Integrand size = 22, antiderivative size = 189 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}-2 a^3 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)+\frac {7}{6} a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-i a^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+i a^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
-1/3*(-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^3-2*a^3*arctan((-a*x+1)^(1/2)/(a*x+ 1)^(1/2))*arctanh(a*x)+7/6*a^3*arctanh((-a^2*x^2+1)^(1/2))-I*a^3*polylog(2 ,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))+I*a^3*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1) ^(1/2))-1/6*a*(-a^2*x^2+1)^(1/2)/x^2+a^2*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x
Time = 1.13 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.27 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=-a^3 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a x}+i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )-i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-\log \left (\cosh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \left (8 \text {arctanh}(a x)+2 \sinh (2 \text {arctanh}(a x))+\frac {\left (\log \left (\cosh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )\right ) \left (3 a x-\sqrt {1-a^2 x^2} \sinh (3 \text {arctanh}(a x))\right )}{\sqrt {1-a^2 x^2}}\right )}{24 x^3} \]
-(a^3*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(a*x)) + I*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - Log[Cosh[Ar cTanh[a*x]/2]] + Log[Sinh[ArcTanh[a*x]/2]] + I*PolyLog[2, (-I)/E^ArcTanh[a *x]] - I*PolyLog[2, I/E^ArcTanh[a*x]])) - ((1 - a^2*x^2)^(3/2)*(8*ArcTanh[ a*x] + 2*Sinh[2*ArcTanh[a*x]] + ((Log[Cosh[ArcTanh[a*x]/2]] - Log[Sinh[Arc Tanh[a*x]/2]])*(3*a*x - Sqrt[1 - a^2*x^2]*Sinh[3*ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(24*x^3)
Time = 1.01 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6576, 6570, 243, 51, 73, 221, 6576, 6512, 6570, 243, 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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^4}dx-a^2 \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle a^2 \left (-\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\right )+\frac {1}{3} a \int \frac {\sqrt {1-a^2 x^2}}{x^3}dx-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle a^2 \left (-\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\right )+\frac {1}{6} a \int \frac {\sqrt {1-a^2 x^2}}{x^4}dx^2-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle a^2 \left (-\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\right )+\frac {1}{6} a \left (-\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle a^2 \left (-\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\right )+\frac {1}{6} a \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle a^2 \left (-\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2}dx\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle -\left (a^2 \left (\int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx-a^2 \int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 6512 |
\(\displaystyle -\left (a^2 \left (\int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx-a^2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle -\left (a^2 \left (a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\left (a^2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\left (a^2 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\left (a^2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\left (a^2 \left (-\left (a^2 \left (-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}\right )\right )-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{3 x^3}\) |
-1/3*((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^3 + (a*(-(Sqrt[1 - a^2*x^2]/x^2) + a^2*ArcTanh[Sqrt[1 - a^2*x^2]]))/6 - a^2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[ a*x])/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]] - a^2*((-2*ArcTan[Sqrt[1 - a*x]/Sq rt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a))
3.5.55.3.1 Defintions of rubi rules used
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_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol ] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
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]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Time = 0.24 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {\left (8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-a x -2 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{6 x^{3}}-\frac {7 a^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{6}+\frac {7 a^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{6}-i a^{3} \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+i a^{3} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-i a^{3} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+i a^{3} \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(218\) |
1/6*(8*a^2*x^2*arctanh(a*x)-a*x-2*arctanh(a*x))*(-a^2*x^2+1)^(1/2)/x^3-7/6 *a^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+7/6*a^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1 /2))-I*a^3*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+I*a^3*arctanh(a*x)*ln(1-I *(a*x+1)/(-a^2*x^2+1)^(1/2))-I*a^3*arctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1 )^(1/2))+I*a^3*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{4}} \,d x } \]
\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{4}} \,d x } \]
Exception generated. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^4} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^4} \,d x \]