Integrand size = 12, antiderivative size = 134 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {4 a^4 (1-a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}-\frac {51}{8} a^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
4*a^4*(-a*x+1)/(-a^2*x^2+1)^(1/2)-1/4*(-a^2*x^2+1)^(1/2)/x^4+a*(-a^2*x^2+1 )^(1/2)/x^3-19/8*a^2*(-a^2*x^2+1)^(1/2)/x^2+6*a^3*(-a^2*x^2+1)^(1/2)/x-51/ 8*a^4*arctanh((-a^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {1}{8} \left (\frac {\sqrt {1-a^2 x^2} \left (-2+6 a x-11 a^2 x^2+29 a^3 x^3+80 a^4 x^4\right )}{x^4 (1+a x)}+51 a^4 \log (x)-51 a^4 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*x^5),x]
Output:
((Sqrt[1 - a^2*x^2]*(-2 + 6*a*x - 11*a^2*x^2 + 29*a^3*x^3 + 80*a^4*x^4))/( x^4*(1 + a*x)) + 51*a^4*Log[x] - 51*a^4*Log[1 + Sqrt[1 - a^2*x^2]])/8
Time = 0.97 (sec) , antiderivative size = 135, 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.
| \(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6674 |
| \(\displaystyle \int \frac {(1-a x)^2}{x^5 (a x+1) \sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle \int \left (\frac {1}{x^5 \sqrt {1-a^2 x^2}}-\frac {3 a}{x^4 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x^3 \sqrt {1-a^2 x^2}}-\frac {4 a^5}{(a x+1) \sqrt {1-a^2 x^2}}+\frac {4 a^4}{x \sqrt {1-a^2 x^2}}-\frac {4 a^3}{x^2 \sqrt {1-a^2 x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {51}{8} a^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {4 a^4 \sqrt {1-a^2 x^2}}{a x+1}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*x^5),x]
Output:
-1/4*Sqrt[1 - a^2*x^2]/x^4 + (a*Sqrt[1 - a^2*x^2])/x^3 - (19*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (6*a^3*Sqrt[1 - a^2*x^2])/x + (4*a^4*Sqrt[1 - a^2*x^2] )/(1 + a*x) - (51*a^4*ArcTanh[Sqrt[1 - a^2*x^2]])/8
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 = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {48 a^{5} x^{5}-19 a^{4} x^{4}-40 a^{3} x^{3}+17 a^{2} x^{2}-8 a x +2}{8 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{4} \left (-51 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {32 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a \left (x +\frac {1}{a}\right )}\right )}{8}\) | \(113\) |
| default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}+\frac {23 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 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}\right )}{4}-3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {2 a^{2} \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 )}{3}\right )-10 a^{3} \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 )+15 a^{4} \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 )-a^{2} \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 )^{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 )\right )-5 a^{3} \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 )-15 a^{4} \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 )\) | \(786\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/8*(48*a^5*x^5-19*a^4*x^4-40*a^3*x^3+17*a^2*x^2-8*a*x+2)/x^4/(-a^2*x^2+1 )^(1/2)+1/8*a^4*(-51*arctanh(1/(-a^2*x^2+1)^(1/2))+32/a/(x+1/a)*(-a^2*(x+1 /a)^2+2*a*(x+1/a))^(1/2))
Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {32 \, a^{5} x^{5} + 32 \, a^{4} x^{4} + 51 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a x^{5} + x^{4}\right )}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="fricas")
Output:
1/8*(32*a^5*x^5 + 32*a^4*x^4 + 51*(a^5*x^5 + a^4*x^4)*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (80*a^4*x^4 + 29*a^3*x^3 - 11*a^2*x^2 + 6*a*x - 2)*sqrt(-a^2 *x^2 + 1))/(a*x^5 + x^4)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**5*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{5}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*x^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (115) = 230\).
Time = 0.13 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.43 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {{\left (a^{5} - \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac {160 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}} - \frac {712 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{3} x^{4}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {51 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\frac {200 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm="giac")
Output:
1/64*(a^5 - 7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3/x + 32*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a/x^2 - 160*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a*x^3) - 7 12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^3*x^4))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a)) - 5 1/8*a^5*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs( a) + 1/64*(200*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5*abs(a)/x - 40*(sqrt(-a^ 2*x^2 + 1)*abs(a) + a)^2*a^3*abs(a)/x^2 + 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a )^3*a*abs(a)/x^3 - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)/(a*x^4))/a^4
Time = 22.82 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {a\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {19\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {6\,a^3\,\sqrt {1-a^2\,x^2}}{x}-\frac {4\,a^5\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8} \] Input:
int((1 - a^2*x^2)^(3/2)/(x^5*(a*x + 1)^3),x)
Output:
(a^4*atan((1 - a^2*x^2)^(1/2)*1i)*51i)/8 - (1 - a^2*x^2)^(1/2)/(4*x^4) + ( a*(1 - a^2*x^2)^(1/2))/x^3 - (19*a^2*(1 - a^2*x^2)^(1/2))/(8*x^2) + (6*a^3 *(1 - a^2*x^2)^(1/2))/x - (4*a^5*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) + ( -a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{x^5} \, dx=\frac {a^{4} \left (408 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}+408 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{9}-7 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{8}+32 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{7}-160 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{6}-912 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}+160 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-32 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+7 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )}{64 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x)
Output:
(a**4*(408*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**5 + 408*log(tan(asin(a* x)/2))*tan(asin(a*x)/2)**4 + tan(asin(a*x)/2)**9 - 7*tan(asin(a*x)/2)**8 + 32*tan(asin(a*x)/2)**7 - 160*tan(asin(a*x)/2)**6 - 912*tan(asin(a*x)/2)** 5 + 160*tan(asin(a*x)/2)**3 - 32*tan(asin(a*x)/2)**2 + 7*tan(asin(a*x)/2) - 1))/(64*tan(asin(a*x)/2)**4*(tan(asin(a*x)/2) + 1))