Integrand size = 12, antiderivative size = 172 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {45 x^2}{128 b^2}+\frac {3 x^4}{32}-\frac {45 \cosh ^2(a+b x)}{128 b^4}-\frac {9 x^2 \cosh ^2(a+b x)}{16 b^2}-\frac {3 \cosh ^4(a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {45 x \cosh (a+b x) \sinh (a+b x)}{64 b^3}+\frac {3 x^3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {3 x \cosh ^3(a+b x) \sinh (a+b x)}{32 b^3}+\frac {x^3 \cosh ^3(a+b x) \sinh (a+b x)}{4 b} \] Output:
45/128*x^2/b^2+3/32*x^4-45/128*cosh(b*x+a)^2/b^4-9/16*x^2*cosh(b*x+a)^2/b^ 2-3/128*cosh(b*x+a)^4/b^4-3/16*x^2*cosh(b*x+a)^4/b^2+45/64*x*cosh(b*x+a)*s inh(b*x+a)/b^3+3/8*x^3*cosh(b*x+a)*sinh(b*x+a)/b+3/32*x*cosh(b*x+a)^3*sinh (b*x+a)/b^3+1/4*x^3*cosh(b*x+a)^3*sinh(b*x+a)/b
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.58 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {-192 \left (1+2 b^2 x^2\right ) \cosh (2 (a+b x))-3 \left (1+8 b^2 x^2\right ) \cosh (4 (a+b x))+4 b x \left (24 b^3 x^3+32 \left (3+2 b^2 x^2\right ) \sinh (2 (a+b x))+\left (3+8 b^2 x^2\right ) \sinh (4 (a+b x))\right )}{1024 b^4} \] Input:
Integrate[x^3*Cosh[a + b*x]^4,x]
Output:
(-192*(1 + 2*b^2*x^2)*Cosh[2*(a + b*x)] - 3*(1 + 8*b^2*x^2)*Cosh[4*(a + b* x)] + 4*b*x*(24*b^3*x^3 + 32*(3 + 2*b^2*x^2)*Sinh[2*(a + b*x)] + (3 + 8*b^ 2*x^2)*Sinh[4*(a + b*x)]))/(1024*b^4)
Time = 0.76 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.38, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3792, 3042, 3791, 3042, 3791, 15, 3792, 15, 3042, 3791, 15}
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.
| \(\displaystyle \int x^3 \cosh ^4(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {3 \int x \cosh ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int x^3 \cosh ^2(a+b x)dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int x \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int x \cosh ^2(a+b x)dx-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int x \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \left (\frac {\int xdx}{2}-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{4} \int x^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {3}{4} \left (\frac {3 \int x \cosh ^2(a+b x)dx}{2 b^2}+\frac {\int x^3dx}{2}-\frac {3 x^2 \cosh ^2(a+b x)}{4 b^2}+\frac {x^3 \sinh (a+b x) \cosh (a+b x)}{2 b}\right )-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{4} \left (\frac {3 \int x \cosh ^2(a+b x)dx}{2 b^2}-\frac {3 x^2 \cosh ^2(a+b x)}{4 b^2}+\frac {x^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^4}{8}\right )-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (\frac {3 \int x \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b^2}-\frac {3 x^2 \cosh ^2(a+b x)}{4 b^2}+\frac {x^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^4}{8}\right )-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {3}{4} \left (\frac {3 \left (\frac {\int xdx}{2}-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}\right )}{2 b^2}-\frac {3 x^2 \cosh ^2(a+b x)}{4 b^2}+\frac {x^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^4}{8}\right )-\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3 x^2 \cosh ^4(a+b x)}{16 b^2}+\frac {3 \left (\frac {3}{4} \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )-\frac {\cosh ^4(a+b x)}{16 b^2}+\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \left (-\frac {3 x^2 \cosh ^2(a+b x)}{4 b^2}+\frac {3 \left (-\frac {\cosh ^2(a+b x)}{4 b^2}+\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^2}{4}\right )}{2 b^2}+\frac {x^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^4}{8}\right )+\frac {x^3 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\) |
