Integrand size = 14, antiderivative size = 75 \[ \int (c+d x) \sin ^3(a+b x) \, dx=-\frac {2 (c+d x) \cos (a+b x)}{3 b}+\frac {2 d \sin (a+b x)}{3 b^2}-\frac {(c+d x) \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sin ^3(a+b x)}{9 b^2} \]
-2/3*(d*x+c)*cos(b*x+a)/b+2/3*d*sin(b*x+a)/b^2-1/3*(d*x+c)*cos(b*x+a)*sin( b*x+a)^2/b+1/9*d*sin(b*x+a)^3/b^2
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\frac {-27 b (c+d x) \cos (a+b x)+3 b (c+d x) \cos (3 (a+b x))+d (27 \sin (a+b x)-\sin (3 (a+b x)))}{36 b^2} \]
(-27*b*(c + d*x)*Cos[a + b*x] + 3*b*(c + d*x)*Cos[3*(a + b*x)] + d*(27*Sin [a + b*x] - Sin[3*(a + b*x)]))/(36*b^2)
Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3791, 3042, 3777, 3042, 3117}
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.
\(\displaystyle \int (c+d x) \sin ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x) \sin (a+b x)^3dx\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {2}{3} \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {2}{3} \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
-1/3*((c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/b + (d*Sin[a + b*x]^3)/(9*b^2 ) + (2*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2))/3
3.1.19.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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 = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\frac {3 b \left (d x +c \right ) \cos \left (3 b x +3 a \right )-d \sin \left (3 b x +3 a \right )-27 \left (d x +c \right ) b \cos \left (b x +a \right )-24 c b +27 d \sin \left (b x +a \right )}{36 b^{2}}\) | \(63\) |
risch | \(-\frac {3 \left (d x +c \right ) \cos \left (b x +a \right )}{4 b}+\frac {3 d \sin \left (b x +a \right )}{4 b^{2}}+\frac {\left (d x +c \right ) \cos \left (3 b x +3 a \right )}{12 b}-\frac {d \sin \left (3 b x +3 a \right )}{36 b^{2}}\) | \(64\) |
derivativedivides | \(\frac {\frac {d a \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}-\frac {c \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}+\frac {d \left (-\frac {\left (b x +a \right ) \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}+\frac {\left (\sin ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
default | \(\frac {\frac {d a \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}-\frac {c \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}+\frac {d \left (-\frac {\left (b x +a \right ) \left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3}+\frac {\left (\sin ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
norman | \(\frac {-\frac {4 c \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {4 c}{3 b}+\frac {4 d \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{3 b^{2}}+\frac {32 d \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{9 b^{2}}+\frac {4 d \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b^{2}}-\frac {2 d x}{3 b}-\frac {2 d x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {2 d x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {2 d x \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\) | \(151\) |
1/36*(3*b*(d*x+c)*cos(3*b*x+3*a)-d*sin(3*b*x+3*a)-27*(d*x+c)*b*cos(b*x+a)- 24*c*b+27*d*sin(b*x+a))/b^2
Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\frac {3 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 9 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) - {\left (d \cos \left (b x + a\right )^{2} - 7 \, d\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \]
1/9*(3*(b*d*x + b*c)*cos(b*x + a)^3 - 9*(b*d*x + b*c)*cos(b*x + a) - (d*co s(b*x + a)^2 - 7*d)*sin(b*x + a))/b^2
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.68 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {d x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 d x \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {7 d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {2 d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-c*sin(a + b*x)**2*cos(a + b*x)/b - 2*c*cos(a + b*x)**3/(3*b) - d*x*sin(a + b*x)**2*cos(a + b*x)/b - 2*d*x*cos(a + b*x)**3/(3*b) + 7*d*si n(a + b*x)**3/(9*b**2) + 2*d*sin(a + b*x)*cos(a + b*x)**2/(3*b**2), Ne(b, 0)), ((c*x + d*x**2/2)*sin(a)**3, True))
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.39 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\frac {12 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} c - \frac {12 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} a d}{b} + \frac {{\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 27 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) + 27 \, \sin \left (b x + a\right )\right )} d}{b}}{36 \, b} \]
1/36*(12*(cos(b*x + a)^3 - 3*cos(b*x + a))*c - 12*(cos(b*x + a)^3 - 3*cos( b*x + a))*a*d/b + (3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a ) - sin(3*b*x + 3*a) + 27*sin(b*x + a))*d/b)/b
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\frac {{\left (b d x + b c\right )} \cos \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} - \frac {3 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )}{4 \, b^{2}} - \frac {d \sin \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac {3 \, d \sin \left (b x + a\right )}{4 \, b^{2}} \]
1/12*(b*d*x + b*c)*cos(3*b*x + 3*a)/b^2 - 3/4*(b*d*x + b*c)*cos(b*x + a)/b ^2 - 1/36*d*sin(3*b*x + 3*a)/b^2 + 3/4*d*sin(b*x + a)/b^2
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05 \[ \int (c+d x) \sin ^3(a+b x) \, dx=\frac {7\,d\,\sin \left (a+b\,x\right )}{9\,b^2}-\frac {c\,\cos \left (a+b\,x\right )-\frac {c\,{\cos \left (a+b\,x\right )}^3}{3}+d\,x\,\cos \left (a+b\,x\right )-\frac {d\,x\,{\cos \left (a+b\,x\right )}^3}{3}}{b}-\frac {d\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{9\,b^2} \]