\(\int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 115 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{1+a x}}} \]

[Out]

1/20/a/x^5-1/4*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^4+1/20*(-a*x+1)^(1/2)/a/x^5/(1/(a*x+1))^(1/2)+1/15*a*
(-a*x+1)^(1/2)/x^3/(1/(a*x+1))^(1/2)+2/15*a^3*(-a*x+1)^(1/2)/x/(1/(a*x+1))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 105, 12, 97} \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{a x+1}}}+\frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{a x+1}}} \]

[In]

Int[E^ArcSech[a*x]/x^5,x]

[Out]

1/(20*a*x^5) - E^ArcSech[a*x]/(4*x^4) + Sqrt[1 - a*x]/(20*a*x^5*Sqrt[(1 + a*x)^(-1)]) + (a*Sqrt[1 - a*x])/(15*
x^3*Sqrt[(1 + a*x)^(-1)]) + (2*a^3*Sqrt[1 - a*x])/(15*x*Sqrt[(1 + a*x)^(-1)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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])

Rule 6470

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ArcSech[a*x^p]/(m + 1)), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}-\frac {\int \frac {1}{x^6} \, dx}{4 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 a} \\ & = \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{20 a} \\ & = \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}-\frac {1}{5} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {1}{15} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}-\frac {1}{15} \left (2 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{20 a x^5}-\frac {e^{\text {sech}^{-1}(a x)}}{4 x^4}+\frac {\sqrt {1-a x}}{20 a x^5 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{15 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {2 a^3 \sqrt {1-a x}}{15 x \sqrt {\frac {1}{1+a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {-3+\sqrt {\frac {1-a x}{1+a x}} (1+a x)^2 \left (-3+3 a x-2 a^2 x^2+2 a^3 x^3\right )}{15 a x^5} \]

[In]

Integrate[E^ArcSech[a*x]/x^5,x]

[Out]

(-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)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.55

method	result	size
default	\(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (2 a^{2} x^{2}+3\right )}{15 x^{4}}-\frac {1}{5 a \,x^{5}}\)	\(63\)

[In]

int((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {{\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 3}{15 \, a x^{5}} \]

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {\int \frac {1}{x^{6}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{5}}\, dx}{a} \]

[In]

integrate((1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2))/x**5,x)

[Out]

(Integral(x**(-6), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x**5, x))/a

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {{\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a x^{6}} - \frac {1}{5 \, a x^{5}} \]

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{5}} \,d x } \]

[In]

integrate((1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/x^5,x, algorithm="giac")

[Out]

integrate((sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x))/x^5, x)

Mupad [B] (verification not implemented)

Time = 4.64 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^5} \, dx=\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{15}-\frac {\sqrt {\frac {1}{a\,x}+1}}{5}+\frac {2\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{15}\right )}{x^4}-\frac {1}{5\,a\,x^5} \]

[In]

int(((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x))/x^5,x)

[Out]

((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)