Integrand size = 12, antiderivative size = 172 \[ \int x^3 \cos ^4(a+b x) \, dx=-\frac {45 x^2}{128 b^2}+\frac {3 x^4}{32}-\frac {45 \cos ^2(a+b x)}{128 b^4}+\frac {9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac {3 \cos ^4(a+b x)}{128 b^4}+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {45 x \cos (a+b x) \sin (a+b x)}{64 b^3}+\frac {3 x^3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac {3 x \cos ^3(a+b x) \sin (a+b x)}{32 b^3}+\frac {x^3 \cos ^3(a+b x) \sin (a+b x)}{4 b} \] Output:
-45/128*x^2/b^2+3/32*x^4-45/128*cos(b*x+a)^2/b^4+9/16*x^2*cos(b*x+a)^2/b^2 -3/128*cos(b*x+a)^4/b^4+3/16*x^2*cos(b*x+a)^4/b^2-45/64*x*cos(b*x+a)*sin(b *x+a)/b^3+3/8*x^3*cos(b*x+a)*sin(b*x+a)/b-3/32*x*cos(b*x+a)^3*sin(b*x+a)/b ^3+1/4*x^3*cos(b*x+a)^3*sin(b*x+a)/b
Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.58 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {192 \left (-1+2 b^2 x^2\right ) \cos (2 (a+b x))+3 \left (-1+8 b^2 x^2\right ) \cos (4 (a+b x))+4 b x \left (24 b^3 x^3+32 \left (-3+2 b^2 x^2\right ) \sin (2 (a+b x))+\left (-3+8 b^2 x^2\right ) \sin (4 (a+b x))\right )}{1024 b^4} \] Input:
Integrate[x^3*Cos[a + b*x]^4,x]
Output:
(192*(-1 + 2*b^2*x^2)*Cos[2*(a + b*x)] + 3*(-1 + 8*b^2*x^2)*Cos[4*(a + b*x )] + 4*b*x*(24*b^3*x^3 + 32*(-3 + 2*b^2*x^2)*Sin[2*(a + b*x)] + (-3 + 8*b^ 2*x^2)*Sin[4*(a + b*x)]))/(1024*b^4)
Time = 0.75 (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.
| \(\displaystyle \int x^3 \cos ^4(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {3 \int x \cos ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int x^3 \cos ^2(a+b x)dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int x \sin \left (a+b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {3 \left (\frac {3}{4} \int x \cos ^2(a+b x)dx+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {3}{4} \int x \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {3 \left (\frac {3}{4} \left (\frac {\int xdx}{2}+\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{4} \int x^3 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {3}{4} \left (-\frac {3 \int x \cos ^2(a+b x)dx}{2 b^2}+\frac {\int x^3dx}{2}+\frac {3 x^2 \cos ^2(a+b x)}{4 b^2}+\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{4} \left (-\frac {3 \int x \cos ^2(a+b x)dx}{2 b^2}+\frac {3 x^2 \cos ^2(a+b x)}{4 b^2}+\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (-\frac {3 \int x \sin \left (a+b x+\frac {\pi }{2}\right )^2dx}{2 b^2}+\frac {3 x^2 \cos ^2(a+b x)}{4 b^2}+\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {3}{4} \left (-\frac {3 \left (\frac {\int xdx}{2}+\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {3 x^2 \cos ^2(a+b x)}{4 b^2}+\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )+\frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac {3 \left (\frac {3}{4} \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )+\frac {\cos ^4(a+b x)}{16 b^2}+\frac {x \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \left (\frac {3 x^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 \left (\frac {\cos ^2(a+b x)}{4 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )}{2 b^2}+\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )+\frac {x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
Input:
Int[x^3*Cos[a + b*x]^4,x]
Output:
