Integrand size = 20, antiderivative size = 119 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {3 x}{8 a \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)}{8 a^2}+\frac {3 \text {arctanh}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac {3 x \text {arctanh}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)^3}{4 a^2}+\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )} \] Output:
-3/8*x/a/(-a^2*x^2+1)-3/8*arctanh(a*x)/a^2+3/4*arctanh(a*x)/a^2/(-a^2*x^2+ 1)-3/4*x*arctanh(a*x)^2/a/(-a^2*x^2+1)-1/4*arctanh(a*x)^3/a^2+1/2*arctanh( a*x)^3/a^2/(-a^2*x^2+1)
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {6 a x-12 \text {arctanh}(a x)+12 a x \text {arctanh}(a x)^2-4 \left (1+a^2 x^2\right ) \text {arctanh}(a x)^3+3 \left (-1+a^2 x^2\right ) \log (1-a x)-3 \left (-1+a^2 x^2\right ) \log (1+a x)}{16 a^2 \left (-1+a^2 x^2\right )} \] Input:
Integrate[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2)^2,x]
Output:
(6*a*x - 12*ArcTanh[a*x] + 12*a*x*ArcTanh[a*x]^2 - 4*(1 + a^2*x^2)*ArcTanh [a*x]^3 + 3*(-1 + a^2*x^2)*Log[1 - a*x] - 3*(-1 + a^2*x^2)*Log[1 + a*x])/( 16*a^2*(-1 + a^2*x^2))
Time = 0.49 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6556, 6518, 6556, 215, 219}
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.
| \(\displaystyle \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{2 a}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\) |
Input:
Int[(x*ArcTanh[a*x]^3)/(1 - a^2*x^2)^2,x]
Output:
ArcTanh[a*x]^3/(2*a^2*(1 - a^2*x^2)) - (3*((x*ArcTanh[a*x]^2)/(2*(1 - a^2* x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTanh[a*x]/(2*a^2*(1 - a^2*x^2)) - (x/ (2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a))))/(2*a)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*( (a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q _.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q + 1))), x] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTanh[c* x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
Time = 32.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(-\frac {2 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x^{2}+3 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-6 \operatorname {arctanh}\left (a x \right )^{2} a x +2 \operatorname {arctanh}\left (a x \right )^{3}-3 a x +3 \,\operatorname {arctanh}\left (a x \right )}{8 \left (a^{2} x^{2}-1\right ) a^{2}}\) | \(72\) |
| risch | \(-\frac {\left (a^{2} x^{2}+1\right ) \ln \left (a x +1\right )^{3}}{32 a^{2} \left (a x -1\right ) \left (a x +1\right )}+\frac {3 \left (x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{32 a^{2} \left (a x -1\right ) \left (a x +1\right )}-\frac {3 \left (a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}+4\right ) \ln \left (a x +1\right )}{32 a^{2} \left (a x -1\right ) \left (a x +1\right )}-\frac {-a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 \ln \left (a x +1\right ) a^{2} x^{2}-6 x^{2} \ln \left (-a x +1\right ) a^{2}-6 a \ln \left (-a x +1\right )^{2} x -\ln \left (-a x +1\right )^{3}-12 a x -6 \ln \left (a x +1\right )-6 \ln \left (-a x +1\right )}{32 a^{2} \left (a x -1\right ) \left (a x +1\right )}\) | \(265\) |
| orering | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (9 a^{6} x^{6}+3 a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \operatorname {arctanh}\left (a x \right )^{3}}{2 a^{4} x^{2} \left (-a^{2} x^{2}+1\right )^{2}}-\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (17 a^{4} x^{4}+6 a^{2} x^{2}+2\right ) \left (\frac {\operatorname {arctanh}\left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 x \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {4 x^{2} \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}\right )}{4 a^{4} x^{2}}-\frac {\left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (4 a^{2} x^{2}+1\right ) \left (\frac {6 \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {12 