Integrand size = 12, antiderivative size = 61 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=-\frac {4 a (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}+3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-4*a*(-a*x+1)/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/x+3*a*arctanh((-a^2*x^ 2+1)^(1/2))
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=\sqrt {1-a^2 x^2} \left (-\frac {1}{x}-\frac {4 a}{1+a x}\right )-3 a \log (x)+3 a \log \left (1+\sqrt {1-a^2 x^2}\right ) \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
Sqrt[1 - a^2*x^2]*(-x^(-1) - (4*a)/(1 + a*x)) - 3*a*Log[x] + 3*a*Log[1 + S qrt[1 - a^2*x^2]]
Time = 0.76 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6674, 2353, 2009}
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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {(1-a x)^2}{x^2 (a x+1) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 2353 |
\(\displaystyle \int \left (\frac {4 a^2}{(a x+1) \sqrt {1-a^2 x^2}}-\frac {3 a}{x \sqrt {1-a^2 x^2}}+\frac {1}{x^2 \sqrt {1-a^2 x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {4 a \sqrt {1-a^2 x^2}}{a x+1}-\frac {\sqrt {1-a^2 x^2}}{x}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
-(Sqrt[1 - a^2*x^2]/x) - (4*a*Sqrt[1 - a^2*x^2])/(1 + a*x) + 3*a*ArcTanh[S qrt[1 - a^2*x^2]]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
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 = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}}-a \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {4 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a \left (x +\frac {1}{a}\right )}\right )\) | \(81\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a}-3 a \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {2 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+9 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\) | \(464\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
(a^2*x^2-1)/x/(-a^2*x^2+1)^(1/2)-a*(-3*arctanh(1/(-a^2*x^2+1)^(1/2))+4/a/( x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=-\frac {4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left (a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x + 1\right )}}{a x^{2} + x} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="fricas")
Output:
-(4*a^2*x^2 + 4*a*x + 3*(a^2*x^2 + a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + sqrt(-a^2*x^2 + 1)*(5*a*x + 1))/(a*x^2 + x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (54) = 108\).
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.46 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=\frac {3 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {{\left (a^{2} + \frac {17 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, x {\left | a \right |}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="giac")
Output:
3*a^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/2*(a^2 + 17*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/x)*a^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 1 /2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a))
Time = 14.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=3\,a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {\sqrt {1-a^2\,x^2}}{x}+\frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \] Input:
int((1 - a^2*x^2)^(3/2)/(x^2*(a*x + 1)^3),x)
Output:
3*a*atanh((1 - a^2*x^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/x + (4*a^2*(1 - a^2*x ^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^2} \, dx=\frac {a \left (-6 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+18 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-1\right )}{2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right ) \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x)
Output:
(a*( - 6*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**2 - 6*log(tan(asin(a*x)/2 ))*tan(asin(a*x)/2) + tan(asin(a*x)/2)**3 + 18*tan(asin(a*x)/2)**2 - 1))/( 2*tan(asin(a*x)/2)*(tan(asin(a*x)/2) + 1))