### 3.47 $$\int \frac{e^{\text{csch}^{-1}(a x^2)}}{x^5} \, dx$$

Optimal. Leaf size=31 $-\frac{1}{6} a^2 \left (\frac{1}{a^2 x^4}+1\right )^{3/2}-\frac{1}{6 a x^6}$

[Out]

-(a^2*(1 + 1/(a^2*x^4))^(3/2))/6 - 1/(6*a*x^6)

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Rubi [A]  time = 0.0216713, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6336, 30, 261} $-\frac{1}{6} a^2 \left (\frac{1}{a^2 x^4}+1\right )^{3/2}-\frac{1}{6 a x^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCsch[a*x^2]/x^5,x]

[Out]

-(a^2*(1 + 1/(a^2*x^4))^(3/2))/6 - 1/(6*a*x^6)

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}\left (a x^2\right )}}{x^5} \, dx &=\frac{\int \frac{1}{x^7} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^5} \, dx\\ &=-\frac{1}{6} a^2 \left (1+\frac{1}{a^2 x^4}\right )^{3/2}-\frac{1}{6 a x^6}\\ \end{align*}

Mathematica [A]  time = 0.0534463, size = 39, normalized size = 1.26 $-\frac{a x^2 \sqrt{\frac{1}{a^2 x^4}+1} \left (a^2 x^4+1\right )+1}{6 a x^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCsch[a*x^2]/x^5,x]

[Out]

-(1 + a*Sqrt[1 + 1/(a^2*x^4)]*x^2*(1 + a^2*x^4))/(6*a*x^6)

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Maple [A]  time = 0.273, size = 42, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{x}^{4}+1}{6\,{x}^{4}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}}}-{\frac{1}{6\,{x}^{6}a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x^5,x)

[Out]

-1/6*((a^2*x^4+1)/a^2/x^4)^(1/2)/x^4*(a^2*x^4+1)-1/6/x^6/a

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Maxima [A]  time = 1.03725, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{6} \, a^{2}{\left (\frac{1}{a^{2} x^{4}} + 1\right )}^{\frac{3}{2}} - \frac{1}{6 \, a x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x^5,x, algorithm="maxima")

[Out]

-1/6*a^2*(1/(a^2*x^4) + 1)^(3/2) - 1/6/(a*x^6)

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Fricas [A]  time = 2.51877, size = 105, normalized size = 3.39 \begin{align*} -\frac{a^{3} x^{6} +{\left (a^{3} x^{6} + a x^{2}\right )} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{6 \, a x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x^5,x, algorithm="fricas")

[Out]

-1/6*(a^3*x^6 + (a^3*x^6 + a*x^2)*sqrt((a^2*x^4 + 1)/(a^2*x^4)) + 1)/(a*x^6)

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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**2+(1+1/a**2/x**4)**(1/2))/x**5,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.2093, size = 34, normalized size = 1.1 \begin{align*} -\frac{{\left (a^{2} + \frac{1}{x^{4}}\right )}^{\frac{3}{2}}{\left | a \right |}}{6 \, a^{2}} - \frac{1}{6 \, a x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x^5,x, algorithm="giac")

[Out]

-1/6*(a^2 + 1/x^4)^(3/2)*abs(a)/a^2 - 1/6/(a*x^6)