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {6 x \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {30 x^{2} \operatorname {arctanh}\left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {24 x^{3} \operatorname {arctanh}\left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{4}}\right )}{4 x \,a^{4}}-\frac {\left (a x +1\right )^{4} \left (a x -1\right )^{4} \left (\frac {18 \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {132 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {144 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {12 \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {6 x \,a^{3}}{\left (-a^{2} x^{2}+1\right )^{5}}+\frac {108 x^{2} \operatorname {arctanh}\left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{5}}+\frac {312 x^{3} \operatorname {arctanh}\left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{5}}+\frac {192 x^{4} \operatorname {arctanh}\left (a x \right )^{3} a^{6}}{\left (-a^{2} x^{2}+1\right )^{5}}\right )}{16 a^{4}}\) | \(532\) |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{2 \left (a^{2} x^{2}-1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{8 \left (a x +1\right )}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{8 \left (a x -1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {3 i \pi \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \left (a x -1\right )}{32 \left (a x +1\right )}}{a^{2}}\) | \(746\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{2 \left (a^{2} x^{2}-1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{8 \left (a x +1\right )}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{8 \left (a x -1\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {3 i \pi \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\frac {3 a x}{32}+\frac {3}{32}}{a x -1}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \left (a x -1\right )}{32 \left (a x +1\right )}}{a^{2}}\) | \(746\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{2 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{2 \left (4 a x -4\right )}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{2 \left (4 a x +4\right )}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {3 i \pi \operatorname {arctanh}\left (a x \right )^{2}}{8}+\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right ) \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}{16}+\frac {3 i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1\right )}\right )^{2} \operatorname {arctanh}\left (a x \right )^{2}}{16}-\frac {3 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3} \operatorname {arctanh}\left (a x \right )^{2}}{8}-\frac {\operatorname {arctanh}\left (a x \right )^{3}}{4}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {3 \left (a x +1\right )}{2 \left (16 a x -16\right )}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \left (a x -1\right )}{16 \left (a x +1\right )}-\frac {3 \left (a x -1\right )}{32 \left (a x +1\right )}}{a^{2}}\) | \(751\) |
Input:
int(x*arctanh(a*x)^3/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
-1/8*(2*arctanh(a*x)^3*a^2*x^2+3*a^2*x^2*arctanh(a*x)-6*arctanh(a*x)^2*a*x +2*arctanh(a*x)^3-3*a*x+3*arctanh(a*x))/(a^2*x^2-1)/a^2
Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{32 \, {\left (a^{4} x^{2} - a^{2}\right )}} \] Input:
integrate(x*arctanh(a*x)^3/(-a^2*x^2+1)^2,x, algorithm="fricas")
Output:
1/32*(6*a*x*log(-(a*x + 1)/(a*x - 1))^2 - (a^2*x^2 + 1)*log(-(a*x + 1)/(a* x - 1))^3 + 12*a*x - 6*(a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))/(a^4*x^2 - a^2)
\[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\int \frac {x \operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \] Input:
integrate(x*atanh(a*x)**3/(-a**2*x**2+1)**2,x)
Output:
Integral(x*atanh(a*x)**3/((a*x - 1)**2*(a*x + 1)**2), x)
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (103) = 206\).
