### 3.36 $$\int \frac{e^{\text{csch}^{-1}(a x)}}{x^5} \, dx$$

Optimal. Leaf size=51 $-\frac{1}{5} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{5/2}+\frac{1}{3} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{5 a x^5}$

[Out]

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

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Rubi [A]  time = 0.0384009, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6336, 30, 266, 43} $-\frac{1}{5} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{5/2}+\frac{1}{3} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{5 a x^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6336

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/
(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]

Rule 30

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x^5} \, dx &=\frac{\int \frac{1}{x^6} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^5} \, dx\\ &=-\frac{1}{5 a x^5}-\frac{1}{2} \operatorname{Subst}\left (\int x \sqrt{1+\frac{x}{a^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{5 a x^5}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-a^2 \sqrt{1+\frac{x}{a^2}}+a^2 \left (1+\frac{x}{a^2}\right )^{3/2}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{3} a^4 \left (1+\frac{1}{a^2 x^2}\right )^{3/2}-\frac{1}{5} a^4 \left (1+\frac{1}{a^2 x^2}\right )^{5/2}-\frac{1}{5 a x^5}\\ \end{align*}

Mathematica [A]  time = 0.0419801, size = 46, normalized size = 0.9 $\frac{a x \sqrt{\frac{1}{a^2 x^2}+1} \left (2 a^4 x^4-a^2 x^2-3\right )-3}{15 a x^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.197, size = 52, normalized size = 1. \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( 2\,{a}^{2}{x}^{2}-3 \right ) }{15\,{x}^{4}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{5\,a{x}^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988547, size = 55, normalized size = 1.08 \begin{align*} -\frac{1}{5} \, a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - \frac{1}{5 \, a x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.43858, size = 124, normalized size = 2.43 \begin{align*} \frac{2 \, a^{5} x^{5} +{\left (2 \, a^{5} x^{5} - a^{3} x^{3} - 3 \, a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 3}{15 \, a x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.04264, size = 65, normalized size = 1.27 \begin{align*} \frac{2 a^{3} \sqrt{a^{2} x^{2} + 1}}{15 x} - \frac{a \sqrt{a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{a^{2} x^{2} + 1}}{5 a x^{5}} - \frac{1}{5 a x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*sqrt(a**2*x**2 + 1)/(15*x) - a*sqrt(a**2*x**2 + 1)/(15*x**3) - sqrt(a**2*x**2 + 1)/(5*a*x**5) - 1/(5*a*
x**5)

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Giac [B]  time = 1.25469, size = 167, normalized size = 3.27 \begin{align*} \frac{4 \,{\left (15 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{6} a^{4} \mathrm{sgn}\left (x\right ) + 5 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{4} a^{4} \mathrm{sgn}\left (x\right ) + 5 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} a^{4} \mathrm{sgn}\left (x\right ) - a^{4} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{5}} - \frac{1}{5 \, a x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*(15*(x*abs(a) - sqrt(a^2*x^2 + 1))^6*a^4*sgn(x) + 5*(x*abs(a) - sqrt(a^2*x^2 + 1))^4*a^4*sgn(x) + 5*(x*ab
s(a) - sqrt(a^2*x^2 + 1))^2*a^4*sgn(x) - a^4*sgn(x))/((x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)^5 - 1/5/(a*x^5)