*x^2 + 2*b^3*c*x - 3*b^3)*log(-(c*x + 1)/(c*x - 1))^3 - 6 4*a^3 - 96*a^2*b - 96*a*b^2 - 48*b^3 - 6*(8*a^2*b + 12*a*b^2 + 7*b^3)*c*x + 3*((4*a*b^2 + 3*b^3)*c^2*x^2 - 12*a*b^2 - 5*b^3 + 2*(4*a*b^2 + b^3)*c*x) *log(-(c*x + 1)/(c*x - 1))^2 + 3*((8*a^2*b + 12*a*b^2 + 7*b^3)*c^2*x^2 - 2 4*a^2*b - 20*a*b^2 - 9*b^3 + 2*(8*a^2*b + 4*a*b^2 + b^3)*c*x)*log(-(c*x + 1)/(c*x - 1)))/(c^3*x^2 + 2*c^2*x + c)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (c x + 1\right )^{3}}\, dx \] Input:
integrate((a+b*atanh(c*x))**3/(c*x+1)**3,x)
Output:
Integral((a + b*atanh(c*x))**3/(c*x + 1)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (188) = 376\).
Time = 0.06 (sec) , antiderivative size = 796, normalized size of antiderivative = 3.83 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+b*arctanh(c*x))^3/(c*x+1)^3,x, algorithm="maxima")
Output:
-1/2*b^3*arctanh(c*x)^3/(c^3*x^2 + 2*c^2*x + c) - 3/16*(c*(2*(c*x + 2)/(c^ 4*x^2 + 2*c^3*x + c^2) - log(c*x + 1)/c^2 + log(c*x - 1)/c^2) + 8*arctanh( c*x)/(c^3*x^2 + 2*c^2*x + c))*a^2*b - 3/32*(4*c*(2*(c*x + 2)/(c^4*x^2 + 2* c^3*x + c^2) - log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*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*x^2 + 2*c^4*x + c^3))*a*b^2 - 1/128*(24*c*(2*(c*x + 2)/(c^4*x^2 + 2*c^3*x + c^2) - log( c*x + 1)/c^2 + log(c*x - 1)/c^2)*arctanh(c*x)^2 - ((2*(c^2*x^2 + 2*c*x + 1 )*log(c*x + 1)^3 - 2*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1)^3 - 3*(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)^2 - 9*(c^2 *x^2 + 2*c*x + 1)*log(c*x - 1)^2 - 42*c*x + 3*(7*c^2*x^2 + 2*(c^2*x^2 + 2* c*x + 1)*log(c*x - 1)^2 + 14*c*x + 6*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) + 7)*log(c*x + 1) - 21*(c^2*x^2 + 2*c*x + 1)*log(c*x - 1) - 48)*c^2/(c^6*x^2 + 2*c^5*x + c^4) - 12*((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*arctanh(c*x)/(c^5*x^2 + 2*c^4*x + c^3))*c)*b^3 - 3/2*a*b^2*arcta nh(c*x)^2/(c^3*x^2 + 2*c^2*x + c) - 1/2*a^3/(c^3*x^2 + 2*c^2*x + c)
Time = 0.13 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=\frac {1}{256} \, {\left (\frac {4 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {4 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {6 \, {\left (\frac {16 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 8 \, a^{2} b + \frac {16 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 4 \, a b^{2} + \frac {8 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - b^{3}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2}} + \frac {{\left (\frac {64 \, {\left (c x + 1\right )} a^{3}}{c x - 1} - 32 \, a^{3} + \frac {96 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} - 24 \, a^{2} b + \frac {96 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} - 12 \, a b^{2} + \frac {48 \, {\left (c x + 1\right )} b^{3}}{c x - 1} - 3 \, b^{3}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{2}}\right )} c \] Input:
