Integrand size = 6, antiderivative size = 37 \[ \int x^5 \cosh (x) \, dx=-120 \cosh (x)-60 x^2 \cosh (x)-5 x^4 \cosh (x)+120 x \sinh (x)+20 x^3 \sinh (x)+x^5 \sinh (x) \] Output:
-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)
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^5 \cosh (x) \, dx=-5 \left (24+12 x^2+x^4\right ) \cosh (x)+x \left (120+20 x^2+x^4\right ) \sinh (x) \] Input:
Integrate[x^5*Cosh[x],x]
Output:
-5*(24 + 12*x^2 + x^4)*Cosh[x] + x*(120 + 20*x^2 + x^4)*Sinh[x]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 3.000, Rules used = {3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^5 \cosh (x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^5 \sin \left (\frac {\pi }{2}+i x\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle x^5 \sinh (x)-5 i \int -i x^4 \sinh (x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)-5 \int x^4 \sinh (x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x^5 \sinh (x)-5 \int -i x^4 \sin (i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)+5 i \int x^4 \sin (i x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \int x^3 \cosh (x)dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \int x^3 \sin \left (i x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)-3 i \int -i x^2 \sinh (x)dx\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)-3 \int x^2 \sinh (x)dx\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)-3 \int -i x^2 \sin (i x)dx\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \int x^2 \sin (i x)dx\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i \int x \cosh (x)dx\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i \int x \sin \left (i x+\frac {\pi }{2}\right )dx\right )\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i (x \sinh (x)-i \int -i \sinh (x)dx)\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i (x \sinh (x)-\int \sinh (x)dx)\right )\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i (x \sinh (x)-\int -i \sin (i x)dx)\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i (x \sinh (x)+i \int \sin (i x)dx)\right )\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle x^5 \sinh (x)+5 i \left (i x^4 \cosh (x)-4 i \left (x^3 \sinh (x)+3 i \left (i x^2 \cosh (x)-2 i (x \sinh (x)-\cosh (x))\right )\right )\right )\) |
Input:
Int[x^5*Cosh[x],x]
Output:
x^5*Sinh[x] + (5*I)*(I*x^4*Cosh[x] - (4*I)*(x^3*Sinh[x] + (3*I)*(I*x^2*Cos h[x] - (2*I)*(-Cosh[x] + x*Sinh[x]))))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\left (-5 x^{4}-60 x^{2}-120\right ) \cosh \left (x \right )-120+\left (x^{5}+20 x^{3}+120 x \right ) \sinh \left (x \right )\) | \(33\) |
default | \(-120 \cosh \left (x \right )-60 x^{2} \cosh \left (x \right )-5 x^{4} \cosh \left (x \right )+120 x \sinh \left (x \right )+20 x^{3} \sinh \left (x \right )+x^{5} \sinh \left (x \right )\) | \(38\) |
parts | \(-120 \cosh \left (x \right )-60 x^{2} \cosh \left (x \right )-5 x^{4} \cosh \left (x \right )+120 x \sinh \left (x \right )+20 x^{3} \sinh \left (x \right )+x^{5} \sinh \left (x \right )\) | \(38\) |
orering | \(-10 \left (x^{4}+16 x^{2}+72\right ) \cosh \left (x \right )+\frac {\left (x^{4}+20 x^{2}+120\right ) \left (5 x^{4} \cosh \left (x \right )+x^{5} \sinh \left (x \right )\right )}{x^{4}}\) | \(44\) |
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 | \(\left (10 x^{3}-30 x^{2}+60 x -60-\frac {5}{2} x^{4}+\frac {1}{2} x^{5}\right ) {\mathrm e}^{x}+\left (-10 x^{3}-30 x^{2}-60 x -60-\frac {5}{2} x^{4}-\frac {1}{2} x^{5}\right ) {\mathrm e}^{-x}\) | \(60\) |
Input:
int(x^5*cosh(x),x,method=_RETURNVERBOSE)
Output:
(-5*x^4-60*x^2-120)*cosh(x)-120+(x^5+20*x^3+120*x)*sinh(x)
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int x^5 \cosh (x) \, dx=-5 \, {\left (x^{4} + 12 \, x^{2} + 24\right )} \cosh \left (x\right ) + {\left (x^{5} + 20 \, x^{3} + 120 \, x\right )} \sinh \left (x\right ) \] Input:
integrate(x^5*cosh(x),x, algorithm="fricas")
Output:
-5*(x^4 + 12*x^2 + 24)*cosh(x) + (x^5 + 20*x^3 + 120*x)*sinh(x)
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int x^5 \cosh (x) \, dx=x^{5} \sinh {\left (x \right )} - 5 x^{4} \cosh {\left (x \right )} + 20 x^{3} \sinh {\left (x \right )} - 60 x^{2} \cosh {\left (x \right )} + 120 x \sinh {\left (x \right )} - 120 \cosh {\left (x \right )} \] Input:
integrate(x**5*cosh(x),x)
Output:
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)
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00 \[ \int x^5 \cosh (x) \, dx=\frac {1}{6} \, x^{6} \cosh \left (x\right ) - \frac {1}{12} \, {\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} - \frac {1}{12} \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} \] Input:
integrate(x^5*cosh(x),x, algorithm="maxima")
Output:
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 + 30*x^4 - 120*x^3 + 360*x^2 - 720*x + 72 0)*e^x
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int x^5 \cosh (x) \, dx=-\frac {1}{2} \, {\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} + \frac {1}{2} \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} \] Input:
integrate(x^5*cosh(x),x, algorithm="giac")
Output:
-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
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x^5 \cosh (x) \, dx=20\,x^3\,\mathrm {sinh}\left (x\right )-60\,x^2\,\mathrm {cosh}\left (x\right )-5\,x^4\,\mathrm {cosh}\left (x\right )-120\,\mathrm {cosh}\left (x\right )+x^5\,\mathrm {sinh}\left (x\right )+120\,x\,\mathrm {sinh}\left (x\right ) \] Input:
int(x^5*cosh(x),x)
Output:
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)
Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x^5 \cosh (x) \, dx=-5 \cosh \left (x \right ) x^{4}-60 \cosh \left (x \right ) x^{2}-120 \cosh \left (x \right )+\sinh \left (x \right ) x^{5}+20 \sinh \left (x \right ) x^{3}+120 \sinh \left (x \right ) x \] Input:
int(x^5*cosh(x),x)
Output:
- 5*cosh(x)*x**4 - 60*cosh(x)*x**2 - 120*cosh(x) + sinh(x)*x**5 + 20*sinh (x)*x**3 + 120*sinh(x)*x