Time = 0.04 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.50 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {3 \, {\left (\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2}}{8 \, a} - \frac {\frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{5} x^{2} - a^{3}} - \frac {6 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname {artanh}\left (a x\right )}{a^{4} x^{2} - a^{2}}}{32 \, a} - \frac {\operatorname {artanh}\left (a x\right )^{3}}{2 \, {\left (a^{2} x^{2} - 1\right )} a^{2}} \] Input:
integrate(x*arctanh(a*x)^3/(-a^2*x^2+1)^2,x, algorithm="maxima")
Output:
3/8*(2*x/(a^2*x^2 - 1) - log(a*x + 1)/a + log(a*x - 1)/a)*arctanh(a*x)^2/a - 1/32*(((a^2*x^2 - 1)*log(a*x + 1)^3 - 3*(a^2*x^2 - 1)*log(a*x + 1)^2*lo g(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^3 - 12*a*x + 3*(2*a^2*x^2 + (a^2*x ^2 - 1)*log(a*x - 1)^2 - 2)*log(a*x + 1) - 6*(a^2*x^2 - 1)*log(a*x - 1))*a ^2/(a^5*x^2 - a^3) - 6*((a^2*x^2 - 1)*log(a*x + 1)^2 - 2*(a^2*x^2 - 1)*log (a*x + 1)*log(a*x - 1) + (a^2*x^2 - 1)*log(a*x - 1)^2 - 4)*a*arctanh(a*x)/ (a^4*x^2 - a^2))/a - 1/2*arctanh(a*x)^3/((a^2*x^2 - 1)*a^2)
Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.61 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {1}{64} \, {\left ({\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 3 \, {\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} - \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, {\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - \frac {6 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}} + \frac {6 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{3}}\right )} a \] Input:
integrate(x*arctanh(a*x)^3/(-a^2*x^2+1)^2,x, algorithm="giac")
Output:
-1/64*(((a*x + 1)/((a*x - 1)*a^3) + (a*x - 1)/((a*x + 1)*a^3))*log(-(a*x + 1)/(a*x - 1))^3 - 3*((a*x + 1)/((a*x - 1)*a^3) - (a*x - 1)/((a*x + 1)*a^3 ))*log(-(a*x + 1)/(a*x - 1))^2 + 6*((a*x + 1)/((a*x - 1)*a^3) + (a*x - 1)/ ((a*x + 1)*a^3))*log(-(a*x + 1)/(a*x - 1)) - 6*(a*x + 1)/((a*x - 1)*a^3) + 6*(a*x - 1)/((a*x + 1)*a^3))*a
Time = 4.43 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.01 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=-\frac {6\,\ln \left (1-a\,x\right )-6\,\ln \left (a\,x+1\right )+12\,a\,x-{\ln \left (a\,x+1\right )}^3+{\ln \left (1-a\,x\right )}^3-3\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+3\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )-a^2\,x^2\,\left (6\,\ln \left (a\,x+1\right )-6\,\ln \left (1-a\,x\right )\right )-a^2\,x^2\,{\ln \left (a\,x+1\right )}^3+a^2\,x^2\,{\ln \left (1-a\,x\right )}^3+6\,a\,x\,{\ln \left (a\,x+1\right )}^2+6\,a\,x\,{\ln \left (1-a\,x\right )}^2-12\,a\,x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )-3\,a^2\,x^2\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2+3\,a^2\,x^2\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{32\,a^2-32\,a^4\,x^2} \] Input:
int((x*atanh(a*x)^3)/(a^2*x^2 - 1)^2,x)
Output:
-(6*log(1 - a*x) - 6*log(a*x + 1) + 12*a*x - log(a*x + 1)^3 + log(1 - a*x) ^3 - 3*log(a*x + 1)*log(1 - a*x)^2 + 3*log(a*x + 1)^2*log(1 - a*x) - a^2*x ^2*(6*log(a*x + 1) - 6*log(1 - a*x)) - a^2*x^2*log(a*x + 1)^3 + a^2*x^2*lo g(1 - a*x)^3 + 6*a*x*log(a*x + 1)^2 + 6*a*x*log(1 - a*x)^2 - 12*a*x*log(a* x + 1)*log(1 - a*x) - 3*a^2*x^2*log(a*x + 1)*log(1 - a*x)^2 + 3*a^2*x^2*lo g(a*x + 1)^2*log(1 - a*x))/(32*a^2 - 32*a^4*x^2)
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02 \[ \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx=\frac {-4 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-4 \mathit {atanh} \left (a x \right )^{3}+12 \mathit {atanh} \left (a x \right )^{2} a x -12 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-3 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}+3 \,\mathrm {log}\left (a^{2} x -a \right )+3 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}-3 \,\mathrm {log}\left (a^{2} x +a \right )+6 a x}{16 a^{2} \left (a^{2} x^{2}-1\right )} \] Input:
int(x*atanh(a*x)^3/(-a^2*x^2+1)^2,x)
Output:
( - 4*atanh(a*x)**3*a**2*x**2 - 4*atanh(a*x)**3 + 12*atanh(a*x)**2*a*x - 1 2*atanh(a*x)*a**2*x**2 - 3*log(a**2*x - a)*a**2*x**2 + 3*log(a**2*x - a) + 3*log(a**2*x + a)*a**2*x**2 - 3*log(a**2*x + a) + 6*a*x)/(16*a**2*(a**2*x **2 - 1))