integrate((a+b*arctanh(c*x))^3/(c*x+1)^3,x, algorithm="giac")
Output:
1/256*(4*(2*(c*x + 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))^3/((c*x + 1)^2*c^2) + 6*(8*(c*x + 1)*a*b^2/(c*x - 1) - 4*a*b^2 + 4* (c*x + 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*log(-(c*x + 1)/(c*x - 1))^2/((c *x + 1)^2*c^2) + 6*(16*(c*x + 1)*a^2*b/(c*x - 1) - 8*a^2*b + 16*(c*x + 1)* a*b^2/(c*x - 1) - 4*a*b^2 + 8*(c*x + 1)*b^3/(c*x - 1) - b^3)*(c*x - 1)^2*l og(-(c*x + 1)/(c*x - 1))/((c*x + 1)^2*c^2) + (64*(c*x + 1)*a^3/(c*x - 1) - 32*a^3 + 96*(c*x + 1)*a^2*b/(c*x - 1) - 24*a^2*b + 96*(c*x + 1)*a*b^2/(c* x - 1) - 12*a*b^2 + 48*(c*x + 1)*b^3/(c*x - 1) - 3*b^3)*(c*x - 1)^2/((c*x + 1)^2*c^2))*c
Time = 6.02 (sec) , antiderivative size = 930, normalized size of antiderivative = 4.47 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx =\text {Too large to display} \] Input:
int((a + b*atanh(c*x))^3/(c*x + 1)^3,x)
Output:
(102*b^3*log(1 - c*x) - 102*b^3*log(c*x + 1) - 96*a*b^2 - 96*a^2*b - 15*b^ 3*log(c*x + 1)^2 - 6*b^3*log(c*x + 1)^3 - 15*b^3*log(1 - c*x)^2 + 6*b^3*lo g(1 - c*x)^3 + 150*b^3*atanh(c*x) - 64*a^3 - 48*b^3 + 144*a*b^2*atanh(c*x) + 48*a^2*b*atanh(c*x) + 30*b^3*log(c*x + 1)*log(1 - c*x) - 132*a*b^2*log( c*x + 1) - 96*a^2*b*log(c*x + 1) + 132*a*b^2*log(1 - c*x) + 96*a^2*b*log(1 - c*x) - 18*b^3*log(c*x + 1)*log(1 - c*x)^2 + 18*b^3*log(c*x + 1)^2*log(1 - c*x) - 36*a*b^2*log(c*x + 1)^2 - 36*a*b^2*log(1 - c*x)^2 - 42*b^3*c*x - 144*b^3*c*x*log(c*x + 1) + 144*b^3*c*x*log(1 - c*x) + 9*b^3*c^2*x^2*log(c *x + 1)^2 + 2*b^3*c^2*x^2*log(c*x + 1)^3 + 9*b^3*c^2*x^2*log(1 - c*x)^2 - 2*b^3*c^2*x^2*log(1 - c*x)^3 + 150*b^3*c^2*x^2*atanh(c*x) - 72*a*b^2*c*x - 48*a^2*b*c*x + 6*b^3*c*x*log(c*x + 1)^2 + 4*b^3*c*x*log(c*x + 1)^3 + 6*b^ 3*c*x*log(1 - c*x)^2 - 4*b^3*c*x*log(1 - c*x)^3 + 72*a*b^2*log(c*x + 1)*lo g(1 - c*x) + 300*b^3*c*x*atanh(c*x) - 54*b^3*c^2*x^2*log(c*x + 1) + 54*b^3 *c^2*x^2*log(1 - c*x) - 12*b^3*c*x*log(c*x + 1)*log(1 - c*x) - 36*a*b^2*c^ 2*x^2*log(c*x + 1) + 36*a*b^2*c^2*x^2*log(1 - c*x) + 6*b^3*c^2*x^2*log(c*x + 1)*log(1 - c*x)^2 - 6*b^3*c^2*x^2*log(c*x + 1)^2*log(1 - c*x) - 120*a*b ^2*c*x*log(c*x + 1) + 120*a*b^2*c*x*log(1 - c*x) + 12*b^3*c*x*log(c*x + 1) *log(1 - c*x)^2 - 12*b^3*c*x*log(c*x + 1)^2*log(1 - c*x) + 12*a*b^2*c^2*x^ 2*log(c*x + 1)^2 + 12*a*b^2*c^2*x^2*log(1 - c*x)^2 + 144*a*b^2*c^2*x^2*ata nh(c*x) + 48*a^2*b*c^2*x^2*atanh(c*x) + 24*a*b^2*c*x*log(c*x + 1)^2 + 2...
