Integrand size = 22, antiderivative size = 94 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}-\frac {3}{40} a^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
3/40*a^3*(-a^2*x^2+1)^(1/2)/x^2-1/20*a*(-a^2*x^2+1)^(3/2)/x^4-1/5*(-a^2*x^ 2+1)^(5/2)*arctanh(a*x)/x^5-3/40*a^5*arctanh((-a^2*x^2+1)^(1/2))
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\left (-\frac {a}{20 x^4}+\frac {a^3}{8 x^2}\right ) \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)}{5 x^5}+\frac {3}{40} a^5 \log (x)-\frac {3}{40} a^5 \log \left (1+\sqrt {1-a^2 x^2}\right ) \] Input:
Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^6,x]
Output:
(-1/20*a/x^4 + a^3/(8*x^2))*Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*(-1 + a ^2*x^2)^2*ArcTanh[a*x])/(5*x^5) + (3*a^5*Log[x])/40 - (3*a^5*Log[1 + Sqrt[ 1 - a^2*x^2]])/40
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6570, 243, 51, 51, 73, 221}
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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle \frac {1}{5} a \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^5}dx-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{10} a \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^6}dx^2-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \int \frac {\sqrt {1-a^2 x^2}}{x^4}dx^2-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (-\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{10} a \left (-\frac {3}{4} a^2 \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (1-a^2 x^2\right )^{5/2} \text {arctanh}(a x)}{5 x^5}\) |
Input:
Int[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^6,x]
Output:
-1/5*((1 - a^2*x^2)^(5/2)*ArcTanh[a*x])/x^5 + (a*(-1/2*(1 - a^2*x^2)^(3/2) /x^4 - (3*a^2*(-(Sqrt[1 - a^2*x^2]/x^2) + a^2*ArcTanh[Sqrt[1 - a^2*x^2]])) /4))/10
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Time = 1.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (8 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-5 a^{3} x^{3}-16 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+2 a x +8 \,\operatorname {arctanh}\left (a x \right )\right )}{40 x^{5}}-\frac {3 a^{5} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{40}+\frac {3 a^{5} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{40}\) | \(116\) |
Input:
int((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/40*(-(a*x-1)*(a*x+1))^(1/2)*(8*a^4*x^4*arctanh(a*x)-5*a^3*x^3-16*a^2*x^ 2*arctanh(a*x)+2*a*x+8*arctanh(a*x))/x^5-3/40*a^5*ln(1+(a*x+1)/(-a^2*x^2+1 )^(1/2))+3/40*a^5*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\frac {3 \, a^{5} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (5 \, a^{3} x^{3} - 2 \, a x - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, x^{5}} \] Input:
integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^6,x, algorithm="fricas")
Output:
1/40*(3*a^5*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (5*a^3*x^3 - 2*a*x - 4*( a^4*x^4 - 2*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1))/x^ 5
\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{6}}\, dx \] Input:
integrate((-a**2*x**2+1)**(3/2)*atanh(a*x)/x**6,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x)/x**6, x)
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.34 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\frac {1}{40} \, {\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4} - 3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{4} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}{x^{2}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4}}\right )} a - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )}{5 \, x^{5}} \] Input:
integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^6,x, algorithm="maxima")
Output:
1/40*((-a^2*x^2 + 1)^(3/2)*a^4 - 3*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2 /abs(x)) + 3*sqrt(-a^2*x^2 + 1)*a^4 + (-a^2*x^2 + 1)^(5/2)*a^2/x^2 - 2*(-a ^2*x^2 + 1)^(5/2)/x^4)*a - 1/5*(-a^2*x^2 + 1)^(5/2)*arctanh(a*x)/x^5
Exception generated. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^6,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^6} \,d x \] Input:
int((atanh(a*x)*(1 - a^2*x^2)^(3/2))/x^6,x)
Output:
int((atanh(a*x)*(1 - a^2*x^2)^(3/2))/x^6, x)
\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^6} \, dx=\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )}{x^{6}}d x -\left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )}{x^{4}}d x \right ) a^{2} \] Input:
int((-a^2*x^2+1)^(3/2)*atanh(a*x)/x^6,x)
Output:
int((sqrt( - a**2*x**2 + 1)*atanh(a*x))/x**6,x) - int((sqrt( - a**2*x**2 + 1)*atanh(a*x))/x**4,x)*a**2