Optimal. Leaf size=27 \[ \frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 2638} \[ \frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3296
Rubi steps
\begin {align*} \int (c+d x) \cos (a+b x) \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 26, normalized size = 0.96 \[ \frac {b (c+d x) \sin (a+b x)+d \cos (a+b x)}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 28, normalized size = 1.04 \[ \frac {d \cos \left (b x + a\right ) + {\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.43, size = 30, normalized size = 1.11 \[ \frac {d \cos \left (b x + a\right )}{b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 51, normalized size = 1.89 \[ \frac {\frac {d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}-\frac {d a \sin \left (b x +a \right )}{b}+c \sin \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 50, normalized size = 1.85 \[ \frac {c \sin \left (b x + a\right ) - \frac {a d \sin \left (b x + a\right )}{b} + \frac {{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} d}{b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 34, normalized size = 1.26 \[ \frac {c\,\sin \left (a+b\,x\right )+d\,x\,\sin \left (a+b\,x\right )}{b}+\frac {d\,\cos \left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.20, size = 46, normalized size = 1.70 \[ \begin {cases} \frac {c \sin {\left (a + b x \right )}}{b} + \frac {d x \sin {\left (a + b x \right )}}{b} + \frac {d \cos {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________