Integrand size = 17, antiderivative size = 134 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1}{36 a \left (1-a^2 x^2\right )^3}-\frac {5}{96 a \left (1-a^2 x^2\right )^2}-\frac {5}{32 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \text {arctanh}(a x)}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \text {arctanh}(a x)}{16 \left (1-a^2 x^2\right )}+\frac {5 \text {arctanh}(a x)^2}{32 a} \] Output:
-1/36/a/(-a^2*x^2+1)^3-5/96/a/(-a^2*x^2+1)^2-5/32/a/(-a^2*x^2+1)+1/6*x*arc tanh(a*x)/(-a^2*x^2+1)^3+5/24*x*arctanh(a*x)/(-a^2*x^2+1)^2+5*x*arctanh(a* x)/(-16*a^2*x^2+16)+5/32*arctanh(a*x)^2/a
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\frac {68-105 a^2 x^2+45 a^4 x^4-6 a x \left (33-40 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)+45 \left (-1+a^2 x^2\right )^3 \text {arctanh}(a x)^2}{288 a \left (-1+a^2 x^2\right )^3} \] Input:
Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^4,x]
Output:
(68 - 105*a^2*x^2 + 45*a^4*x^4 - 6*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcT anh[a*x] + 45*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2)/(288*a*(-1 + a^2*x^2)^3)
Time = 0.45 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6522, 6522, 6518, 241}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\) |
Input:
Int[ArcTanh[a*x]/(1 - a^2*x^2)^4,x]
Output:
-1/36*1/(a*(1 - a^2*x^2)^3) + (x*ArcTanh[a*x])/(6*(1 - a^2*x^2)^3) + (5*(- 1/16*1/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*(-1 /4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x] ^2/(4*a)))/4))/6
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Time = 0.69 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93
| method | result | size |
| parallelrisch | \(-\frac {-45 \operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}+198 a x \,\operatorname {arctanh}\left (a x \right )+90 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-99 a^{2} x^{2}-68 a^{6} x^{6}+159 a^{4} x^{4}+45 \operatorname {arctanh}\left (a x \right )^{2}-240 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+135 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}-135 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{288 \left (a^{2} x^{2}-1\right )^{3} a}\) | \(124\) |
| derivativedivides | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{48 \left (a x -1\right )^{3}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {\operatorname {arctanh}\left (a x \right )}{48 \left (a x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{288 \left (a x -1\right )^{3}}-\frac {7}{384 \left (a x -1\right )^{2}}+\frac {37}{384 \left (a x -1\right )}-\frac {1}{288 \left (a x +1\right )^{3}}-\frac {7}{384 \left (a x +1\right )^{2}}-\frac {37}{384 \left (a x +1\right )}}{a}\) | \(222\) |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (a x \right )}{48 \left (a x -1\right )^{3}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32}-\frac {\operatorname {arctanh}\left (a x \right )}{48 \left (a x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32}-\frac {5 \ln \left (a x -1\right )^{2}}{128}+\frac {5 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{64}+\frac {5 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{64}-\frac {5 \ln \left (a x +1\right )^{2}}{128}+\frac {1}{288 \left (a x -1\right )^{3}}-\frac {7}{384 \left (a x -1\right )^{2}}+\frac {37}{384 \left (a x -1\right )}-\frac {1}{288 \left (a x +1\right )^{3}}-\frac {7}{384 \left (a x +1\right )^{2}}-\frac {37}{384 \left (a x +1\right )}}{a}\) | \(222\) |
| risch | \(\frac {5 \ln \left (a x +1\right )^{2}}{128 a}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 x^{4} \ln \left (-a x +1\right ) a^{4}-80 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+66 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{192 \left (a^{2} x^{2}-1\right )^{3} a}+\frac {45 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+180 x^{5} \ln \left (-a x +1\right ) a^{5}-135 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+180 a^{4} x^{4}-480 a^{3} x^{3} \ln \left (-a x +1\right )+135 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-420 a^{2} x^{2}+396 a x \ln \left (-a x +1\right )-45 \ln \left (-a x +1\right )^{2}+272}{1152 \left (a x +1\right ) a \left (a x -1\right ) \left (a^{2} x^{2}-1\right )^{2}}\) | \(263\) |
