Integrand size = 12, antiderivative size = 134 \[ \int x^2 \cos ^4(a+b x) \, dx=-\frac {15 x}{64 b^2}+\frac {x^3}{8}+\frac {3 x \cos ^2(a+b x)}{8 b^2}+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {15 \cos (a+b x) \sin (a+b x)}{64 b^3}+\frac {3 x^2 \cos (a+b x) \sin (a+b x)}{8 b}-\frac {\cos ^3(a+b x) \sin (a+b x)}{32 b^3}+\frac {x^2 \cos ^3(a+b x) \sin (a+b x)}{4 b} \] Output:
-15/64*x/b^2+1/8*x^3+3/8*x*cos(b*x+a)^2/b^2+1/8*x*cos(b*x+a)^4/b^2-15/64*c os(b*x+a)*sin(b*x+a)/b^3+3/8*x^2*cos(b*x+a)*sin(b*x+a)/b-1/32*cos(b*x+a)^3 *sin(b*x+a)/b^3+1/4*x^2*cos(b*x+a)^3*sin(b*x+a)/b
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.69 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {32 b^3 x^3+64 b x \cos (2 (a+b x))+4 b x \cos (4 (a+b x))-32 \sin (2 (a+b x))+64 b^2 x^2 \sin (2 (a+b x))-\sin (4 (a+b x))+8 b^2 x^2 \sin (4 (a+b x))}{256 b^3} \] Input:
Integrate[x^2*Cos[a + b*x]^4,x]
Output:
(32*b^3*x^3 + 64*b*x*Cos[2*(a + b*x)] + 4*b*x*Cos[4*(a + b*x)] - 32*Sin[2* (a + b*x)] + 64*b^2*x^2*Sin[2*(a + b*x)] - Sin[4*(a + b*x)] + 8*b^2*x^2*Si n[4*(a + b*x)])/(256*b^3)
Time = 0.62 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.35, 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, 3115, 3042, 3115, 24, 3792, 15, 3042, 3115, 24}
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.
| \(\displaystyle \int x^2 \cos ^4(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {\int \cos ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int x^2 \cos ^2(a+b x)dx+\frac {x \cos ^4(a+b x)}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {x \cos ^4(a+b x)}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {x \cos ^4(a+b x)}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {x \cos ^4(a+b x)}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {x \cos ^4(a+b x)}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{4} \int x^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {3}{4} \left (-\frac {\int \cos ^2(a+b x)dx}{2 b^2}+\frac {\int x^2dx}{2}+\frac {x \cos ^2(a+b x)}{2 b^2}+\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{4} \left (-\frac {\int \cos ^2(a+b x)dx}{2 b^2}+\frac {x \cos ^2(a+b x)}{2 b^2}+\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^3}{6}\right )+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \left (-\frac {\int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx}{2 b^2}+\frac {x \cos ^2(a+b x)}{2 b^2}+\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^3}{6}\right )+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{4} \left (-\frac {\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}}{2 b^2}+\frac {x \cos ^2(a+b x)}{2 b^2}+\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^3}{6}\right )+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{4} \left (\frac {x \cos ^2(a+b x)}{2 b^2}-\frac {\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}}{2 b^2}+\frac {x^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^3}{6}\right )+\frac {x \cos ^4(a+b x)}{8 b^2}-\frac {\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\) |
Input:
Int[x^2*Cos[a + b*x]^4,x]
Output:
(x*Cos[a + b*x]^4)/(8*b^2) + (x^2*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) - ((C os[a + b*x]^3*Sin[a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*Sin[a + b*x])/ (2*b)))/4)/(8*b^2) + (3*(x^3/6 + (x*Cos[a + b*x]^2)/(2*b^2) + (x^2*Cos[a + b*x]*Sin[a + b*x])/(2*b) - (x/2 + (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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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 = 2.42 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {\left (64 x^{2} b^{2}-32\right ) \sin \left (2 b x +2 a \right )+\left (8 x^{2} b^{2}-1\right ) \sin \left (4 b x +4 a \right )+32 b x \left (x^{2} b^{2}+2 \cos \left (2 b x +2 a \right )+\frac {\cos \left (4 b x +4 a \right )}{8}\right )}{256 b^{3}}\) | \(81\) |
| risch | \(\frac {x^{3}}{8}+\frac {x \cos \left (4 b x +4 a \right )}{64 b^{2}}+\frac {\left (8 x^{2} b^{2}-1\right ) \sin \left (4 b x +4 a \right )}{256 b^{3}}+\frac {x \cos \left (2 b x +2 a \right )}{4 b^{2}}+\frac {\left (2 x^{2} b^{2}-1\right ) \sin \left (2 b x +2 a \right )}{8 b^{3}}\) | \(85\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{64}\right )+\left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{8}-\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{32}-\frac {15 b x}{64}-\frac {15 a}{64}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{8}-\frac {3 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}-\frac {\left (b x +a \right )^{3}}{4}}{b^{3}}\) | \(237\) |
