Integrand size = 14, antiderivative size = 49 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {2 d^2 \sin (a+b x)}{b^3}+\frac {(c+d x)^2 \sin (a+b x)}{b} \]
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {2 b d (c+d x) \cos (a+b x)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a+b x)}{b^3} \]
Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3777, 25, 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.
\(\displaystyle \int (c+d x)^2 \cos (a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {2 d \int -((c+d x) \sin (a+b x))dx}{b}+\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\) |
((c + d*x)^2*Sin[a + b*x])/b - (2*d*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Si n[a + b*x])/b^2))/b
3.1.3.3.1 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]
Time = 0.90 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22
method | result | size |
risch | \(\frac {2 d \left (d x +c \right ) \cos \left (b x +a \right )}{b^{2}}+\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (b x +a \right )}{b^{3}}\) | \(60\) |
parallelrisch | \(\frac {-2 d^{2} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) x b +\left (2 \left (d x +c \right )^{2} b^{2}-4 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+4 b d \left (\frac {d x}{2}+c \right )}{b^{3} \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\) | \(77\) |
parts | \(\frac {\sin \left (b x +a \right ) x^{2} d^{2}}{b}+\frac {2 \sin \left (b x +a \right ) c d x}{b}+\frac {\sin \left (b x +a \right ) c^{2}}{b}-\frac {2 d \left (\frac {d a \cos \left (b x +a \right )}{b}-c \cos \left (b x +a \right )+\frac {d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b}\right )}{b^{2}}\) | \(98\) |
norman | \(\frac {\frac {4 c d}{b^{2}}+\frac {2 d^{2} x}{b^{2}}+\frac {2 \left (b^{2} c^{2}-2 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {2 d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {2 d^{2} x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {4 c d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(118\) |
derivativedivides | \(\frac {\frac {a^{2} d^{2} \sin \left (b x +a \right )}{b^{2}}-\frac {2 a c d \sin \left (b x +a \right )}{b}-\frac {2 a \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+c^{2} \sin \left (b x +a \right )+\frac {2 c d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(143\) |
default | \(\frac {\frac {a^{2} d^{2} \sin \left (b x +a \right )}{b^{2}}-\frac {2 a c d \sin \left (b x +a \right )}{b}-\frac {2 a \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+c^{2} \sin \left (b x +a \right )+\frac {2 c d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(143\) |
meijerg | \(\frac {4 d^{2} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {x \left (b^{2}\right )^{\frac {3}{2}} \cos \left (b x \right )}{2 \sqrt {\pi }\, b^{2}}-\frac {\left (b^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{6 \sqrt {\pi }\, b^{3}}\right )}{b^{2} \sqrt {b^{2}}}-\frac {4 d^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} b^{2}}{2}+1\right ) \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}+\frac {4 d c \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {4 d c \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c^{2} \cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {c^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(225\) |
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{b^{3}} \]
(2*(b*d^2*x + b*c*d)*cos(b*x + a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (48) = 96\).
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.29 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\begin {cases} \frac {c^{2} \sin {\left (a + b x \right )}}{b} + \frac {2 c d x \sin {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sin {\left (a + b x \right )}}{b} + \frac {2 c d \cos {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} x \cos {\left (a + b x \right )}}{b^{2}} - \frac {2 d^{2} \sin {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**2*sin(a + b*x)/b + 2*c*d*x*sin(a + b*x)/b + d**2*x**2*sin(a + b*x)/b + 2*c*d*cos(a + b*x)/b**2 + 2*d**2*x*cos(a + b*x)/b**2 - 2*d**2*s in(a + b*x)/b**3, Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*cos(a), Tr ue))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (49) = 98\).
Time = 0.33 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.78 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {c^{2} \sin \left (b x + a\right ) - \frac {2 \, a c d \sin \left (b x + a\right )}{b} + \frac {a^{2} d^{2} \sin \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} c d}{b} - \frac {2 \, {\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left (2 \, {\left (b x + a\right )} \cos \left (b x + a\right ) + {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{b} \]
(c^2*sin(b*x + a) - 2*a*c*d*sin(b*x + a)/b + a^2*d^2*sin(b*x + a)/b^2 + 2* ((b*x + a)*sin(b*x + a) + cos(b*x + a))*c*d/b - 2*((b*x + a)*sin(b*x + a) + cos(b*x + a))*a*d^2/b^2 + (2*(b*x + a)*cos(b*x + a) + ((b*x + a)^2 - 2)* sin(b*x + a))*d^2/b^2)/b
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{b^{3}} + \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{b^{3}} \]
2*(b*d^2*x + b*c*d)*cos(b*x + a)/b^3 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^ 2 - 2*d^2)*sin(b*x + a)/b^3
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.71 \[ \int (c+d x)^2 \cos (a+b x) \, dx=\frac {d^2\,x^2\,\sin \left (a+b\,x\right )}{b}-\frac {\sin \left (a+b\,x\right )\,\left (2\,d^2-b^2\,c^2\right )}{b^3}+\frac {2\,c\,d\,\cos \left (a+b\,x\right )}{b^2}+\frac {2\,d^2\,x\,\cos \left (a+b\,x\right )}{b^2}+\frac {2\,c\,d\,x\,\sin \left (a+b\,x\right )}{b} \]