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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 = 0.00, 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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh ^5(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sinh[x]^5,x]

[Out]

Could not integrate

________________________________________________________________________________________

fricas [B]  time = 0.95, size = 42, normalized size = 2.21 \[ \frac {1}{80} \, \cosh \relax (x)^{5} + \frac {1}{16} \, \cosh \relax (x) \sinh \relax (x)^{4} - \frac {5}{48} \, \cosh \relax (x)^{3} + \frac {1}{16} \, {\left (2 \, \cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + \frac {5}{8} \, \cosh \relax (x) \]

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)

________________________________________________________________________________________

giac [B]  time = 0.59, size = 37, normalized size = 1.95 \[ \frac {1}{480} \, {\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{x} \]

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

________________________________________________________________________________________

maple [A]  time = 0.34, size = 18, normalized size = 0.95

method	result	size
default	\(\left (\frac {8}{15}+\frac {\left (\sinh ^{4}\relax (x )\right )}{5}-\frac {4 \left (\sinh ^{2}\relax (x )\right )}{15}\right ) \cosh \relax (x )\)	\(18\)
risch	\(\frac {{\mathrm e}^{5 x}}{160}-\frac {5 \,{\mathrm e}^{3 x}}{96}+\frac {5 \,{\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{-x}}{16}-\frac {5 \,{\mathrm e}^{-3 x}}{96}+\frac {{\mathrm e}^{-5 x}}{160}\)	\(36\)
meijerg	error in int/gbinthm/express: improper op or subscript selector\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^5,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 35, normalized size = 1.84 \[ \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{\left (-x\right )} - \frac {5}{96} \, e^{\left (-3 \, x\right )} + \frac {1}{160} \, e^{\left (-5 \, x\right )} + \frac {5}{16} \, e^{x} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 15, normalized size = 0.79 \[ \frac {{\mathrm {cosh}\relax (x)}^5}{5}-\frac {2\,{\mathrm {cosh}\relax (x)}^3}{3}+\mathrm {cosh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^5,x)

[Out]

cosh(x) - (2*cosh(x)^3)/3 + cosh(x)^5/5

________________________________________________________________________________________

sympy [A]  time = 1.12, size = 29, normalized size = 1.53 \[ \sinh ^{4}{\relax (x )} \cosh {\relax (x )} - \frac {4 \sinh ^{2}{\relax (x )} \cosh ^{3}{\relax (x )}}{3} + \frac {8 \cosh ^{5}{\relax (x )}}{15} \]

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

________________________________________________________________________________________