### 3.34 $$\int \frac{e^{\text{csch}^{-1}(a x)}}{x^3} \, dx$$

Optimal. Leaf size=31 $-\frac{1}{3} a^2 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{3 a x^3}$

[Out]

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

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Rubi [A]  time = 0.0219248, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {6336, 30, 261} $-\frac{1}{3} a^2 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{3 a x^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCsch[a*x]/x^3,x]

[Out]

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

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 261

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

Rubi steps

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

Mathematica [A]  time = 0.0336995, size = 37, normalized size = 1.19 $-\frac{a x \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+1}{3 a x^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCsch[a*x]/x^3,x]

[Out]

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

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Maple [A]  time = 0.179, size = 42, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{x}^{2}+1}{3\,{x}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{3\,{x}^{3}a}} \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^3,x)

[Out]

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

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Maxima [A]  time = 1.03324, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{3} \, a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - \frac{1}{3 \, a x^{3}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.41047, size = 103, normalized size = 3.32 \begin{align*} -\frac{a^{3} x^{3} +{\left (a^{3} x^{3} + a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{3 \, a x^{3}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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**3,x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.19317, size = 93, normalized size = 3. \begin{align*} \frac{2 \,{\left (3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{4} a^{2} \mathrm{sgn}\left (x\right ) + a^{2} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac{1}{3 \, a x^{3}} \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^3,x, algorithm="giac")

[Out]

2/3*(3*(x*abs(a) - sqrt(a^2*x^2 + 1))^4*a^2*sgn(x) + a^2*sgn(x))/((x*abs(a) - sqrt(a^2*x^2 + 1))^2 - 1)^3 - 1/
3/(a*x^3)