Integrand size = 14, antiderivative size = 75 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\frac {2 d \cos (a+b x)}{3 b^2}+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sin (a+b x)}{3 b}+\frac {(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b} \]
2/3*d*cos(b*x+a)/b^2+1/9*d*cos(b*x+a)^3/b^2+2/3*(d*x+c)*sin(b*x+a)/b+1/3*( d*x+c)*cos(b*x+a)^2*sin(b*x+a)/b
Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.69 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\frac {27 d \cos (a+b x)+d \cos (3 (a+b x))+3 b (c+d x) (9 \sin (a+b x)+\sin (3 (a+b x)))}{36 b^2} \]
(27*d*Cos[a + b*x] + d*Cos[3*(a + b*x)] + 3*b*(c + d*x)*(9*Sin[a + b*x] + Sin[3*(a + b*x)]))/(36*b^2)
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3791, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int (c+d x) \cos ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {2}{3} \int (c+d x) \cos (a+b x)dx+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {2}{3} \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {2}{3} \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\) |
(d*Cos[a + b*x]^3)/(9*b^2) + ((c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2*((d*Cos[a + b*x])/b^2 + ((c + d*x)*Sin[a + b*x])/b))/3
3.1.19.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
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]
Time = 1.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(\frac {3 b \left (d x +c \right ) \sin \left (3 b x +3 a \right )+d \cos \left (3 b x +3 a \right )+27 \left (d x +c \right ) b \sin \left (b x +a \right )+27 d \cos \left (b x +a \right )+28 d}{36 b^{2}}\) | \(61\) |
risch | \(\frac {3 d \cos \left (b x +a \right )}{4 b^{2}}+\frac {3 \left (d x +c \right ) \sin \left (b x +a \right )}{4 b}+\frac {d \cos \left (3 b x +3 a \right )}{36 b^{2}}+\frac {\left (d x +c \right ) \sin \left (3 b x +3 a \right )}{12 b}\) | \(64\) |
derivativedivides | \(\frac {-\frac {d a \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}+\frac {c \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cos \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
default | \(\frac {-\frac {d a \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}+\frac {c \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cos \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
norman | \(\frac {\frac {2 d \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {14 d}{9 b^{2}}+\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {4 c \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 c \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {8 d \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b^{2}}+\frac {2 d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {4 d x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 d x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\) | \(159\) |
1/36*(3*b*(d*x+c)*sin(3*b*x+3*a)+d*cos(3*b*x+3*a)+27*(d*x+c)*b*sin(b*x+a)+ 27*d*cos(b*x+a)+28*d)/b^2
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\frac {d \cos \left (b x + a\right )^{3} + 6 \, d \cos \left (b x + a\right ) + 3 \, {\left (2 \, b d x + {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + 2 \, b c\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \]
1/9*(d*cos(b*x + a)^3 + 6*d*cos(b*x + a) + 3*(2*b*d*x + (b*d*x + b*c)*cos( b*x + a)^2 + 2*b*c)*sin(b*x + a))/b^2
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.68 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\begin {cases} \frac {2 c \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {c \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {d x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {7 d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((2*c*sin(a + b*x)**3/(3*b) + c*sin(a + b*x)*cos(a + b*x)**2/b + 2*d*x*sin(a + b*x)**3/(3*b) + d*x*sin(a + b*x)*cos(a + b*x)**2/b + 2*d*sin (a + b*x)**2*cos(a + b*x)/(3*b**2) + 7*d*cos(a + b*x)**3/(9*b**2), Ne(b, 0 )), ((c*x + d*x**2/2)*cos(a)**3, True))
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.37 \[ \int (c+d x) \cos ^3(a+b x) \, dx=-\frac {12 \, {\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} c - \frac {12 \, {\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} a d}{b} - \frac {{\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 27 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) + 27 \, \cos \left (b x + a\right )\right )} d}{b}}{36 \, b} \]
-1/36*(12*(sin(b*x + a)^3 - 3*sin(b*x + a))*c - 12*(sin(b*x + a)^3 - 3*sin (b*x + a))*a*d/b - (3*(b*x + a)*sin(3*b*x + 3*a) + 27*(b*x + a)*sin(b*x + a) + cos(3*b*x + 3*a) + 27*cos(b*x + a))*d/b)/b
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\frac {d \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac {3 \, d \cos \left (b x + a\right )}{4 \, b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} + \frac {3 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )}{4 \, b^{2}} \]
1/36*d*cos(3*b*x + 3*a)/b^2 + 3/4*d*cos(b*x + a)/b^2 + 1/12*(b*d*x + b*c)* sin(3*b*x + 3*a)/b^2 + 3/4*(b*d*x + b*c)*sin(b*x + a)/b^2
Time = 14.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int (c+d x) \cos ^3(a+b x) \, dx=\frac {\frac {3\,c\,\sin \left (a+b\,x\right )}{4}+\frac {c\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {d\,x\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,d\,x\,\sin \left (a+b\,x\right )}{4}}{b}+\frac {d\,\cos \left (3\,a+3\,b\,x\right )}{36\,b^2}+\frac {3\,d\,\cos \left (a+b\,x\right )}{4\,b^2} \]