3.67 \(\int \frac {\cosh ^{\frac {2}{3}}(x)}{\sinh ^{\frac {8}{3}}(x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac {3 \cosh ^{\frac {5}{3}}(x)}{5 \sinh ^{\frac {5}{3}}(x)} \]

[Out]

-3/5*cosh(x)^(5/3)/sinh(x)^(5/3)

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2563} \[ -\frac {3 \cosh ^{\frac {5}{3}}(x)}{5 \sinh ^{\frac {5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^(2/3)/Sinh[x]^(8/3),x]

[Out]

(-3*Cosh[x]^(5/3))/(5*Sinh[x]^(5/3))

Rule 2563

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[((a*Sin[e +
f*x])^(m + 1)*(b*Cos[e + f*x])^(n + 1))/(a*b*f*(m + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2,
0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\cosh ^{\frac {2}{3}}(x)}{\sinh ^{\frac {8}{3}}(x)} \, dx &=-\frac {3 \cosh ^{\frac {5}{3}}(x)}{5 \sinh ^{\frac {5}{3}}(x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ -\frac {3 \cosh ^{\frac {5}{3}}(x)}{5 \sinh ^{\frac {5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^(2/3)/Sinh[x]^(8/3),x]

[Out]

(-3*Cosh[x]^(5/3))/(5*Sinh[x]^(5/3))

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fricas [B]  time = 0.43, size = 93, normalized size = 5.81 \[ -\frac {6 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)\right )} \cosh \relax (x)^{\frac {2}{3}} \sinh \relax (x)^{\frac {1}{3}}}{5 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^(2/3)/sinh(x)^(8/3),x, algorithm="fricas")

[Out]

-6/5*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 + 1)*sinh(x) + cosh(x))*cosh(x)^(2/3)*sinh(x)
^(1/3)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)
^3 - cosh(x))*sinh(x) + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \relax (x)^{\frac {2}{3}}}{\sinh \relax (x)^{\frac {8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^(2/3)/sinh(x)^(8/3),x, algorithm="giac")

[Out]

integrate(cosh(x)^(2/3)/sinh(x)^(8/3), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{\frac {2}{3}}\relax (x )}{\sinh \relax (x )^{\frac {8}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^(2/3)/sinh(x)^(8/3),x)

[Out]

int(cosh(x)^(2/3)/sinh(x)^(8/3),x)

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maxima [B]  time = 0.49, size = 61, normalized size = 3.81 \[ \frac {3 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {2}{3}} e^{\left (-4 \, x\right )}}{5 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {8}{3}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {8}{3}}} - \frac {3 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {2}{3}}}{5 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {8}{3}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^(2/3)/sinh(x)^(8/3),x, algorithm="maxima")

[Out]

3/5*(e^(-2*x) + 1)^(2/3)*e^(-4*x)/((e^(-x) + 1)^(8/3)*(-e^(-x) + 1)^(8/3)) - 3/5*(e^(-2*x) + 1)^(2/3)/((e^(-x)
+ 1)^(8/3)*(-e^(-x) + 1)^(8/3))

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mupad [B]  time = 1.54, size = 6, normalized size = 0.38 \[ -\frac {3\,{\mathrm {coth}\relax (x)}^{5/3}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^(2/3)/sinh(x)^(8/3),x)

[Out]

-(3*coth(x)^(5/3))/5

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**(2/3)/sinh(x)**(8/3),x)

[Out]

Timed out

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