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]

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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]

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*}

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Mathematica [A]  time = 0.05, 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]

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

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fricas [A]  time = 0.56, size = 49, normalized size = 1.58 \[ -\frac {a^{3} x^{6} + {\left (a^{3} x^{6} + a x^{2}\right )} \sqrt {\frac {a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{6 \, a x^{6}} \]

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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giac [B]  time = 0.14, size = 71, normalized size = 2.29 \[ \frac {\frac {2 \, {\left (3 \, {\left (x^{2} {\left | a \right |} - \sqrt {a^{2} x^{4} + 1}\right )}^{4} a^{4} + a^{4}\right )}}{{\left ({\left (x^{2} {\left | a \right |} - \sqrt {a^{2} x^{4} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {a}{x^{6}}}{6 \, a^{2}} \]

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

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maple [A]  time = 0.14, size = 42, normalized size = 1.35 \[ -\frac {\sqrt {\frac {a^{2} x^{4}+1}{a^{2} x^{4}}}\, \left (a^{2} x^{4}+1\right )}{6 x^{4}}-\frac {1}{6 x^{6} a} \]

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 = 0.32, size = 25, normalized size = 0.81 \[ -\frac {1}{6} \, a^{2} {\left (\frac {1}{a^{2} x^{4}} + 1\right )}^{\frac {3}{2}} - \frac {1}{6 \, a x^{6}} \]

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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mupad [B]  time = 2.16, size = 44, normalized size = 1.42 \[ -\frac {\frac {1}{6\,a}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^4}+1}}{6}}{x^6}-\frac {a^2\,\sqrt {\frac {1}{a^2\,x^4}+1}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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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