Integrand size = 10, antiderivative size = 114 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {2 a^3 \sqrt {1-a^2 x^2}}{3 x}-\frac {3}{8} a^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
-3/8*a^4*arctanh((-a^2*x^2+1)^(1/2))-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
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=\frac {1}{24} \left (-\frac {\sqrt {1-a^2 x^2} \left (6+8 a x+9 a^2 x^2+16 a^3 x^3\right )}{x^4}+9 a^4 \log (x)-9 a^4 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \]
(-((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*L og[x] - 9*a^4*Log[1 + Sqrt[1 - a^2*x^2]])/24
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6674, 539, 25, 27, 539, 25, 27, 539, 25, 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 |
| \(\displaystyle \int \frac {a x+1}{x^5 \sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {1}{4} \int -\frac {a (3 a x+4)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \frac {a (3 a x+4)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} a \int \frac {3 a x+4}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {1}{4} a \left (-\frac {1}{3} \int -\frac {a (8 a x+9)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} \int \frac {a (8 a x+9)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} a \int \frac {8 a x+9}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} a \left (-\frac {1}{2} \int -\frac {a (9 a x+16)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} \int \frac {a (9 a x+16)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \int \frac {9 a x+16}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-9 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {16 \sqrt {1-a^2 x^2}}{x}\right )-\frac {9 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {4 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{4 x^4}\) |
-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
3.1.10.3.1 Defintions of rubi rules used
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 = 0.08 (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 | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}+\frac {3 a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}+a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )\) | \(100\) |
| meijerg | \(-\frac {a \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {a^{4} \left (\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{2 \sqrt {\pi }}\) | \(166\) |
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 = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=\frac {9 \, a^{4} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 8 \, a x + 6\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \]
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
Result contains complex when optimal does not.
Time = 3.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=a \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases} \]
a*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/( 3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(- a**2*x**2 + 1)/(3*x**3), True)) + Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3* a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4* asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqr t(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))
Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=-\frac {3}{8} \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{3}}{3 \, x} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{8 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} a}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{4 \, x^{4}} \]
-3/8*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/3*sqrt(-a^2*x^2 + 1)*a^3/x - 3/8*sqrt(-a^2*x^2 + 1)*a^2/x^2 - 1/3*sqrt(-a^2*x^2 + 1)*a/x^3 - 1/4*sqrt(-a^2*x^2 + 1)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (94) = 188\).
Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.39 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=\frac {{\left (3 \, a^{5} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} + \frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {3 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {72 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \]
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 = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^5} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {a\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {3\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {2\,a^3\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8} \]