Integrand size = 12, antiderivative size = 89 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\frac {4 a^2 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}+\frac {3 a \sqrt {1-a^2 x^2}}{x}-\frac {9}{2} a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
4*a^2*(-a*x+1)/(-a^2*x^2+1)^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/x^2+3*a*(-a^2*x^2 +1)^(1/2)/x-9/2*a^2*arctanh((-a^2*x^2+1)^(1/2))
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\sqrt {1-a^2 x^2} \left (-\frac {1}{2 x^2}+\frac {3 a}{x}+\frac {4 a^2}{1+a x}\right )+\frac {9}{2} a^2 \log (x)-\frac {9}{2} a^2 \log \left (1+\sqrt {1-a^2 x^2}\right ) \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*x^3),x]
Output:
Sqrt[1 - a^2*x^2]*(-1/2*1/x^2 + (3*a)/x + (4*a^2)/(1 + a*x)) + (9*a^2*Log[ x])/2 - (9*a^2*Log[1 + Sqrt[1 - a^2*x^2]])/2
Time = 0.82 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {(1-a x)^2}{x^3 (a x+1) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 2353 |
\(\displaystyle \int \left (\frac {4 a^2}{x \sqrt {1-a^2 x^2}}-\frac {3 a}{x^2 \sqrt {1-a^2 x^2}}+\frac {1}{x^3 \sqrt {1-a^2 x^2}}-\frac {4 a^3}{(a x+1) \sqrt {1-a^2 x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9}{2} a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {4 a^2 \sqrt {1-a^2 x^2}}{a x+1}+\frac {3 a \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*x^3),x]
Output:
-1/2*Sqrt[1 - a^2*x^2]/x^2 + (3*a*Sqrt[1 - a^2*x^2])/x + (4*a^2*Sqrt[1 - a ^2*x^2])/(1 + a*x) - (9*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/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 = 97, normalized size of antiderivative = 1.09
method | result | size |
risch | \(-\frac {6 a^{3} x^{3}-a^{2} x^{2}-6 a x +1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{2} \left (-9 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {8 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a \left (x +\frac {1}{a}\right )}\right )}{2}\) | \(97\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {9 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}-3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )-6 a^{2} \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\) | \(448\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(6*a^3*x^3-a^2*x^2-6*a*x+1)/x^2/(-a^2*x^2+1)^(1/2)+1/2*a^2*(-9*arctan h(1/(-a^2*x^2+1)^(1/2))+8/a/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\frac {8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 \, {\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (14 \, a^{2} x^{2} + 5 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a x^{3} + x^{2}\right )}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm="fricas")
Output:
1/2*(8*a^3*x^3 + 8*a^2*x^2 + 9*(a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (14*a^2*x^2 + 5*a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*x^3 + x^2)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**3*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{3}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (76) = 152\).
Time = 0.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.40 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\frac {{\left (a^{3} - \frac {11 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {76 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {9 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x, algorithm="giac")
Output:
1/8*(a^3 - 11*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 76*(sqrt(-a^2*x^2 + 1) *abs(a) + a)^2/(a*x^2))*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*((sqrt( -a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 9/2*a^3*log(1/2*abs(-2*sq rt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/8*(12*(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 = 14.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\frac {3\,a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {4\,a^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{2} \] Input:
int((1 - a^2*x^2)^(3/2)/(x^3*(a*x + 1)^3),x)
Output:
(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*9i)/2 - (1 - a^2*x^2)^(1/2)/(2*x^2) + (3 *a*(1 - a^2*x^2)^(1/2))/x - (4*a^3*(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 = 105, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^3} \, dx=\frac {a^{2} \left (36 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+36 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}-11 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}-88 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+11 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )}{8 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^3,x)
Output:
(a**2*(36*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**3 + 36*log(tan(asin(a*x) /2))*tan(asin(a*x)/2)**2 + tan(asin(a*x)/2)**5 - 11*tan(asin(a*x)/2)**4 - 88*tan(asin(a*x)/2)**3 + 11*tan(asin(a*x)/2) - 1))/(8*tan(asin(a*x)/2)**2* (tan(asin(a*x)/2) + 1))