Integrand size = 12, antiderivative size = 63 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {a \sqrt {1-a^2 x^2}}{x}-\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-1/2*(-a^2*x^2+1)^(1/2)/x^2+a*(-a^2*x^2+1)^(1/2)/x-1/2*a^2*arctanh((-a^2*x ^2+1)^(1/2))
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\frac {1}{2} \left (\frac {(-1+2 a x) \sqrt {1-a^2 x^2}}{x^2}+a^2 \log (x)-a^2 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \] Input:
Integrate[1/(E^ArcTanh[a*x]*x^3),x]
Output:
(((-1 + 2*a*x)*Sqrt[1 - a^2*x^2])/x^2 + a^2*Log[x] - a^2*Log[1 + Sqrt[1 - a^2*x^2]])/2
Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6674, 539, 27, 534, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {1-a x}{x^3 \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -\frac {1}{2} \int \frac {a (2-a x)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} a \int \frac {2-a x}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {1}{2} a \left (-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {1}{2} a \left (-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {1}{2} a \left (\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {1}{2} a \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
Input:
Int[1/(E^ArcTanh[a*x]*x^3),x]
Output:
-1/2*Sqrt[1 - a^2*x^2]/x^2 - (a*((-2*Sqrt[1 - a^2*x^2])/x + a*ArcTanh[Sqrt [1 - a^2*x^2]]))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x )^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / ; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {2 a^{3} x^{3}-a^{2} x^{2}-2 a x +1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(59\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}+\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )-a^{2} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\) | \(188\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(2*a^3*x^3-a^2*x^2-2*a*x+1)/x^2/(-a^2*x^2+1)^(1/2)-1/2*a^2*arctanh(1/ (-a^2*x^2+1)^(1/2))
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\frac {a^{2} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )}}{2 \, x^{2}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm="fricas")
Output:
1/2*(a^2*x^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(2*a*x - 1))/x^2
\[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**3,x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} x^{3}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (53) = 106\).
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.52 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\frac {{\left (a^{3} - \frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} - \frac {a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x, algorithm="giac")
Output:
1/8*(a^3 - 4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x)*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)) - 1/2*a^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs (a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/8*(4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a *abs(a)/x - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)/(a*x^2))/a^2
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\frac {a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {a^2\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{2} \] Input:
int((1 - a^2*x^2)^(1/2)/(x^3*(a*x + 1)),x)
Output:
(a*(1 - a^2*x^2)^(1/2))/x - (1 - a^2*x^2)^(1/2)/(2*x^2) - (a^2*atanh((1 - a^2*x^2)^(1/2)))/2
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^3} \, dx=\frac {2 \sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}}{2 x^{2}} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^3,x)
Output:
(2*sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1) + log(tan(asin(a*x) /2))*a**2*x**2)/(2*x**2)