Optimal. Leaf size=27 \[ \frac{\cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0100385, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2633
Rubi steps
\begin{align*} \int \sin ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos (a+b x)}{b}+\frac{\cos ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0096789, size = 29, normalized size = 1.07 \[ \frac{\cos (3 (a+b x))}{12 b}-\frac{3 \cos (a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 22, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.952396, size = 30, normalized size = 1.11 \begin{align*} \frac{\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52438, size = 55, normalized size = 2.04 \begin{align*} \frac{\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.495649, size = 37, normalized size = 1.37 \begin{align*} \begin{cases} - \frac{\sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 \cos ^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10015, size = 34, normalized size = 1.26 \begin{align*} \frac{\cos \left (b x + a\right )^{3}}{3 \, b} - \frac{\cos \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]