Integrand size = 12, antiderivative size = 114 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3}{8} a^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-1/4*(-a^2*x^2+1)^(1/2)/x^4+1/3*a*(-a^2*x^2+1)^(1/2)/x^3-3/8*a^2*(-a^2*x^2 +1)^(1/2)/x^2+2/3*a^3*(-a^2*x^2+1)^(1/2)/x-3/8*a^4*arctanh((-a^2*x^2+1)^(1 /2))
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\frac {1}{24} \left (\frac {\sqrt {1-a^2 x^2} \left (-6+8 a x-9 a^2 x^2+16 a^3 x^3\right )}{x^4}+9 a^4 \log (x)-9 a^4 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \] Input:
Integrate[1/(E^ArcTanh[a*x]*x^5),x]
Output:
((Sqrt[1 - a^2*x^2]*(-6 + 8*a*x - 9*a^2*x^2 + 16*a^3*x^3))/x^4 + 9*a^4*Log [x] - 9*a^4*Log[1 + Sqrt[1 - a^2*x^2]])/24
Time = 0.53 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6674, 539, 27, 539, 27, 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.
| \(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6674 |
| \(\displaystyle \int \frac {1-a x}{x^5 \sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {1}{4} \int \frac {a (4-3 a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} a \int \frac {4-3 a x}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} \int \frac {a (9-8 a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \int \frac {9-8 a x}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} \int \frac {a (16-9 a x)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \int \frac {16-9 a x}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (-9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {16 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (-\frac {9}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {16 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (\frac {9 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {16 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {1}{4} a \left (-\frac {1}{3} a \left (-\frac {1}{2} a \left (9 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {16 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
Input:
Int[1/(E^ArcTanh[a*x]*x^5),x]
Output:
-1/4*Sqrt[1 - a^2*x^2]/x^4 - (a*((-4*Sqrt[1 - a^2*x^2])/(3*x^3) - (a*((-9* Sqrt[1 - a^2*x^2])/(2*x^2) - (a*((-16*Sqrt[1 - a^2*x^2])/x + 9*a*ArcTanh[S qrt[1 - a^2*x^2]]))/2))/3))/4
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 = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {16 a^{5} x^{5}-9 a^{4} x^{4}-8 a^{3} x^{3}+3 a^{2} x^{2}-8 a x +6}{24 x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}\) | \(75\) |
| default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {5 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+a^{4} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}-a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )-a^{4} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\) | \(263\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/24*(16*a^5*x^5-9*a^4*x^4-8*a^3*x^3+3*a^2*x^2-8*a*x+6)/x^4/(-a^2*x^2+1)^ (1/2)-3/8*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\frac {9 \, a^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (16 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 8 \, a x - 6\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x, algorithm="fricas")
Output:
1/24*(9*a^4*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (16*a^3*x^3 - 9*a^2*x^2 + 8*a*x - 6)*sqrt(-a^2*x^2 + 1))/x^4
\[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{5} \left (a x + 1\right )}\, dx \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**5,x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**5*(a*x + 1)), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} x^{5}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x, algorithm="maxima")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*x^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (94) = 188\).
Time = 0.13 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.39 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\frac {{\left (3 \, a^{5} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {3 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} - \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x, algorithm="giac")
Output:
1/192*(3*a^5 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3/x + 24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a/x^2 - 72*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a*x^3))* a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)) - 3/8*a^5*log(1/2*abs(- 2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/192*(72*(sqrt( -a^2*x^2 + 1)*abs(a) + a)*a^5*abs(a)/x - 24*(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 - 3* (sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*abs(a)/(a*x^4))/a^4
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\frac {a\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {3\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8} \] Input:
int((1 - a^2*x^2)^(1/2)/(x^5*(a*x + 1)),x)
Output:
(a^4*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/8 - (1 - a^2*x^2)^(1/2)/(4*x^4) + (a *(1 - a^2*x^2)^(1/2))/(3*x^3) - (3*a^2*(1 - a^2*x^2)^(1/2))/(8*x^2) + (2*a ^3*(1 - a^2*x^2)^(1/2))/(3*x)
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\text {arctanh}(a x)}}{x^5} \, dx=\frac {16 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-9 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}+9 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}}{24 x^{4}} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^5,x)
Output:
(16*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 9*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 8*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x**2 + 1) + 9*log(tan(asin (a*x)/2))*a**4*x**4)/(24*x**4)