3.577 \(\int \sinh ^5(x) \, dx\)

Optimal. Leaf size=19 \[ \frac{\cosh ^5(x)}{5}-\frac{2 \cosh ^3(x)}{3}+\cosh (x) \]

[Out]

Cosh[x] - (2*Cosh[x]^3)/3 + Cosh[x]^5/5

________________________________________________________________________________________

Rubi [A]  time = 0.01108, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2633} \[ \frac{\cosh ^5(x)}{5}-\frac{2 \cosh ^3(x)}{3}+\cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^5,x]

[Out]

Cosh[x] - (2*Cosh[x]^3)/3 + Cosh[x]^5/5

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \sinh ^5(x) \, dx &=\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\frac{2 \cosh ^3(x)}{3}+\frac{\cosh ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0019877, size = 23, normalized size = 1.21 \[ \frac{5 \cosh (x)}{8}-\frac{5}{48} \cosh (3 x)+\frac{1}{80} \cosh (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^5,x]

[Out]

(5*Cosh[x])/8 - (5*Cosh[3*x])/48 + Cosh[5*x]/80

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 18, normalized size = 1. \begin{align*} \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^5,x)

[Out]

(8/15+1/5*sinh(x)^4-4/15*sinh(x)^2)*cosh(x)

________________________________________________________________________________________

Maxima [B]  time = 0.93214, size = 47, normalized size = 2.47 \begin{align*} \frac{1}{160} \, e^{\left (5 \, x\right )} - \frac{5}{96} \, e^{\left (3 \, x\right )} + \frac{5}{16} \, e^{\left (-x\right )} - \frac{5}{96} \, e^{\left (-3 \, x\right )} + \frac{1}{160} \, e^{\left (-5 \, x\right )} + \frac{5}{16} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^5,x, algorithm="maxima")

[Out]

1/160*e^(5*x) - 5/96*e^(3*x) + 5/16*e^(-x) - 5/96*e^(-3*x) + 1/160*e^(-5*x) + 5/16*e^x

________________________________________________________________________________________

Fricas [B]  time = 2.10196, size = 155, normalized size = 8.16 \begin{align*} \frac{1}{80} \, \cosh \left (x\right )^{5} + \frac{1}{16} \, \cosh \left (x\right ) \sinh \left (x\right )^{4} - \frac{5}{48} \, \cosh \left (x\right )^{3} + \frac{1}{16} \,{\left (2 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac{5}{8} \, \cosh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^5,x, algorithm="fricas")

[Out]

1/80*cosh(x)^5 + 1/16*cosh(x)*sinh(x)^4 - 5/48*cosh(x)^3 + 1/16*(2*cosh(x)^3 - 5*cosh(x))*sinh(x)^2 + 5/8*cosh
(x)

________________________________________________________________________________________

Sympy [A]  time = 1.12174, size = 29, normalized size = 1.53 \begin{align*} \sinh ^{4}{\left (x \right )} \cosh{\left (x \right )} - \frac{4 \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac{8 \cosh ^{5}{\left (x \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**5,x)

[Out]

sinh(x)**4*cosh(x) - 4*sinh(x)**2*cosh(x)**3/3 + 8*cosh(x)**5/15

________________________________________________________________________________________

Giac [B]  time = 1.10703, size = 50, normalized size = 2.63 \begin{align*} \frac{1}{480} \,{\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{160} \, e^{\left (5 \, x\right )} - \frac{5}{96} \, e^{\left (3 \, x\right )} + \frac{5}{16} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^5,x, algorithm="giac")

[Out]

1/480*(150*e^(4*x) - 25*e^(2*x) + 3)*e^(-5*x) + 1/160*e^(5*x) - 5/96*e^(3*x) + 5/16*e^x