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

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

[Out]

3/5*sinh(x)^(5/3)/cosh(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} \[ \frac {3 \sinh ^{\frac {5}{3}}(x)}{5 \cosh ^{\frac {5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sinh[x]^(5/3))/(5*Cosh[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 {\sinh ^{\frac {2}{3}}(x)}{\cosh ^{\frac {8}{3}}(x)} \, dx &=\frac {3 \sinh ^{\frac {5}{3}}(x)}{5 \cosh ^{\frac {5}{3}}(x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 1.00 \[ \frac {3 \sinh ^{\frac {5}{3}}(x)}{5 \cosh ^{\frac {5}{3}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 93, normalized size = 5.81 \[ \frac {6 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)\right )} \cosh \relax (x)^{\frac {1}{3}} \sinh \relax (x)^{\frac {2}{3}}}{5 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^(2/3)/cosh(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)^(1/3)*sinh(x)^
(2/3)/(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 \[ \int \frac {\sinh \relax (x)^{\frac {2}{3}}}{\cosh \relax (x)^{\frac {8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.46, size = 61, normalized size = 3.81 \[ -\frac {3 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {2}{3}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {2}{3}} e^{\left (-4 \, x\right )}}{5 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {8}{3}}} + \frac {3 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {2}{3}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {2}{3}}}{5 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\mathrm {sinh}\relax (x)}^{2/3}}{{\mathrm {cosh}\relax (x)}^{8/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinh(x)^(2/3)/cosh(x)^(8/3), x)

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

[Out]

Timed out

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