Input:
Int[x^3*Cosh[a + b*x]^4,x]
Output:
(-3*x^2*Cosh[a + b*x]^4)/(16*b^2) + (x^3*Cosh[a + b*x]^3*Sinh[a + b*x])/(4 *b) + (3*(-1/16*Cosh[a + b*x]^4/b^2 + (x*Cosh[a + b*x]^3*Sinh[a + b*x])/(4 *b) + (3*(x^2/4 - Cosh[a + b*x]^2/(4*b^2) + (x*Cosh[a + b*x]*Sinh[a + b*x] )/(2*b)))/4))/(8*b^2) + (3*(x^4/8 - (3*x^2*Cosh[a + b*x]^2)/(4*b^2) + (x^3 *Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (3*(x^2/4 - Cosh[a + b*x]^2/(4*b^2) + (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)))/(2*b^2)))/4
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Time = 1.41 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {\left (-384 x^{2} b^{2}-192\right ) \cosh \left (2 b x +2 a \right )+\left (-24 x^{2} b^{2}-3\right ) \cosh \left (4 b x +4 a \right )+\left (256 x^{3} b^{3}+384 b x \right ) \sinh \left (2 b x +2 a \right )+\left (32 x^{3} b^{3}+12 b x \right ) \sinh \left (4 b x +4 a \right )+96 x^{4} b^{4}+195}{1024 b^{4}}\) | \(102\) |
| risch | \(\frac {3 x^{4}}{32}+\frac {\left (32 x^{3} b^{3}-24 x^{2} b^{2}+12 b x -3\right ) {\mathrm e}^{4 b x +4 a}}{2048 b^{4}}+\frac {\left (4 x^{3} b^{3}-6 x^{2} b^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}-\frac {\left (4 x^{3} b^{3}+6 x^{2} b^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}-\frac {\left (32 x^{3} b^{3}+24 x^{2} b^{2}+12 b x +3\right ) {\mathrm e}^{-4 b x -4 a}}{2048 b^{4}}\) | \(151\) |
| derivativedivides | \(\frac {-a^{3} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {45 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}-\frac {45 \cosh \left (b x +a \right )^{2}}{128}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(400\) |
| default | \(\frac {-a^{3} \left (\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}-\frac {3 \cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{8}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {15 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {15 b x}{64}+\frac {15 a}{64}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {3 \left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}+\frac {45 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}-\frac {45 \cosh \left (b x +a \right )^{2}}{128}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(400\) |
| orering | \(\frac {\left (32 x^{6} b^{6}-240 x^{4} b^{4}-330 x^{2} b^{2}+945\right ) \cosh \left (b x +a \right )^{4}}{128 x^{2} b^{6}}+\frac {5 \left (40 x^{4} b^{4}+24 x^{2} b^{2}-243\right ) \left (3 x^{2} \cosh \left (b x +a \right )^{4}+4 x^{3} \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )\right )}{256 x^{4} b^{6}}-\frac {5 \left (4 x^{4} b^{4}-9 x^{2} b^{2}-69\right ) \left (6 x \cosh \left (b x +a \right )^{4}+24 x^{2} \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )+12 x^{3} \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+4 x^{3} \cosh \left (b x +a \right )^{4} b^{2}\right )}{256 x^{3} b^{6}}-\frac {\left (32 x^{2} b^{2}+105\right ) \left (6 \cosh \left (b x +a \right )^{4}+72 x \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )+108 x^{2} \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+36 x^{2} \cosh \left (b x +a \right )^{4} b^{2}+24 x^{3} \cosh \left (b x +a \right ) b^{3} \sinh \left (b x +a \right )^{3}+40 x^{3} \cosh \left (b x +a \right )^{3} b^{3} \sinh \left (b x +a \right )\right )}{512 x^{2} b^{6}}+\frac {\left (4 x^{2} b^{2}+15\right ) \left (96 \cosh \left (b x +a \right )^{3} b \sinh \left (b x +a \right )+432 x \cosh \left (b x +a \right )^{2} b^{2} \sinh \left (b x +a \right )^{2}+144 x \cosh \left (b x +a \right )^{4} b^{2}+288 x^{2} \cosh \left (b x +a \right ) b^{3} \sinh \left (b x +a \right )^{3}+480 x^{2} \cosh \left (b x +a \right )^{3} b^{3} \sinh \left (b x +a \right )+24 x^{3} b^{4} \sinh \left (b x +a \right )^{4}+192 x^{3} \cosh \left (b x +a \right )^{2} b^{4} \sinh \left (b x +a \right )^{2}+40 x^{3} \cosh \left (b x +a \right )^{4} b^{4}\right )}{1024 x \,b^{6}}\) | \(505\) |
Input:
int(x^3*cosh(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/1024*((-384*b^2*x^2-192)*cosh(2*b*x+2*a)+(-24*b^2*x^2-3)*cosh(4*b*x+4*a) +(256*b^3*x^3+384*b*x)*sinh(2*b*x+2*a)+(32*b^3*x^3+12*b*x)*sinh(4*b*x+4*a) +96*x^4*b^4+195)/b^4
Time = 0.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.13 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {96 \, b^{4} x^{4} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} + 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 192 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 6 \, {\left (64 \, b^{2} x^{2} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} + 32\right )} \sinh \left (b x + a\right )^{2} + 16 \, {\left ({\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{1024 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^4,x, algorithm="fricas")
Output:
1/1024*(96*b^4*x^4 - 3*(8*b^2*x^2 + 1)*cosh(b*x + a)^4 + 16*(8*b^3*x^3 + 3 *b*x)*cosh(b*x + a)*sinh(b*x + a)^3 - 3*(8*b^2*x^2 + 1)*sinh(b*x + a)^4 - 192*(2*b^2*x^2 + 1)*cosh(b*x + a)^2 - 6*(64*b^2*x^2 + 3*(8*b^2*x^2 + 1)*co sh(b*x + a)^2 + 32)*sinh(b*x + a)^2 + 16*((8*b^3*x^3 + 3*b*x)*cosh(b*x + a )^3 + 16*(2*b^3*x^3 + 3*b*x)*cosh(b*x + a))*sinh(b*x + a))/b^4
Time = 0.53 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.47 \[ \int x^3 \cosh ^4(a+b x) \, dx=\begin {cases} \frac {3 x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} - \frac {3 x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac {3 x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} - \frac {3 x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {5 x^{3} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac {45 x^{2} \sinh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {9 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {51 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {45 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {51 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac {45 \sinh ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac {51 \cosh ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cosh ^{4}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*cosh(b*x+a)**4,x)
Output:
