3.3.100 \(\int x^5 \cosh (x) \, dx\) [300]

Optimal. Leaf size=37 \[ -120 \cosh (x)-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x) \]

[Out]

-120*cosh(x)-60*x^2*cosh(x)-5*x^4*cosh(x)+120*x*sinh(x)+20*x^3*sinh(x)+x^5*sinh(x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3377, 2718} \begin {gather*} x^5 \sinh (x)-5 x^4 \cosh (x)+20 x^3 \sinh (x)-60 x^2 \cosh (x)+120 x \sinh (x)-120 \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Cosh[x],x]

[Out]

-120*Cosh[x] - 60*x^2*Cosh[x] - 5*x^4*Cosh[x] + 120*x*Sinh[x] + 20*x^3*Sinh[x] + x^5*Sinh[x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rubi steps

\begin {align*} \int x^5 \cosh (x) \, dx &=x^5 \sinh (x)-5 \int x^4 \sinh (x) \, dx\\ &=-5 x^4 \cosh (x)+x^5 \sinh (x)+20 \int x^3 \cosh (x) \, dx\\ &=-5 x^4 \cosh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)-60 \int x^2 \sinh (x) \, dx\\ &=-60 x^2 \cosh (x)-5 x^4 \cosh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)+120 \int x \cosh (x) \, dx\\ &=-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)-120 \int \sinh (x) \, dx\\ &=-120 \cosh (x)-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 29, normalized size = 0.78 \begin {gather*} -5 \left (24+12 x^2+x^4\right ) \cosh (x)+x \left (120+20 x^2+x^4\right ) \sinh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Cosh[x],x]

[Out]

-5*(24 + 12*x^2 + x^4)*Cosh[x] + x*(120 + 20*x^2 + x^4)*Sinh[x]

________________________________________________________________________________________

Mathics [A]
time = 2.50, size = 37, normalized size = 1.00 \begin {gather*} 120 x \text {Sinh}\left [x\right ]-60 x^2 \text {Cosh}\left [x\right ]+20 x^3 \text {Sinh}\left [x\right ]-5 x^4 \text {Cosh}\left [x\right ]+x^5 \text {Sinh}\left [x\right ]-120 \text {Cosh}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^5*Cosh[x],x]')

[Out]

120 x Sinh[x] - 60 x ^ 2 Cosh[x] + 20 x ^ 3 Sinh[x] - 5 x ^ 4 Cosh[x] + x ^ 5 Sinh[x] - 120 Cosh[x]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 38, normalized size = 1.03

method	result	size
default	\(-120 \cosh \left (x \right )-60 x^{2} \cosh \left (x \right )-5 x^{4} \cosh \left (x \right )+120 x \sinh \left (x \right )+20 x^{3} \sinh \left (x \right )+x^{5} \sinh \left (x \right )\)	\(38\)
meijerg	\(-32 \sqrt {\pi }\, \left (-\frac {15}{4 \sqrt {\pi }}+\frac {\left (\frac {15}{8} x^{4}+\frac {45}{2} x^{2}+45\right ) \cosh \left (x \right )}{12 \sqrt {\pi }}-\frac {x \left (\frac {3}{8} x^{4}+\frac {15}{2} x^{2}+45\right ) \sinh \left (x \right )}{12 \sqrt {\pi }}\right )\)	\(51\)
risch	\(\left (10 x^{3}-30 x^{2}+60 x -60-\frac {5}{2} x^{4}+\frac {1}{2} x^{5}\right ) {\mathrm e}^{x}+\left (-10 x^{3}-30 x^{2}-60 x -60-\frac {5}{2} x^{4}-\frac {1}{2} x^{5}\right ) {\mathrm e}^{-x}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cosh(x),x,method=_RETURNVERBOSE)

[Out]

-120*cosh(x)-60*x^2*cosh(x)-5*x^4*cosh(x)+120*x*sinh(x)+20*x^3*sinh(x)+x^5*sinh(x)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 74, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, x^{6} \cosh \left (x\right ) - \frac {1}{12} \, {\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} - \frac {1}{12} \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(x),x, algorithm="maxima")

[Out]

1/6*x^6*cosh(x) - 1/12*(x^6 + 6*x^5 + 30*x^4 + 120*x^3 + 360*x^2 + 720*x + 720)*e^(-x) - 1/12*(x^6 - 6*x^5 + 3
0*x^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^x

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 30, normalized size = 0.81 \begin {gather*} -5 \, {\left (x^{4} + 12 \, x^{2} + 24\right )} \cosh \left (x\right ) + {\left (x^{5} + 20 \, x^{3} + 120 \, x\right )} \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(x),x, algorithm="fricas")

[Out]

-5*(x^4 + 12*x^2 + 24)*cosh(x) + (x^5 + 20*x^3 + 120*x)*sinh(x)

________________________________________________________________________________________

Sympy [A]
time = 0.32, size = 42, normalized size = 1.14 \begin {gather*} x^{5} \sinh {\left (x \right )} - 5 x^{4} \cosh {\left (x \right )} + 20 x^{3} \sinh {\left (x \right )} - 60 x^{2} \cosh {\left (x \right )} + 120 x \sinh {\left (x \right )} - 120 \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*cosh(x),x)

[Out]

x**5*sinh(x) - 5*x**4*cosh(x) + 20*x**3*sinh(x) - 60*x**2*cosh(x) + 120*x*sinh(x) - 120*cosh(x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 64, normalized size = 1.73 \begin {gather*} \frac {\left (x^{5}-5 x^{4}+20 x^{3}-60 x^{2}+120 x-120\right ) \mathrm {e}^{x}+\left (-x^{5}-5 x^{4}-20 x^{3}-60 x^{2}-120 x-120\right ) \mathrm {e}^{-x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*cosh(x),x)

[Out]

1/2*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x - 1/2*(x^5 + 5*x^4 + 20*x^3 + 60*x^2 + 120*x + 120)/e^x

________________________________________________________________________________________

Mupad [B]
time = 0.21, size = 37, normalized size = 1.00 \begin {gather*} 20\,x^3\,\mathrm {sinh}\left (x\right )-60\,x^2\,\mathrm {cosh}\left (x\right )-5\,x^4\,\mathrm {cosh}\left (x\right )-120\,\mathrm {cosh}\left (x\right )+x^5\,\mathrm {sinh}\left (x\right )+120\,x\,\mathrm {sinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*cosh(x),x)

[Out]

20*x^3*sinh(x) - 60*x^2*cosh(x) - 5*x^4*cosh(x) - 120*cosh(x) + x^5*sinh(x) + 120*x*sinh(x)

________________________________________________________________________________________