3.24 \(\int \frac{\cosh (\frac{1}{x^5})}{x^6} \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{5} \sinh \left (\frac{1}{x^5}\right ) \]

[Out]

-Sinh[x^(-5)]/5

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Rubi [A]  time = 0.0120885, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5321, 2637} \[ -\frac{1}{5} \sinh \left (\frac{1}{x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x^(-5)]/x^6,x]

[Out]

-Sinh[x^(-5)]/5

Rule 5321

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cosh \left (\frac{1}{x^5}\right )}{x^6} \, dx &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \cosh (x) \, dx,x,\frac{1}{x^5}\right )\right )\\ &=-\frac{1}{5} \sinh \left (\frac{1}{x^5}\right )\\ \end{align*}

Mathematica [A]  time = 0.0044277, size = 8, normalized size = 1. \[ -\frac{1}{5} \sinh \left (\frac{1}{x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x^(-5)]/x^6,x]

[Out]

-Sinh[x^(-5)]/5

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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \begin{align*} -{\frac{\sinh \left ({x}^{-5} \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(1/x^5)/x^6,x)

[Out]

-1/5*sinh(1/x^5)

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Maxima [A]  time = 1.14956, size = 8, normalized size = 1. \begin{align*} -\frac{1}{5} \, \sinh \left (\frac{1}{x^{5}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(1/x^5)/x^6,x, algorithm="maxima")

[Out]

-1/5*sinh(x^(-5))

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Fricas [A]  time = 1.67049, size = 26, normalized size = 3.25 \begin{align*} -\frac{1}{5} \, \sinh \left (\frac{1}{x^{5}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(1/x^5)/x^6,x, algorithm="fricas")

[Out]

-1/5*sinh(x^(-5))

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Sympy [A]  time = 43.3938, size = 8, normalized size = 1. \begin{align*} - \frac{\sinh{\left (\frac{1}{x^{5}} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(1/x**5)/x**6,x)

[Out]

-sinh(x**(-5))/5

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Giac [B]  time = 1.2444, size = 20, normalized size = 2.5 \begin{align*} \frac{1}{10} \, e^{\left (-\frac{1}{x^{5}}\right )} - \frac{1}{10} \, e^{\left (\frac{1}{x^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(1/x^5)/x^6,x, algorithm="giac")

[Out]

1/10*e^(-1/x^5) - 1/10*e^(x^(-5))