Integrand size = 17, antiderivative size = 90 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {3 x}{128}+\frac {3 \cos (a+b x) \sin (a+b x)}{128 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac {\cos ^5(a+b x) \sin ^3(a+b x)}{8 b} \] Output:
3/128*x+3/128*cos(b*x+a)*sin(b*x+a)/b+1/64*cos(b*x+a)^3*sin(b*x+a)/b-1/16* cos(b*x+a)^5*sin(b*x+a)/b-1/8*cos(b*x+a)^5*sin(b*x+a)^3/b
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.37 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {24 (a+b x)-8 \sin (4 (a+b x))+\sin (8 (a+b x))}{1024 b} \] Input:
Integrate[Cos[a + b*x]^4*Sin[a + b*x]^4,x]
Output:
(24*(a + b*x) - 8*Sin[4*(a + b*x)] + Sin[8*(a + b*x)])/(1024*b)
Time = 0.75 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 3048, 3042, 3048, 3042, 3115, 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.
\(\displaystyle \int \sin ^4(a+b x) \cos ^4(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^4 \cos (a+b x)^4dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{8} \int \cos ^4(a+b x) \sin ^2(a+b x)dx-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \int \cos (a+b x)^4 \sin (a+b x)^2dx-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \int \cos ^4(a+b x)dx-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}\) |
Input:
Int[Cos[a + b*x]^4*Sin[a + b*x]^4,x]
Output:
-1/8*(Cos[a + b*x]^5*Sin[a + b*x]^3)/b + (3*(-1/6*(Cos[a + b*x]^5*Sin[a + b*x])/b + ((Cos[a + b*x]^3*Sin[a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*S in[a + b*x])/(2*b)))/4)/6))/8
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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]
Time = 7.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.34
method | result | size |
parallelrisch | \(\frac {24 b x +\sin \left (8 b x +8 a \right )-8 \sin \left (4 b x +4 a \right )}{1024 b}\) | \(31\) |
risch | \(\frac {3 x}{128}+\frac {\sin \left (8 b x +8 a \right )}{1024 b}-\frac {\sin \left (4 b x +4 a \right )}{128 b}\) | \(33\) |
derivativedivides | \(\frac {-\frac {\cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3}}{8}-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{5}}{16}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{64}+\frac {3 b x}{128}+\frac {3 a}{128}}{b}\) | \(72\) |
default | \(\frac {-\frac {\cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3}}{8}-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{5}}{16}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{64}+\frac {3 b x}{128}+\frac {3 a}{128}}{b}\) | \(72\) |
norman | \(\frac {\frac {3 x}{128}-\frac {3 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}-\frac {23 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{64 b}+\frac {333 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{64 b}-\frac {671 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{64 b}+\frac {671 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}}{64 b}-\frac {333 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{11}}{64 b}+\frac {23 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{13}}{64 b}+\frac {3 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{15}}{64 b}+\frac {3 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{16}+\frac {21 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{32}+\frac {21 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{16}+\frac {105 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{64}+\frac {21 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{16}+\frac {21 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{12}}{32}+\frac {3 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{14}}{16}+\frac {3 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{16}}{128}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{8}}\) | \(259\) |
orering | \(x \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4}-\frac {5 \left (-4 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{5} b +4 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b \right )}{64 b^{2}}+\frac {5 x \left (12 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{2}-40 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{2}+12 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{2}\right )}{64 b^{2}}-\frac {-24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7} b^{3}+232 \sin \left (b x +a \right )^{5} b^{3} \cos \left (b x +a \right )^{3}-232 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right )^{3} b^{3}+24 \cos \left (b x +a \right )^{7} \sin \left (b x +a \right ) b^{3}}{1024 b^{4}}+\frac {x \left (24 \sin \left (b x +a \right )^{8} b^{4}-864 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{6} b^{4}+2320 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{4} b^{4}-864 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{2} b^{4}+24 \cos \left (b x +a \right )^{8} b^{4}\right )}{1024 b^{4}}\) | \(316\) |
Input:
int(cos(b*x+a)^4*sin(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/1024*(24*b*x+sin(8*b*x+8*a)-8*sin(4*b*x+4*a))/b
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {3 \, b x + {\left (16 \, \cos \left (b x + a\right )^{7} - 24 \, \cos \left (b x + a\right )^{5} + 2 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^4,x, algorithm="fricas")
Output:
1/128*(3*b*x + (16*cos(b*x + a)^7 - 24*cos(b*x + a)^5 + 2*cos(b*x + a)^3 + 3*cos(b*x + a))*sin(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (80) = 160\).
Time = 0.65 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.10 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {3 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {3 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {9 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {3 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac {3 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} + \frac {11 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} - \frac {11 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{128 b} - \frac {3 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**4*sin(b*x+a)**4,x)
Output:
Piecewise((3*x*sin(a + b*x)**8/128 + 3*x*sin(a + b*x)**6*cos(a + b*x)**2/3 2 + 9*x*sin(a + b*x)**4*cos(a + b*x)**4/64 + 3*x*sin(a + b*x)**2*cos(a + b *x)**6/32 + 3*x*cos(a + b*x)**8/128 + 3*sin(a + b*x)**7*cos(a + b*x)/(128* b) + 11*sin(a + b*x)**5*cos(a + b*x)**3/(128*b) - 11*sin(a + b*x)**3*cos(a + b*x)**5/(128*b) - 3*sin(a + b*x)*cos(a + b*x)**7/(128*b), Ne(b, 0)), (x *sin(a)**4*cos(a)**4, True))
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.37 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {24 \, b x + 24 \, a + \sin \left (8 \, b x + 8 \, a\right ) - 8 \, \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^4,x, algorithm="maxima")
Output:
1/1024*(24*b*x + 24*a + sin(8*b*x + 8*a) - 8*sin(4*b*x + 4*a))/b
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.36 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {3}{128} \, x + \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^4,x, algorithm="giac")
Output:
3/128*x + 1/1024*sin(8*b*x + 8*a)/b - 1/128*sin(4*b*x + 4*a)/b
Time = 26.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {3\,x}{128}-\frac {-\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}-\frac {11\,{\mathrm {tan}\left (a+b\,x\right )}^5}{128}+\frac {11\,{\mathrm {tan}\left (a+b\,x\right )}^3}{128}+\frac {3\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \] Input:
int(cos(a + b*x)^4*sin(a + b*x)^4,x)
Output:
(3*x)/128 - ((3*tan(a + b*x))/128 + (11*tan(a + b*x)^3)/128 - (11*tan(a + b*x)^5)/128 - (3*tan(a + b*x)^7)/128)/(b*(4*tan(a + b*x)^2 + 6*tan(a + b*x )^4 + 4*tan(a + b*x)^6 + tan(a + b*x)^8 + 1))
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80 \[ \int \cos ^4(a+b x) \sin ^4(a+b x) \, dx=\frac {-16 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7}+24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5}-2 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}-3 \cos \left (b x +a \right ) \sin \left (b x +a \right )+3 b x}{128 b} \] Input:
int(cos(b*x+a)^4*sin(b*x+a)^4,x)
Output:
( - 16*cos(a + b*x)*sin(a + b*x)**7 + 24*cos(a + b*x)*sin(a + b*x)**5 - 2* cos(a + b*x)*sin(a + b*x)**3 - 3*cos(a + b*x)*sin(a + b*x) + 3*b*x)/(128*b )