Piecewise((3*x**4*sinh(a + b*x)**4/32 - 3*x**4*sinh(a + b*x)**2*cosh(a + b *x)**2/16 + 3*x**4*cosh(a + b*x)**4/32 - 3*x**3*sinh(a + b*x)**3*cosh(a + b*x)/(8*b) + 5*x**3*sinh(a + b*x)*cosh(a + b*x)**3/(8*b) + 45*x**2*sinh(a + b*x)**4/(128*b**2) - 9*x**2*sinh(a + b*x)**2*cosh(a + b*x)**2/(64*b**2) - 51*x**2*cosh(a + b*x)**4/(128*b**2) - 45*x*sinh(a + b*x)**3*cosh(a + b*x )/(64*b**3) + 51*x*sinh(a + b*x)*cosh(a + b*x)**3/(64*b**3) + 45*sinh(a + b*x)**4/(256*b**4) - 51*cosh(a + b*x)**4/(256*b**4), Ne(b, 0)), (x**4*cosh (a)**4/4, True))
Time = 0.05 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^4,x, algorithm="maxima")
Output:
3/32*x^4 + 1/2048*(32*b^3*x^3*e^(4*a) - 24*b^2*x^2*e^(4*a) + 12*b*x*e^(4*a ) - 3*e^(4*a))*e^(4*b*x)/b^4 + 1/32*(4*b^3*x^3*e^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^(2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 - 1/32*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^4 - 1/2048*(32*b^3*x^3 + 24*b^2*x^2 + 12*b *x + 3)*e^(-4*b*x - 4*a)/b^4
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.87 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^4,x, algorithm="giac")
Output:
3/32*x^4 + 1/2048*(32*b^3*x^3 - 24*b^2*x^2 + 12*b*x - 3)*e^(4*b*x + 4*a)/b ^4 + 1/32*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*e^(2*b*x + 2*a)/b^4 - 1/32*( 4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^4 - 1/2048*(32*b^3*x ^3 + 24*b^2*x^2 + 12*b*x + 3)*e^(-4*b*x - 4*a)/b^4
Time = 0.39 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {3\,x^4}{32}-\frac {\frac {3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{16}+\frac {3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{1024}+b^2\,\left (\frac {3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{8}+\frac {3\,x^2\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{128}\right )-b\,\left (\frac {3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8}+\frac {3\,x\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}\right )-b^3\,\left (\frac {x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}+\frac {x^3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}\right )}{b^4} \] Input:
int(x^3*cosh(a + b*x)^4,x)
Output:
(3*x^4)/32 - ((3*cosh(2*a + 2*b*x))/16 + (3*cosh(4*a + 4*b*x))/1024 + b^2* ((3*x^2*cosh(2*a + 2*b*x))/8 + (3*x^2*cosh(4*a + 4*b*x))/128) - b*((3*x*si nh(2*a + 2*b*x))/8 + (3*x*sinh(4*a + 4*b*x))/256) - b^3*((x^3*sinh(2*a + 2 *b*x))/4 + (x^3*sinh(4*a + 4*b*x))/32))/b^4
Time = 0.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.41 \[ \int x^3 \cosh ^4(a+b x) \, dx=\frac {32 e^{8 b x +8 a} b^{3} x^{3}-24 e^{8 b x +8 a} b^{2} x^{2}+12 e^{8 b x +8 a} b x -3 e^{8 b x +8 a}+256 e^{6 b x +6 a} b^{3} x^{3}-384 e^{6 b x +6 a} b^{2} x^{2}+384 e^{6 b x +6 a} b x -192 e^{6 b x +6 a}+192 e^{4 b x +4 a} b^{4} x^{4}-256 e^{2 b x +2 a} b^{3} x^{3}-384 e^{2 b x +2 a} b^{2} x^{2}-384 e^{2 b x +2 a} b x -192 e^{2 b x +2 a}-32 b^{3} x^{3}-24 b^{2} x^{2}-12 b x -3}{2048 e^{4 b x +4 a} b^{4}} \] Input:
int(x^3*cosh(b*x+a)^4,x)
Output:
(32*e**(8*a + 8*b*x)*b**3*x**3 - 24*e**(8*a + 8*b*x)*b**2*x**2 + 12*e**(8* a + 8*b*x)*b*x - 3*e**(8*a + 8*b*x) + 256*e**(6*a + 6*b*x)*b**3*x**3 - 384 *e**(6*a + 6*b*x)*b**2*x**2 + 384*e**(6*a + 6*b*x)*b*x - 192*e**(6*a + 6*b *x) + 192*e**(4*a + 4*b*x)*b**4*x**4 - 256*e**(2*a + 2*b*x)*b**3*x**3 - 38 4*e**(2*a + 2*b*x)*b**2*x**2 - 384*e**(2*a + 2*b*x)*b*x - 192*e**(2*a + 2* b*x) - 32*b**3*x**3 - 24*b**2*x**2 - 12*b*x - 3)/(2048*e**(4*a + 4*b*x)*b* *4)