Integrand size = 17, antiderivative size = 67 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {x}{16}+\frac {\cos (a+b x) \sin (a+b x)}{16 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{24 b}-\frac {\cos ^5(a+b x) \sin (a+b x)}{6 b} \] Output:
1/16*x+1/16*cos(b*x+a)*sin(b*x+a)/b+1/24*cos(b*x+a)^3*sin(b*x+a)/b-1/6*cos (b*x+a)^5*sin(b*x+a)/b
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=-\frac {-12 b x-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))}{192 b} \] Input:
Integrate[Cos[a + b*x]^4*Sin[a + b*x]^2,x]
Output:
-1/192*(-12*b*x - 3*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x )])/b
Time = 0.57 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {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.
\(\displaystyle \int \sin ^2(a+b x) \cos ^4(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^2 \cos (a+b x)^4dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {1}{6} \int \cos ^4(a+b x)dx-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )-\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\) |
Input:
Int[Cos[a + b*x]^4*Sin[a + b*x]^2,x]
Output:
-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]*Sin[a + b*x])/(2*b)))/4)/6
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 = 10.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(\frac {12 b x -\sin \left (6 b x +6 a \right )-3 \sin \left (4 b x +4 a \right )+3 \sin \left (2 b x +2 a \right )}{192 b}\) | \(44\) |
risch | \(\frac {x}{16}-\frac {\sin \left (6 b x +6 a \right )}{192 b}-\frac {\sin \left (4 b x +4 a \right )}{64 b}+\frac {\sin \left (2 b x +2 a \right )}{64 b}\) | \(47\) |
derivativedivides | \(\frac {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{5}}{6}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{24}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(54\) |
default | \(\frac {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{5}}{6}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{24}+\frac {b x}{16}+\frac {a}{16}}{b}\) | \(54\) |
norman | \(\frac {\frac {x}{16}-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {47 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{24 b}-\frac {13 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{4 b}+\frac {13 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}}{4 b}-\frac {47 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}}{24 b}+\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{11}}{8 b}+\frac {3 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{8}+\frac {15 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{16}+\frac {5 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{4}+\frac {15 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{16}+\frac {3 x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{8}+\frac {x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{12}}{16}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{6}}\) | \(199\) |
orering | \(x \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{2}-\frac {49 \left (-4 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3} b +2 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right ) b \right )}{144 b^{2}}+\frac {49 x \left (12 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4} b^{2}-22 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{2} b^{2}+2 \cos \left (b x +a \right )^{6} b^{2}\right )}{144 b^{2}}-\frac {7 \left (-24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5} b^{3}+136 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3} b^{3}-56 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right ) b^{3}\right )}{288 b^{4}}+\frac {7 x \left (24 \sin \left (b x +a \right )^{6} b^{4}-528 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4} b^{4}+688 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{2} b^{4}-56 \cos \left (b x +a \right )^{6} b^{4}\right )}{288 b^{4}}-\frac {1200 \sin \left (b x +a \right )^{5} b^{5} \cos \left (b x +a \right )-4864 \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3} b^{5}+1712 \cos \left (b x +a \right )^{5} \sin \left (b x +a \right ) b^{5}}{2304 b^{6}}+\frac {x \left (20592 \sin \left (b x +a \right )^{4} b^{6} \cos \left (b x +a \right )^{2}-1200 \sin \left (b x +a \right )^{6} b^{6}-23152 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{2} b^{6}+1712 \cos \left (b x +a \right )^{6} b^{6}\right )}{2304 b^{6}}\) | \(404\) |
Input:
int(cos(b*x+a)^4*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/192*(12*b*x-sin(6*b*x+6*a)-3*sin(4*b*x+4*a)+3*sin(2*b*x+2*a))/b
Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {3 \, b x - {\left (8 \, \cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{48 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/48*(3*b*x - (8*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. 136 vs. \(2 (56) = 112\).
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.03 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {x \sin ^{6}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {x \cos ^{6}{\left (a + b x \right )}}{16} + \frac {\sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b} + \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{6 b} - \frac {\sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**4*sin(b*x+a)**2,x)
Output:
Piecewise((x*sin(a + b*x)**6/16 + 3*x*sin(a + b*x)**4*cos(a + b*x)**2/16 + 3*x*sin(a + b*x)**2*cos(a + b*x)**4/16 + x*cos(a + b*x)**6/16 + sin(a + b *x)**5*cos(a + b*x)/(16*b) + sin(a + b*x)**3*cos(a + b*x)**3/(6*b) - sin(a + b*x)*cos(a + b*x)**5/(16*b), Ne(b, 0)), (x*sin(a)**2*cos(a)**4, True))
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.55 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {4 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 12 \, b x + 12 \, a - 3 \, \sin \left (4 \, b x + 4 \, a\right )}{192 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/192*(4*sin(2*b*x + 2*a)^3 + 12*b*x + 12*a - 3*sin(4*b*x + 4*a))/b
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{16} \, x - \frac {\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{64 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \] Input:
integrate(cos(b*x+a)^4*sin(b*x+a)^2,x, algorithm="giac")
Output:
1/16*x - 1/192*sin(6*b*x + 6*a)/b - 1/64*sin(4*b*x + 4*a)/b + 1/64*sin(2*b *x + 2*a)/b
Time = 25.85 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {x}{16}-\frac {\frac {\sin \left (4\,a+4\,b\,x\right )}{64}-\frac {\sin \left (2\,a+2\,b\,x\right )}{64}+\frac {\sin \left (6\,a+6\,b\,x\right )}{192}}{b} \] Input:
int(cos(a + b*x)^4*sin(a + b*x)^2,x)
Output:
x/16 - (sin(4*a + 4*b*x)/64 - sin(2*a + 2*b*x)/64 + sin(6*a + 6*b*x)/192)/ b
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx=\frac {-8 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5}+14 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}-3 \cos \left (b x +a \right ) \sin \left (b x +a \right )+3 b x}{48 b} \] Input:
int(cos(b*x+a)^4*sin(b*x+a)^2,x)
Output:
( - 8*cos(a + b*x)*sin(a + b*x)**5 + 14*cos(a + b*x)*sin(a + b*x)**3 - 3*c os(a + b*x)*sin(a + b*x) + 3*b*x)/(48*b)