Integrand size = 22, antiderivative size = 53 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^2}{16 d}-\frac {d \cos (4 a+4 b x)}{128 b^2}-\frac {(c+d x) \sin (4 a+4 b x)}{32 b} \]
Time = 0.60 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {8 (a+b x) (-2 b c+a d-b d x)+d \cos (4 (a+b x))+4 b (c+d x) \sin (4 (a+b x))}{128 b^2} \]
-1/128*(8*(a + b*x)*(-2*b*c + a*d - b*d*x) + d*Cos[4*(a + b*x)] + 4*b*(c + d*x)*Sin[4*(a + b*x)])/b^2
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4906, 2009}
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.
\(\displaystyle \int (c+d x) \sin ^2(a+b x) \cos ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \int \left (\frac {1}{8} (c+d x)-\frac {1}{8} (c+d x) \cos (4 a+4 b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \cos (4 a+4 b x)}{128 b^2}-\frac {(c+d x) \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^2}{16 d}\) |
3.1.83.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Time = 0.97 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {d \,x^{2}}{16}+\frac {c x}{8}-\frac {d \cos \left (4 x b +4 a \right )}{128 b^{2}}-\frac {\left (d x +c \right ) \sin \left (4 x b +4 a \right )}{32 b}\) | \(46\) |
derivativedivides | \(\frac {-\frac {d a \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}+c \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {d \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}}{b}\) | \(200\) |
default | \(\frac {-\frac {d a \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )}{b}+c \left (-\frac {\cos \left (x b +a \right )^{3} \sin \left (x b +a \right )}{4}+\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{8}+\frac {x b}{8}+\frac {a}{8}\right )+\frac {d \left (\left (x b +a \right ) \left (-\frac {\cos \left (x b +a \right ) \sin \left (x b +a \right )}{2}+\frac {x b}{2}+\frac {a}{2}\right )-\frac {\left (x b +a \right )^{2}}{16}+\frac {\sin \left (x b +a \right )^{2}}{4}-\left (x b +a \right ) \left (-\frac {\left (\sin \left (x b +a \right )^{3}+\frac {3 \sin \left (x b +a \right )}{2}\right ) \cos \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )-\frac {\left (2 \cos \left (x b +a \right )^{2}-5\right )^{2}}{64}\right )}{b}}{b}\) | \(200\) |
norman | \(\frac {\frac {c x}{8}+\frac {d \,x^{2}}{16}-\frac {c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b}+\frac {7 c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b}-\frac {7 c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{4 b}+\frac {c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{4 b}+\frac {c x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2}+\frac {3 c x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{4}+\frac {c x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{2}+\frac {c x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{8}+\frac {d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{4}+\frac {3 d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8}+\frac {d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{4}+\frac {d \,x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{16}-\frac {d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{2 b^{2}}+\frac {d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{4 b^{2}}+\frac {d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{4 b^{2}}-\frac {d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b}+\frac {7 d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b}-\frac {7 d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{4 b}+\frac {d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{7}}{4 b}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{4}}\) | \(343\) |
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.60 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {b^{2} d x^{2} - d \cos \left (b x + a\right )^{4} + 2 \, b^{2} c x + d \cos \left (b x + a\right )^{2} - 2 \, {\left (2 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - {\left (b d x + b c\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{16 \, b^{2}} \]
1/16*(b^2*d*x^2 - d*cos(b*x + a)^4 + 2*b^2*c*x + d*cos(b*x + a)^2 - 2*(2*( b*d*x + b*c)*cos(b*x + a)^3 - (b*d*x + b*c)*cos(b*x + a))*sin(b*x + a))/b^ 2
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (46) = 92\).
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.49 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {c x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {d x^{2} \sin ^{4}{\left (a + b x \right )}}{16} + \frac {d x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8} + \frac {d x^{2} \cos ^{4}{\left (a + b x \right )}}{16} + \frac {c \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {c \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} + \frac {d x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {d x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac {d \sin ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac {d \cos ^{4}{\left (a + b x \right )}}{32 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin ^{2}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c*x*sin(a + b*x)**4/8 + c*x*sin(a + b*x)**2*cos(a + b*x)**2/4 + c*x*cos(a + b*x)**4/8 + d*x**2*sin(a + b*x)**4/16 + d*x**2*sin(a + b*x)** 2*cos(a + b*x)**2/8 + d*x**2*cos(a + b*x)**4/16 + c*sin(a + b*x)**3*cos(a + b*x)/(8*b) - c*sin(a + b*x)*cos(a + b*x)**3/(8*b) + d*x*sin(a + b*x)**3* cos(a + b*x)/(8*b) - d*x*sin(a + b*x)*cos(a + b*x)**3/(8*b) - d*sin(a + b* x)**4/(32*b**2) - d*cos(a + b*x)**4/(32*b**2), Ne(b, 0)), ((c*x + d*x**2/2 )*sin(a)**2*cos(a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (47) = 94\).
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.81 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {4 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c - \frac {4 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a d}{b} + \frac {{\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} d}{b}}{128 \, b} \]
1/128*(4*(4*b*x + 4*a - sin(4*b*x + 4*a))*c - 4*(4*b*x + 4*a - sin(4*b*x + 4*a))*a*d/b + (8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*d/b)/b
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{16} \, d x^{2} + \frac {1}{8} \, c x - \frac {d \cos \left (4 \, b x + 4 \, a\right )}{128 \, b^{2}} - \frac {{\left (b d x + b c\right )} \sin \left (4 \, b x + 4 \, a\right )}{32 \, b^{2}} \]
1/16*d*x^2 + 1/8*c*x - 1/128*d*cos(4*b*x + 4*a)/b^2 - 1/32*(b*d*x + b*c)*s in(4*b*x + 4*a)/b^2
Time = 23.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08 \[ \int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {c\,x}{8}+\frac {d\,x^2}{16}-\frac {d\,\cos \left (4\,a+4\,b\,x\right )}{128\,b^2}-\frac {c\,\sin \left (4\,a+4\,b\,x\right )}{32\,b}-\frac {d\,x\,\sin \left (4\,a+4\,b\,x\right )}{32\,b} \]