Time = 0.17 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^3} \, dx=\frac {-24 \,\mathrm {log}\left (c x -1\right ) a^{2} b -12 \,\mathrm {log}\left (c x -1\right ) a \,b^{2}+24 \,\mathrm {log}\left (c x +1\right ) a^{2} b +12 \,\mathrm {log}\left (c x +1\right ) a \,b^{2}-48 \mathit {atanh} \left (c x \right )^{3} b^{3}-60 \mathit {atanh} \left (c x \right ) b^{3}-72 a^{2} b -60 a \,b^{2}-48 \,\mathrm {log}\left (c x -1\right ) a^{2} b c x -24 \,\mathrm {log}\left (c x -1\right ) a \,b^{2} c x +48 \,\mathrm {log}\left (c x +1\right ) a^{2} b c x +24 \,\mathrm {log}\left (c x +1\right ) a \,b^{2} c x -6 \,\mathrm {log}\left (c x -1\right ) b^{3} c x +6 \,\mathrm {log}\left (c x +1\right ) b^{3} c x -3 \,\mathrm {log}\left (c x -1\right ) b^{3} c^{2} x^{2}+3 \,\mathrm {log}\left (c x +1\right ) b^{3} c^{2} x^{2}-12 \,\mathrm {log}\left (c x -1\right ) a \,b^{2} c^{2} x^{2}+24 \,\mathrm {log}\left (c x +1\right ) a^{2} b \,c^{2} x^{2}+12 \,\mathrm {log}\left (c x +1\right ) a \,b^{2} c^{2} x^{2}+32 \mathit {atanh} \left (c x \right )^{3} b^{3} c x +36 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{2} x^{2}+24 a^{2} b \,c^{2} x^{2}+21 b^{3} c^{2} x^{2}-64 a^{3}-3 \,\mathrm {log}\left (c x -1\right ) b^{3}+3 \,\mathrm {log}\left (c x +1\right ) b^{3}-24 \,\mathrm {log}\left (c x -1\right ) a^{2} b \,c^{2} x^{2}+24 \mathit {atanh} \left (c x \right )^{2} b^{3} c x +36 \mathit {atanh} \left (c x \right ) b^{3} c^{2} x^{2}+36 a \,b^{2} c^{2} x^{2}+16 \mathit {atanh} \left (c x \right )^{3} b^{3} c^{2} x^{2}+96 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c x +48 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{2} x^{2}-60 \mathit {atanh} \left (c x \right )^{2} b^{3}-144 \mathit {atanh} \left (c x \right )^{2} a \,b^{2}-192 \mathit {atanh} \left (c x \right ) a^{2} b -144 \mathit {atanh} \left (c x \right ) a \,b^{2}-27 b^{3}+48 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{2} x^{2}}{128 c \left (c^{2} x^{2}+2 c x +1\right )} \] Input:
int((a+b*atanh(c*x))^3/(c*x+1)^3,x)
Output:
(16*atanh(c*x)**3*b**3*c**2*x**2 + 32*atanh(c*x)**3*b**3*c*x - 48*atanh(c* x)**3*b**3 + 48*atanh(c*x)**2*a*b**2*c**2*x**2 + 96*atanh(c*x)**2*a*b**2*c *x - 144*atanh(c*x)**2*a*b**2 + 36*atanh(c*x)**2*b**3*c**2*x**2 + 24*atanh (c*x)**2*b**3*c*x - 60*atanh(c*x)**2*b**3 - 192*atanh(c*x)*a**2*b + 48*ata nh(c*x)*a*b**2*c**2*x**2 - 144*atanh(c*x)*a*b**2 + 36*atanh(c*x)*b**3*c**2 *x**2 - 60*atanh(c*x)*b**3 - 24*log(c*x - 1)*a**2*b*c**2*x**2 - 48*log(c*x - 1)*a**2*b*c*x - 24*log(c*x - 1)*a**2*b - 12*log(c*x - 1)*a*b**2*c**2*x* *2 - 24*log(c*x - 1)*a*b**2*c*x - 12*log(c*x - 1)*a*b**2 - 3*log(c*x - 1)* b**3*c**2*x**2 - 6*log(c*x - 1)*b**3*c*x - 3*log(c*x - 1)*b**3 + 24*log(c* x + 1)*a**2*b*c**2*x**2 + 48*log(c*x + 1)*a**2*b*c*x + 24*log(c*x + 1)*a** 2*b + 12*log(c*x + 1)*a*b**2*c**2*x**2 + 24*log(c*x + 1)*a*b**2*c*x + 12*l og(c*x + 1)*a*b**2 + 3*log(c*x + 1)*b**3*c**2*x**2 + 6*log(c*x + 1)*b**3*c *x + 3*log(c*x + 1)*b**3 - 64*a**3 + 24*a**2*b*c**2*x**2 - 72*a**2*b + 36* a*b**2*c**2*x**2 - 60*a*b**2 + 21*b**3*c**2*x**2 - 27*b**3)/(128*c*(c**2*x **2 + 2*c*x + 1))