\(\int \sinh ^5(x) \, dx\) [577]

Optimal result

Integrand size = 4, antiderivative size = 19 \[ \int \sinh ^5(x) \, dx=\cosh (x)-\frac {2 \cosh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \sinh ^5(x) \, dx=\frac {\cosh ^5(x)}{5}-\frac {2 \cosh ^3(x)}{3}+\cosh (x) \]

[In]

Int[Sinh[x]^5,x]

[Out]

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

Rule 2713

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*} \text {integral}& = \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \sinh ^5(x) \, dx=\frac {5 \cosh (x)}{8}-\frac {5}{48} \cosh (3 x)+\frac {1}{80} \cosh (5 x) \]

[In]

Integrate[Sinh[x]^5,x]

[Out]

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

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
default	\(\left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (x \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (x \right )\right )}{15}\right ) \cosh \left (x \right )\)	\(18\)
risch	\(\frac {{\mathrm e}^{5 x}}{160}-\frac {5 \,{\mathrm e}^{3 x}}{96}+\frac {5 \,{\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{-x}}{16}-\frac {5 \,{\mathrm e}^{-3 x}}{96}+\frac {{\mathrm e}^{-5 x}}{160}\)	\(36\)
parallelrisch	\(\frac {-18 \cosh \left (7 x \right )+90 \cosh \left (5 x \right )-162 \cosh \left (3 x \right )+90 \cosh \left (x \right )+3 \cosh \left (8 x \right )-180 \cosh \left (4 x \right )+20 \cosh \left (6 x \right )+492 \cosh \left (2 x \right )-335}{480 \cosh \left (3 x \right )+7200 \cosh \left (x \right )-2880 \cosh \left (2 x \right )-4800}\)	\(70\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \sinh ^5(x) \, dx=\frac {1}{80} \, \cosh \left (x\right )^{5} + \frac {1}{16} \, \cosh \left (x\right ) \sinh \left (x\right )^{4} - \frac {5}{48} \, \cosh \left (x\right )^{3} + \frac {1}{16} \, {\left (2 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac {5}{8} \, \cosh \left (x\right ) \]

[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] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \sinh ^5(x) \, dx=\sinh ^{4}{\left (x \right )} \cosh {\left (x \right )} - \frac {4 \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac {8 \cosh ^{5}{\left (x \right )}}{15} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \sinh ^5(x) \, dx=\frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{\left (-x\right )} - \frac {5}{96} \, e^{\left (-3 \, x\right )} + \frac {1}{160} \, e^{\left (-5 \, x\right )} + \frac {5}{16} \, e^{x} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \sinh ^5(x) \, dx=\frac {1}{480} \, {\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \sinh ^5(x) \, dx=\frac {{\mathrm {cosh}\left (x\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (x\right )}^3}{3}+\mathrm {cosh}\left (x\right ) \]

[In]

int(sinh(x)^5,x)

[Out]

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