Integrand size = 10, antiderivative size = 80 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=-\frac {2}{15} a^4 \left (-1+\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}-\frac {1}{5 a x^5}-\frac {a^2 \left (-1+\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 x^2} \] Output:
-2/15*a^4*(-1+1/a/x)^(3/2)*(1+1/a/x)^(3/2)-1/5/a/x^5-1/5*a^2*(-1+1/a/x)^(3 /2)*(1+1/a/x)^(3/2)/x^2
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {-3+\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-3+3 a x-2 a^2 x^2+2 a^3 x^3\right )}{15 a x^5} \] Input:
Integrate[E^ArcSech[a*x]/x^5,x]
Output:
(-3 + Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-3 + 3*a*x - 2*a^2*x^2 + 2*a^ 3*x^3))/(15*a*x^5)
Result contains higher order function than in optimal. Order 3 vs. order 2 in optimal.
Time = 0.50 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.76, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6889, 15, 114, 27, 114, 27, 106}
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.
\(\displaystyle \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle -\frac {\int \frac {1}{x^6}dx}{4 a}-\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {a x+1}}dx}{4 a}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {a x+1}}dx}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {1}{5} \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {4}{5} a^2 \left (-\frac {1}{3} \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {a x+1}}dx-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle -\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a x} \sqrt {a x+1}}{3 x}-\frac {\sqrt {1-a x} \sqrt {a x+1}}{3 x^3}\right )-\frac {\sqrt {1-a x} \sqrt {a x+1}}{5 x^5}\right )}{4 a}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}\) |
Input:
Int[E^ArcSech[a*x]/x^5,x]
Output:
1/(20*a*x^5) - E^ArcSech[a*x]/(4*x^4) - (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x ]*(-1/5*(Sqrt[1 - a*x]*Sqrt[1 + a*x])/x^5 + (4*a^2*(-1/3*(Sqrt[1 - a*x]*Sq rt[1 + a*x])/x^3 - (2*a^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(3*x)))/5))/(4*a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ ArcSech[a*x^p]/(m + 1)), x] + (Simp[p/(a*(m + 1)) Int[x^(m - p), x], x] + Simp[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)] Int[x^(m - p)/( Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (2 a^{2} x^{2}+3\right )}{15 x^{4}}-\frac {1}{5 a \,x^{5}}\) | \(63\) |
orering | \(\frac {\left (\frac {4}{5} x^{5} a^{4}-\frac {2}{5} a^{2} x^{3}-\frac {3}{5} x \right ) \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{x^{5}}+\frac {\left (2 a^{2} x^{2}+1\right ) x^{2} \left (a x -1\right ) \left (a x +1\right ) \left (\frac {-\frac {1}{a \,x^{2}}-\frac {\sqrt {1+\frac {1}{a x}}}{2 \sqrt {-1+\frac {1}{a x}}\, a \,x^{2}}-\frac {\sqrt {-1+\frac {1}{a x}}}{2 \sqrt {1+\frac {1}{a x}}\, a \,x^{2}}}{x^{5}}-\frac {5 \left (\frac {1}{a x}+\sqrt {-1+\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{x^{6}}\right )}{15}\) | \(192\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^5,x,method=_RETURNVERBOSE)
Output:
1/15*(-(a*x-1)/a/x)^(1/2)/x^4*((a*x+1)/a/x)^(1/2)*(a^2*x^2-1)*(2*a^2*x^2+3 )-1/5/a/x^5
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 3}{15 \, a x^{5}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="frica s")
Output:
1/15*((2*a^5*x^5 + a^3*x^3 - 3*a*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/ (a*x)) - 3)/(a*x^5)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {\int \frac {1}{x^{6}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{5}}\, dx}{a} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))/x**5,x)
Output:
(Integral(x**(-6), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x* *5, x))/a
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {{\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a x^{6}} - \frac {1}{5 \, a x^{5}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="maxim a")
Output:
1/15*(2*a^4*x^5 + a^2*x^3 - 3*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a*x^6) - 1/ 5/(a*x^5)
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{5}} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="giac" )
Output:
integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^5, x)
Time = 24.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{15}-\frac {\sqrt {\frac {1}{a\,x}+1}}{5}+\frac {2\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{15}\right )}{x^4}-\frac {1}{5\,a\,x^5} \] Input:
int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^5,x)
Output:
((1/(a*x) - 1)^(1/2)*((a^2*x^2*(1/(a*x) + 1)^(1/2))/15 - (1/(a*x) + 1)^(1/ 2)/5 + (2*a^4*x^4*(1/(a*x) + 1)^(1/2))/15))/x^4 - 1/(5*a*x^5)
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{4} x^{4}+\sqrt {a x +1}\, \sqrt {-a x +1}\, a^{2} x^{2}-3 \sqrt {a x +1}\, \sqrt {-a x +1}-3}{15 a \,x^{5}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/x^5,x)
Output:
(2*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**4*x**4 + sqrt(a*x + 1)*sqrt( - a*x + 1)*a**2*x**2 - 3*sqrt(a*x + 1)*sqrt( - a*x + 1) - 3)/(15*a*x**5)