(3*x^2*Cos[a + b*x]^4)/(16*b^2) + (x^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) - (3*(Cos[a + b*x]^4/(16*b^2) + (x*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) + (3 *(x^2/4 + Cos[a + b*x]^2/(4*b^2) + (x*Cos[a + b*x]*Sin[a + b*x])/(2*b)))/4 ))/(8*b^2) + (3*(x^4/8 + (3*x^2*Cos[a + b*x]^2)/(4*b^2) + (x^3*Cos[a + b*x ]*Sin[a + b*x])/(2*b) - (3*(x^2/4 + Cos[a + b*x]^2/(4*b^2) + (x*Cos[a + b* x]*Sin[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 = 2.74 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {\left (384 x^{2} b^{2}-192\right ) \cos \left (2 b x +2 a \right )+\left (24 x^{2} b^{2}-3\right ) \cos \left (4 b x +4 a \right )+\left (256 b^{3} x^{3}-384 b x \right ) \sin \left (2 b x +2 a \right )+\left (32 b^{3} x^{3}-12 b x \right ) \sin \left (4 b x +4 a \right )+96 x^{4} b^{4}+195}{1024 b^{4}}\) | \(102\) |
| risch | \(\frac {3 x^{4}}{32}+\frac {3 \left (8 x^{2} b^{2}-1\right ) \cos \left (4 b x +4 a \right )}{1024 b^{4}}+\frac {x \left (8 x^{2} b^{2}-3\right ) \sin \left (4 b x +4 a \right )}{256 b^{3}}+\frac {3 \left (2 x^{2} b^{2}-1\right ) \cos \left (2 b x +2 a \right )}{16 b^{4}}+\frac {x \left (2 x^{2} b^{2}-3\right ) \sin \left (2 b x +2 a \right )}{8 b^{3}}\) | \(105\) |
| derivativedivides | \(\frac {-a^{3} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{64}\right )-3 a \left (\left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{8}-\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{32}-\frac {15 b x}{64}-\frac {15 a}{64}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{8}-\frac {3 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}-\frac {\left (b x +a \right )^{3}}{4}\right )+\left (b x +a \right )^{3} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {3 \left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{16}-\frac {3 \left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{8}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{512}+\frac {9 \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{16}-\frac {9 \left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{8}+\frac {9 \sin \left (b x +a \right )^{2}}{32}-\frac {9 \left (b x +a \right )^{4}}{32}}{b^{4}}\) | \(432\) |
| default | \(\frac {-a^{3} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+3 a^{2} \left (\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{64}\right )-3 a \left (\left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{8}-\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{32}-\frac {15 b x}{64}-\frac {15 a}{64}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{8}-\frac {3 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}-\frac {\left (b x +a \right )^{3}}{4}\right )+\left (b x +a \right )^{3} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {3 \left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{16}-\frac {3 \left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{8}+\frac {45 \left (b x +a \right )^{2}}{128}-\frac {3 \left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{512}+\frac {9 \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{16}-\frac {9 \left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{8}+\frac {9 \sin \left (b x +a \right )^{2}}{32}-\frac {9 \left (b x +a \right )^{4}}{32}}{b^{4}}\) | \(432\) |
| orering | \(\frac {\left (32 b^{6} x^{6}+240 x^{4} b^{4}-330 x^{2} b^{2}-945\right ) \cos \left (b x +a \right )^{4}}{128 b^{6} x^{2}}-\frac {5 \left (40 x^{4} b^{4}-24 x^{2} b^{2}-243\right ) \left (3 x^{2} \cos \left (b x +a \right )^{4}-4 x^{3} \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )\right )}{256 x^{4} b^{6}}+\frac {5 \left (4 x^{4} b^{4}+9 x^{2} b^{2}-69\right ) \left (6 x \cos \left (b x +a \right )^{4}-24 x^{2} \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )+12 x^{3} \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-4 x^{3} \cos \left (b x +a \right )^{4} b^{2}\right )}{256 x^{3} b^{6}}-\frac {\left (32 x^{2} b^{2}-105\right ) \left (6 \cos \left (b x +a \right )^{4}-72 x \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )+108 x^{2} \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-36 x^{2} \cos \left (b x +a \right )^{4} b^{2}-24 x^{3} \cos \left (b x +a \right ) b^{3} \sin \left (b x +a \right )^{3}+40 x^{3} \cos \left (b x +a \right )^{3} b^{3} \sin \left (b x +a \right )\right )}{512 x^{2} b^{6}}+\frac {\left (4 x^{2} b^{2}-15\right ) \left (-96 \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )+432 x \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-144 x \cos \left (b x +a \right )^{4} b^{2}-288 x^{2} \cos \left (b x +a \right ) b^{3} \sin \left (b x +a \right )^{3}+480 x^{2} \cos \left (b x +a \right )^{3} b^{3} \sin \left (b x +a \right )+24 x^{3} b^{4} \sin \left (b x +a \right )^{4}-192 x^{3} \cos \left (b x +a \right )^{2} b^{4} \sin \left (b x +a \right )^{2}+40 x^{3} \cos \left (b x +a \right )^{4} b^{4}\right )}{1024 x \,b^{6}}\) | \(505\) |
Input:
int(x^3*cos(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/1024*((384*b^2*x^2-192)*cos(2*b*x+2*a)+(24*b^2*x^2-3)*cos(4*b*x+4*a)+(25 6*b^3*x^3-384*b*x)*sin(2*b*x+2*a)+(32*b^3*x^3-12*b*x)*sin(4*b*x+4*a)+96*x^ 4*b^4+195)/b^4
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.67 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {12 \, b^{4} x^{4} + 3 \, {\left (8 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right )^{4} - 45 \, b^{2} x^{2} + 9 \, {\left (8 \, b^{2} x^{2} - 5\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (2 \, {\left (8 \, b^{3} x^{3} - 3 \, b x\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{3} x^{3} - 15 \, b x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{4}} \] Input:
integrate(x^3*cos(b*x+a)^4,x, algorithm="fricas")
Output:
1/128*(12*b^4*x^4 + 3*(8*b^2*x^2 - 1)*cos(b*x + a)^4 - 45*b^2*x^2 + 9*(8*b ^2*x^2 - 5)*cos(b*x + a)^2 + 2*(2*(8*b^3*x^3 - 3*b*x)*cos(b*x + a)^3 + 3*( 8*b^3*x^3 - 15*b*x)*cos(b*x + a))*sin(b*x + a))/b^4
Time = 0.55 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.47 \[ \int x^3 \cos ^4(a+b x) \, dx=\begin {cases} \frac {3 x^{4} \sin ^{4}{\left (a + b x \right )}}{32} + \frac {3 x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x^{4} \cos ^{4}{\left (a + b x \right )}}{32} + \frac {3 x^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {5 x^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {45 x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {9 x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{2}} + \frac {51 x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {45 x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} - \frac {51 x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac {45 \sin ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac {51 \cos ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos ^{4}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*cos(b*x+a)**4,x)
Output:
Piecewise((3*x**4*sin(a + b*x)**4/32 + 3*x**4*sin(a + b*x)**2*cos(a + b*x) **2/16 + 3*x**4*cos(a + b*x)**4/32 + 3*x**3*sin(a + b*x)**3*cos(a + b*x)/( 8*b) + 5*x**3*sin(a + b*x)*cos(a + b*x)**3/(8*b) - 45*x**2*sin(a + b*x)**4 /(128*b**2) - 9*x**2*sin(a + b*x)**2*cos(a + b*x)**2/(64*b**2) + 51*x**2*c os(a + b*x)**4/(128*b**2) - 45*x*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) - 51*x*sin(a + b*x)*cos(a + b*x)**3/(64*b**3) + 45*sin(a + b*x)**4/(256*b**4 ) - 51*cos(a + b*x)**4/(256*b**4), Ne(b, 0)), (x**4*cos(a)**4/4, True))
