\(\int \frac {\cosh (\frac {1}{x^5})}{x^6} \, dx\) [24]

Optimal result

Integrand size = 8, antiderivative size = 8 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=-\frac {1}{5} \sinh \left (\frac {1}{x^5}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5429, 2717} \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=-\frac {1}{5} \sinh \left (\frac {1}{x^5}\right ) \]

[In]

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

[Out]

-1/5*Sinh[x^(-5)]

Rule 2717

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

Rule 5429

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]))

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{5} \text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=-\frac {1}{5} \sinh \left (\frac {1}{x^5}\right ) \]

[In]

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

[Out]

-1/5*Sinh[x^(-5)]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {\sinh \left (\frac {1}{x^{5}}\right )}{5}\)	\(7\)
default	\(-\frac {\sinh \left (\frac {1}{x^{5}}\right )}{5}\)	\(7\)
meijerg	\(-\frac {\sinh \left (\frac {1}{x^{5}}\right )}{5}\)	\(7\)
parallelrisch	\(-\frac {\sinh \left (\frac {1}{x^{5}}\right )}{5}\)	\(7\)
risch	\(-\frac {{\mathrm e}^{\frac {1}{x^{5}}}}{10}+\frac {{\mathrm e}^{-\frac {1}{x^{5}}}}{10}\)	\(16\)

[In]

int(cosh(1/x^5)/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=-\frac {1}{5} \, \sinh \left (\frac {1}{x^{5}}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=- \frac {\sinh {\left (\frac {1}{x^{5}} \right )}}{5} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=-\frac {1}{5} \, \sinh \left (\frac {1}{x^{5}}\right ) \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (6) = 12\).

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.88 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=\frac {1}{10} \, e^{\left (-\frac {1}{x^{5}}\right )} - \frac {1}{10} \, e^{\left (\frac {1}{x^{5}}\right )} \]

[In]

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

[Out]

1/10*e^(-1/x^5) - 1/10*e^(x^(-5))

Mupad [B] (verification not implemented)

Time = 1.56 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.88 \[ \int \frac {\cosh \left (\frac {1}{x^5}\right )}{x^6} \, dx=\frac {{\mathrm {e}}^{-\frac {1}{x^5}}}{10}-\frac {{\mathrm {e}}^{\frac {1}{x^5}}}{10} \]

[In]

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

[Out]

exp(-1/x^5)/10 - exp(1/x^5)/10