Optimal. Leaf size=30 \[ \frac{\sin (a+b x)}{2 b}-\frac{\sin (3 a+3 b x)}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.011129, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4282} \[ \frac{\sin (a+b x)}{2 b}-\frac{\sin (3 a+3 b x)}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4282
Rubi steps
\begin{align*} \int \sin (a+b x) \sin (2 a+2 b x) \, dx &=\frac{\sin (a+b x)}{2 b}-\frac{\sin (3 a+3 b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0319888, size = 15, normalized size = 0.5 \[ \frac{2 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 27, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( bx+a \right ) }{2\,b}}-{\frac{\sin \left ( 3\,bx+3\,a \right ) }{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09884, size = 35, normalized size = 1.17 \begin{align*} -\frac{\sin \left (3 \, b x + 3 \, a\right )}{6 \, b} + \frac{\sin \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.47103, size = 57, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.12469, size = 51, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{2 \sin{\left (a + b x \right )} \cos{\left (2 a + 2 b x \right )}}{3 b} + \frac{\sin{\left (2 a + 2 b x \right )} \cos{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} \sin{\left (2 a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30566, size = 35, normalized size = 1.17 \begin{align*} -\frac{\sin \left (3 \, b x + 3 \, a\right )}{6 \, b} + \frac{\sin \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]