Optimal. Leaf size=15 \[ -\frac{\cos ^3(a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0206154, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2565, 30} \[ -\frac{\cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0045087, size = 15, normalized size = 1. \[ -\frac{\cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 14, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.993427, size = 18, normalized size = 1.2 \begin{align*} -\frac{\cos \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80328, size = 31, normalized size = 2.07 \begin{align*} -\frac{\cos \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.505304, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} - \frac{\cos ^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} \cos ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12636, size = 18, normalized size = 1.2 \begin{align*} -\frac{\cos \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]