Integrand size = 19, antiderivative size = 133 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {4}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {8}{15 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \text {arctanh}(a x)}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \text {arctanh}(a x)}{15 \sqrt {1-a^2 x^2}} \] Output:
-1/25/a/(-a^2*x^2+1)^(5/2)-4/45/a/(-a^2*x^2+1)^(3/2)-8/15/a/(-a^2*x^2+1)^( 1/2)+1/5*x*arctanh(a*x)/(-a^2*x^2+1)^(5/2)+4/15*x*arctanh(a*x)/(-a^2*x^2+1 )^(3/2)+8/15*x*arctanh(a*x)/(-a^2*x^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.49 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\frac {-149+260 a^2 x^2-120 a^4 x^4+15 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \text {arctanh}(a x)}{225 a \left (1-a^2 x^2\right )^{5/2}} \] Input:
Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(7/2),x]
Output:
(-149 + 260*a^2*x^2 - 120*a^4*x^4 + 15*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*A rcTanh[a*x])/(225*a*(1 - a^2*x^2)^(5/2))
Time = 0.49 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6522, 6522, 6520}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {4}{5} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\) |
Input:
Int[ArcTanh[a*x]/(1 - a^2*x^2)^(7/2),x]
Output:
-1/25*1/(a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x])/(5*(1 - a^2*x^2)^(5/2)) + (4*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3*(1 - a^2*x^2)^ (3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^ 2]))/3))/5
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 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 = 1.55 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.59
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (120 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-120 a^{4} x^{4}-300 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+260 a^{2} x^{2}+225 a x \,\operatorname {arctanh}\left (a x \right )-149\right )}{225 a \left (a^{2} x^{2}-1\right )^{3}}\) | \(79\) |
orering | \(\frac {\left (-\frac {64}{15} a^{6} x^{7}+\frac {616}{45} a^{4} x^{5}-\frac {3388}{225} a^{2} x^{3}+\frac {1268}{225} x \right ) \operatorname {arctanh}\left (a x \right )}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}-\frac {\left (120 a^{4} x^{4}-260 a^{2} x^{2}+149\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {a}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {7 \,\operatorname {arctanh}\left (a x \right ) a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}\right )}{225 a^{2}}\) | \(121\) |
Input:
int(arctanh(a*x)/(-a^2*x^2+1)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/225/a*(-a^2*x^2+1)^(1/2)*(120*arctanh(a*x)*a^5*x^5-120*a^4*x^4-300*a^3* x^3*arctanh(a*x)+260*a^2*x^2+225*a*x*arctanh(a*x)-149)/(a^2*x^2-1)^3
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.74 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\frac {{\left (240 \, a^{4} x^{4} - 520 \, a^{2} x^{2} - 15 \, {\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 298\right )} \sqrt {-a^{2} x^{2} + 1}}{450 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(7/2),x, algorithm="fricas")
Output:
1/450*(240*a^4*x^4 - 520*a^2*x^2 - 15*(8*a^5*x^5 - 20*a^3*x^3 + 15*a*x)*lo g(-(a*x + 1)/(a*x - 1)) + 298)*sqrt(-a^2*x^2 + 1)/(a^7*x^6 - 3*a^5*x^4 + 3 *a^3*x^2 - a)
\[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate(atanh(a*x)/(-a**2*x**2+1)**(7/2),x)
Output:
Integral(atanh(a*x)/(-(a*x - 1)*(a*x + 1))**(7/2), x)
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\frac {1}{225} \, a {\left (\frac {120}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {20}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {9}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {3 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(7/2),x, algorithm="maxima")
Output:
-1/225*a*(120/(sqrt(-a^2*x^2 + 1)*a^2) + 20/((-a^2*x^2 + 1)^(3/2)*a^2) + 9 /((-a^2*x^2 + 1)^(5/2)*a^2)) + 1/15*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^ 2 + 1)^(3/2) + 3*x/(-a^2*x^2 + 1)^(5/2))*arctanh(a*x)
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (4 \, {\left (2 \, a^{4} x^{2} - 5 \, a^{2}\right )} x^{2} + 15\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{30 \, {\left (a^{2} x^{2} - 1\right )}^{3}} + \frac {20 \, a^{2} x^{2} - 120 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 29}{225 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} a} \] Input:
integrate(arctanh(a*x)/(-a^2*x^2+1)^(7/2),x, algorithm="giac")
Output:
-1/30*sqrt(-a^2*x^2 + 1)*(4*(2*a^4*x^2 - 5*a^2)*x^2 + 15)*x*log(-(a*x + 1) /(a*x - 1))/(a^2*x^2 - 1)^3 + 1/225*(20*a^2*x^2 - 120*(a^2*x^2 - 1)^2 - 29 )/((a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*a)
Timed out. \[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \] Input:
int(atanh(a*x)/(1 - a^2*x^2)^(7/2),x)
Output:
int(atanh(a*x)/(1 - a^2*x^2)^(7/2), x)
\[ \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x \right ) \] Input:
int(atanh(a*x)/(-a^2*x^2+1)^(7/2),x)
Output:
- int(atanh(a*x)/(sqrt( - a**2*x**2 + 1)*a**6*x**6 - 3*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 3*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1)),x)