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

________________________________________________________________________________________

Rubi [A]  time = 0.0296269, antiderivative size = 16, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0159079, size = 16, normalized size = 1. $\frac{3 \sinh ^{\frac{5}{3}}(x)}{5 \cosh ^{\frac{5}{3}}(x)}$

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))

________________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sinh \left ( x \right ) \right ) ^{{\frac{2}{3}}} \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{8}{3}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [B]  time = 1.63048, size = 82, normalized size = 5.12 \begin{align*} -\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}}} \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.84881, size = 333, normalized size = 20.81 \begin{align*} \frac{6 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )} \cosh \left (x\right )^{\frac{1}{3}} \sinh \left (x\right )^{\frac{2}{3}}}{5 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )^{\frac{2}{3}}}{\cosh \left (x\right )^{\frac{8}{3}}}\,{d x} \end{align*}

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)