| parts | \(-\frac {\operatorname {arctanh}\left (a x \right )}{48 a \left (a x +1\right )^{3}}-\frac {\operatorname {arctanh}\left (a x \right )}{16 a \left (a x +1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 a \left (a x +1\right )}+\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{32 a}-\frac {\operatorname {arctanh}\left (a x \right )}{48 a \left (a x -1\right )^{3}}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a \left (a x -1\right )^{2}}-\frac {5 \,\operatorname {arctanh}\left (a x \right )}{32 a \left (a x -1\right )}-\frac {5 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{32 a}-\frac {a \left (\frac {\frac {15 \ln \left (a x -1\right )^{2}}{4}-\frac {15 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {15 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}}{a^{2}}-\frac {15 \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )}{a^{2}}+\frac {1}{3 a^{2} \left (a x +1\right )^{3}}+\frac {37}{4 a^{2} \left (a x +1\right )}+\frac {7}{4 a^{2} \left (a x +1\right )^{2}}-\frac {1}{3 a^{2} \left (a x -1\right )^{3}}-\frac {37}{4 a^{2} \left (a x -1\right )}+\frac {7}{4 a^{2} \left (a x -1\right )^{2}}\right )}{96}\) | \(294\) |
Input:
int(arctanh(a*x)/(-a^2*x^2+1)^4,x,method=_RETURNVERBOSE)
Output:
-1/288*(-45*arctanh(a*x)^2*a^6*x^6+198*a*x*arctanh(a*x)+90*arctanh(a*x)*a^ 5*x^5-99*a^2*x^2-68*a^6*x^6+159*a^4*x^4+45*arctanh(a*x)^2-240*a^3*x^3*arct anh(a*x)+135*a^4*x^4*arctanh(a*x)^2-135*a^2*x^2*arctanh(a*x)^2)/(a^2*x^2-1 )^3/a
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\frac {180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} + 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 272}{1152 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^4,x, algorithm="fricas")
Output:
1/1152*(180*a^4*x^4 - 420*a^2*x^2 + 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x)*log (-(a*x + 1)/(a*x - 1)) + 272)/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)
\[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \] Input:
integrate(atanh(a*x)/(-a**2*x**2+1)**4,x)
Output:
Integral(atanh(a*x)/((a*x - 1)**4*(a*x + 1)**4), x)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (114) = 228\).
Time = 0.05 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.79 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a}{1152 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^4,x, algorithm="maxima")
Output:
-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a*x - 1)/a)*arctanh(a*x) + 1/1152*(180* a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 272)*a/(a^8*x ^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2)
\[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{4}} \,d x } \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^4,x, algorithm="giac")
Output:
integrate(arctanh(a*x)/(a^2*x^2 - 1)^4, x)
Time = 4.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.54 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\frac {\frac {34}{3\,a}-\frac {35\,a\,x^2}{2}+\frac {15\,a^3\,x^4}{2}}{48\,a^6\,x^6-144\,a^4\,x^4+144\,a^2\,x^2-48}-\ln \left (1-a\,x\right )\,\left (\frac {5\,\ln \left (a\,x+1\right )}{64\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{2\,a^6\,x^6-6\,a^4\,x^4+6\,a^2\,x^2-2}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^2}{128\,a}+\frac {5\,{\ln \left (1-a\,x\right )}^2}{128\,a}-\frac {\ln \left (a\,x+1\right )\,\left (\frac {11\,x}{32\,a}-\frac {5\,a\,x^3}{12}+\frac {5\,a^3\,x^5}{32}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6} \] Input:
int(atanh(a*x)/(a^2*x^2 - 1)^4,x)
Output:
(34/(3*a) - (35*a*x^2)/2 + (15*a^3*x^4)/2)/(144*a^2*x^2 - 144*a^4*x^4 + 48 *a^6*x^6 - 48) - log(1 - a*x)*((5*log(a*x + 1))/(64*a) - ((11*x)/16 - (5*a ^2*x^3)/6 + (5*a^4*x^5)/16)/(6*a^2*x^2 - 6*a^4*x^4 + 2*a^6*x^6 - 2)) + (5* log(a*x + 1)^2)/(128*a) + (5*log(1 - a*x)^2)/(128*a) - (log(a*x + 1)*((11* x)/(32*a) - (5*a*x^3)/12 + (5*a^3*x^5)/32))/(3*a*x^2 - 1/a - 3*a^3*x^4 + a ^5*x^6)
Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4} \, dx=\frac {45 \mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-135 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+135 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-45 \mathit {atanh} \left (a x \right )^{2}-90 \mathit {atanh} \left (a x \right ) a^{5} x^{5}+240 \mathit {atanh} \left (a x \right ) a^{3} x^{3}-198 \mathit {atanh} \left (a x \right ) a x +15 a^{6} x^{6}-60 a^{2} x^{2}+53}{288 a \left (a^{6} x^{6}-3 a^{4} x^{4}+3 a^{2} x^{2}-1\right )} \] Input:
int(atanh(a*x)/(-a^2*x^2+1)^4,x)
Output:
(45*atanh(a*x)**2*a**6*x**6 - 135*atanh(a*x)**2*a**4*x**4 + 135*atanh(a*x) **2*a**2*x**2 - 45*atanh(a*x)**2 - 90*atanh(a*x)*a**5*x**5 + 240*atanh(a*x )*a**3*x**3 - 198*atanh(a*x)*a*x + 15*a**6*x**6 - 60*a**2*x**2 + 53)/(288* a*(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1))