| default | \(\frac {a^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-2 a \left (\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )-\frac {3 \left (b x +a \right )^{2}}{16}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{64}\right )+\left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{8}-\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{32}-\frac {15 b x}{64}-\frac {15 a}{64}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{8}-\frac {3 \cos \left (b x +a \right ) \sin \left (b x +a \right )}{16}-\frac {\left (b x +a \right )^{3}}{4}}{b^{3}}\) | \(237\) |
| norman | \(\frac {\frac {x^{3}}{8}-\frac {17 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{32 b^{3}}-\frac {9 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{32 b^{3}}+\frac {9 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{32 b^{3}}+\frac {17 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{32 b^{3}}+\frac {17 x}{64 b^{2}}+\frac {x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{2}+\frac {3 x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{4}+\frac {x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{2}+\frac {x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{8}-\frac {23 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{16 b^{2}}-\frac {45 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{32 b^{2}}-\frac {23 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16 b^{2}}+\frac {17 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{64 b^{2}}+\frac {5 x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}-\frac {3 x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{4 b}+\frac {3 x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{4 b}-\frac {5 x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{4 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{4}}\) | \(297\) |
| orering | \(\frac {\left (32 b^{6} x^{6}+120 x^{4} b^{4}-30 x^{2} b^{2}-135\right ) \cos \left (b x +a \right )^{4}}{96 b^{6} x^{3}}-\frac {5 \left (112 x^{4} b^{4}-6 x^{2} b^{2}-171\right ) \left (2 x \cos \left (b x +a \right )^{4}-4 x^{2} \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )\right )}{768 x^{4} b^{6}}+\frac {5 \left (16 x^{4} b^{4}+18 x^{2} b^{2}-63\right ) \left (2 \cos \left (b x +a \right )^{4}-16 x \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )+12 x^{2} \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-4 x^{2} \cos \left (b x +a \right )^{4} b^{2}\right )}{768 x^{3} b^{6}}-\frac {\left (88 x^{2} b^{2}-135\right ) \left (-24 \cos \left (b x +a \right )^{3} b \sin \left (b x +a \right )+72 x \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-24 x \cos \left (b x +a \right )^{4} b^{2}-24 x^{2} \cos \left (b x +a \right ) b^{3} \sin \left (b x +a \right )^{3}+40 x^{2} \cos \left (b x +a \right )^{3} b^{3} \sin \left (b x +a \right )\right )}{1536 x^{2} b^{6}}+\frac {\left (8 x^{2} b^{2}-15\right ) \left (144 \cos \left (b x +a \right )^{2} b^{2} \sin \left (b x +a \right )^{2}-48 \cos \left (b x +a \right )^{4} b^{2}-192 x \cos \left (b x +a \right ) b^{3} \sin \left (b x +a \right )^{3}+320 x \cos \left (b x +a \right )^{3} b^{3} \sin \left (b x +a \right )+24 x^{2} b^{4} \sin \left (b x +a \right )^{4}-192 x^{2} \cos \left (b x +a \right )^{2} b^{4} \sin \left (b x +a \right )^{2}+40 x^{2} \cos \left (b x +a \right )^{4} b^{4}\right )}{1536 x \,b^{6}}\) | \(462\) |
Input:
int(x^2*cos(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/256*((64*b^2*x^2-32)*sin(2*b*x+2*a)+(8*b^2*x^2-1)*sin(4*b*x+4*a)+32*b*x* (x^2*b^2+2*cos(2*b*x+2*a)+1/8*cos(4*b*x+4*a)))/b^3
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {8 \, b^{3} x^{3} + 8 \, b x \cos \left (b x + a\right )^{4} + 24 \, b x \cos \left (b x + a\right )^{2} - 15 \, b x + {\left (2 \, {\left (8 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{2} x^{2} - 5\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{64 \, b^{3}} \] Input:
integrate(x^2*cos(b*x+a)^4,x, algorithm="fricas")
Output:
1/64*(8*b^3*x^3 + 8*b*x*cos(b*x + a)^4 + 24*b*x*cos(b*x + a)^2 - 15*b*x + (2*(8*b^2*x^2 - 1)*cos(b*x + a)^3 + 3*(8*b^2*x^2 - 5)*cos(b*x + a))*sin(b* x + a))/b^3
Time = 0.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int x^2 \cos ^4(a+b x) \, dx=\begin {cases} \frac {x^{3} \sin ^{4}{\left (a + b x \right )}}{8} + \frac {x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {x^{3} \cos ^{4}{\left (a + b x \right )}}{8} + \frac {3 x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {5 x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {15 x \sin ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{2}} + \frac {17 x \cos ^{4}{\left (a + b x \right )}}{64 b^{2}} - \frac {15 \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{3}} - \frac {17 \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \cos ^{4}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*cos(b*x+a)**4,x)
Output:
Piecewise((x**3*sin(a + b*x)**4/8 + x**3*sin(a + b*x)**2*cos(a + b*x)**2/4 + x**3*cos(a + b*x)**4/8 + 3*x**2*sin(a + b*x)**3*cos(a + b*x)/(8*b) + 5* x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b) - 15*x*sin(a + b*x)**4/(64*b**2) - 3*x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b**2) + 17*x*cos(a + b*x)**4/(64* b**2) - 15*sin(a + b*x)**3*cos(a + b*x)/(64*b**3) - 17*sin(a + b*x)*cos(a + b*x)**3/(64*b**3), Ne(b, 0)), (x**3*cos(a)**4/3, True))
Time = 0.05 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.40 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {32 \, {\left (b x + a\right )}^{3} + 8 \, {\left (12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} - 4 \, {\left (24 \, {\left (b x + a\right )}^{2} + 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right ) + 16 \, \cos \left (2 \, b x + 2 \, a\right )\right )} a + 4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 64 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{256 \, b^{3}} \] Input:
integrate(x^2*cos(b*x+a)^4,x, algorithm="maxima")
Output:
1/256*(32*(b*x + a)^3 + 8*(12*b*x + 12*a + sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a^2 - 4*(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 + 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))/b^3
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {1}{8} \, x^{3} + \frac {x \cos \left (4 \, b x + 4 \, a\right )}{64 \, b^{2}} + \frac {x \cos \left (2 \, b x + 2 \, a\right )}{4 \, b^{2}} + \frac {{\left (8 \, b^{2} x^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{3}} \] Input:
integrate(x^2*cos(b*x+a)^4,x, algorithm="giac")
Output:
1/8*x^3 + 1/64*x*cos(4*b*x + 4*a)/b^2 + 1/4*x*cos(2*b*x + 2*a)/b^2 + 1/256 *(8*b^2*x^2 - 1)*sin(4*b*x + 4*a)/b^3 + 1/8*(2*b^2*x^2 - 1)*sin(2*b*x + 2* a)/b^3
Time = 44.75 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.78 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {x^3}{8}-\frac {\frac {\sin \left (2\,a+2\,b\,x\right )}{8}+\frac {\sin \left (4\,a+4\,b\,x\right )}{256}+b\,\left (\frac {x\,\left (2\,{\sin \left (a+b\,x\right )}^2-1\right )}{4}+\frac {x\,\left (2\,{\sin \left (2\,a+2\,b\,x\right )}^2-1\right )}{64}\right )-b^2\,\left (\frac {x^2\,\sin \left (2\,a+2\,b\,x\right )}{4}+\frac {x^2\,\sin \left (4\,a+4\,b\,x\right )}{32}\right )}{b^3} \] Input:
int(x^2*cos(a + b*x)^4,x)
Output:
x^3/8 - (sin(2*a + 2*b*x)/8 + sin(4*a + 4*b*x)/256 + b*((x*(2*sin(a + b*x) ^2 - 1))/4 + (x*(2*sin(2*a + 2*b*x)^2 - 1))/64) - b^2*((x^2*sin(2*a + 2*b* x))/4 + (x^2*sin(4*a + 4*b*x))/32))/b^3
Time = 0.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int x^2 \cos ^4(a+b x) \, dx=\frac {-16 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{2} x^{2}+2 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}+40 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} x^{2}-17 \cos \left (b x +a \right ) \sin \left (b x +a \right )+8 \sin \left (b x +a \right )^{4} b x -40 \sin \left (b x +a \right )^{2} b x +8 b^{3} x^{3}+17 b x}{64 b^{3}} \] Input:
int(x^2*cos(b*x+a)^4,x)
Output:
( - 16*cos(a + b*x)*sin(a + b*x)**3*b**2*x**2 + 2*cos(a + b*x)*sin(a + b*x )**3 + 40*cos(a + b*x)*sin(a + b*x)*b**2*x**2 - 17*cos(a + b*x)*sin(a + b* x) + 8*sin(a + b*x)**4*b*x - 40*sin(a + b*x)**2*b*x + 8*b**3*x**3 + 17*b*x )/(64*b**3)