Integrand size = 12, antiderivative size = 27 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3377, 2718} \[ \int (c+d x) \cos (a+b x) \, dx=\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \]
[In]
[Out]
Rule 2718
Rule 3377
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x) \, dx}{b} \\ & = \frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {d \cos (a+b x)+b (c+d x) \sin (a+b x)}{b^2} \]
[In]
[Out]
Time = 0.68 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {d \cos \left (b x +a \right )}{b^{2}}+\frac {\left (d x +c \right ) \sin \left (b x +a \right )}{b}\) | \(28\) |
parallelrisch | \(\frac {\left (d x +c \right ) b \sin \left (b x +a \right )+d \left (\cos \left (b x +a \right )-1\right )}{b^{2}}\) | \(29\) |
parts | \(\frac {\sin \left (b x +a \right ) d x}{b}+\frac {\sin \left (b x +a \right ) c}{b}+\frac {d \cos \left (b x +a \right )}{b^{2}}\) | \(36\) |
derivativedivides | \(\frac {-\frac {d a \sin \left (b x +a \right )}{b}+c \sin \left (b x +a \right )+\frac {d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}}{b}\) | \(51\) |
default | \(\frac {-\frac {d a \sin \left (b x +a \right )}{b}+c \sin \left (b x +a \right )+\frac {d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}}{b}\) | \(51\) |
norman | \(\frac {\frac {2 d}{b^{2}}+\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {2 d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(55\) |
meijerg | \(\frac {2 d \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {2 d \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c \cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {c \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(106\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {d \cos \left (b x + a\right ) + {\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int (c+d x) \cos (a+b x) \, dx=\begin {cases} \frac {c \sin {\left (a + b x \right )}}{b} + \frac {d x \sin {\left (a + b x \right )}}{b} + \frac {d \cos {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {c \sin \left (b x + a\right ) - \frac {a d \sin \left (b x + a\right )}{b} + \frac {{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} d}{b}}{b} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {d \cos \left (b x + a\right )}{b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int (c+d x) \cos (a+b x) \, dx=\frac {c\,\sin \left (a+b\,x\right )+d\,x\,\sin \left (a+b\,x\right )}{b}+\frac {d\,\cos \left (a+b\,x\right )}{b^2} \]
[In]
[Out]