Time = 0.05 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.76 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {96 \, {\left (b x + a\right )}^{4} - 32 \, {\left (12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{3} + 24 \, {\left (24 \, {\left (b x + a\right )}^{2} + 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right ) + 16 \, \cos \left (2 \, b x + 2 \, a\right )\right )} a^{2} - 12 \, {\left (32 \, {\left (b x + a\right )}^{3} + 4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 64 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a + 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 192 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 128 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )}{1024 \, b^{4}} \] Input:
integrate(x^3*cos(b*x+a)^4,x, algorithm="maxima")
Output:
1/1024*(96*(b*x + a)^4 - 32*(12*b*x + 12*a + sin(4*b*x + 4*a) + 8*sin(2*b* x + 2*a))*a^3 + 24*(24*(b*x + a)^2 + 4*(b*x + a)*sin(4*b*x + 4*a) + 32*(b* x + a)*sin(2*b*x + 2*a) + cos(4*b*x + 4*a) + 16*cos(2*b*x + 2*a))*a^2 - 12 *(32*(b*x + a)^3 + 4*(b*x + a)*cos(4*b*x + 4*a) + 64*(b*x + a)*cos(2*b*x + 2*a) + (8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) + 32*(2*(b*x + a)^2 - 1)*sin( 2*b*x + 2*a))*a + 3*(8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) + 192*(2*(b*x + a )^2 - 1)*cos(2*b*x + 2*a) + 4*(8*(b*x + a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4* a) + 128*(2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2*b*x + 2*a))/b^4
Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.63 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {3}{32} \, x^{4} + \frac {3 \, {\left (8 \, b^{2} x^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} + \frac {{\left (8 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} + \frac {{\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} \] Input:
integrate(x^3*cos(b*x+a)^4,x, algorithm="giac")
Output:
3/32*x^4 + 3/1024*(8*b^2*x^2 - 1)*cos(4*b*x + 4*a)/b^4 + 3/16*(2*b^2*x^2 - 1)*cos(2*b*x + 2*a)/b^4 + 1/256*(8*b^3*x^3 - 3*b*x)*sin(4*b*x + 4*a)/b^4 + 1/8*(2*b^3*x^3 - 3*b*x)*sin(2*b*x + 2*a)/b^4
Time = 45.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {\frac {3\,{\sin \left (2\,a+2\,b\,x\right )}^2}{512}-b^2\,\left (\frac {3\,x^2\,\left (2\,{\sin \left (2\,a+2\,b\,x\right )}^2-1\right )}{128}+\frac {3\,x^2\,\left (2\,{\sin \left (a+b\,x\right )}^2-1\right )}{8}\right )-b\,\left (\frac {3\,x\,\sin \left (2\,a+2\,b\,x\right )}{8}+\frac {3\,x\,\sin \left (4\,a+4\,b\,x\right )}{256}\right )+b^3\,\left (\frac {x^3\,\sin \left (2\,a+2\,b\,x\right )}{4}+\frac {x^3\,\sin \left (4\,a+4\,b\,x\right )}{32}\right )+\frac {3\,{\sin \left (a+b\,x\right )}^2}{8}}{b^4}+\frac {3\,x^4}{32} \] Input:
int(x^3*cos(a + b*x)^4,x)
Output:
((3*sin(2*a + 2*b*x)^2)/512 - b^2*((3*x^2*(2*sin(2*a + 2*b*x)^2 - 1))/128 + (3*x^2*(2*sin(a + b*x)^2 - 1))/8) - b*((3*x*sin(2*a + 2*b*x))/8 + (3*x*s in(4*a + 4*b*x))/256) + b^3*((x^3*sin(2*a + 2*b*x))/4 + (x^3*sin(4*a + 4*b *x))/32) + (3*sin(a + b*x)^2)/8)/b^4 + (3*x^4)/32
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int x^3 \cos ^4(a+b x) \, dx=\frac {-32 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{3} x^{3}+12 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b x +80 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} x^{3}-102 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b x +24 \sin \left (b x +a \right )^{4} b^{2} x^{2}-3 \sin \left (b x +a \right )^{4}-120 \sin \left (b x +a \right )^{2} b^{2} x^{2}+51 \sin \left (b x +a \right )^{2}+12 b^{4} x^{4}+51 b^{2} x^{2}-51}{128 b^{4}} \] Input:
int(x^3*cos(b*x+a)^4,x)
Output:
( - 32*cos(a + b*x)*sin(a + b*x)**3*b**3*x**3 + 12*cos(a + b*x)*sin(a + b* x)**3*b*x + 80*cos(a + b*x)*sin(a + b*x)*b**3*x**3 - 102*cos(a + b*x)*sin( a + b*x)*b*x + 24*sin(a + b*x)**4*b**2*x**2 - 3*sin(a + b*x)**4 - 120*sin( a + b*x)**2*b**2*x**2 + 51*sin(a + b*x)**2 + 12*b**4*x**4 + 51*b**2*x**2 - 51)/(128*b**4)