Optimal. Leaf size=27 \[ \frac {\cos ^3(a+b x)}{3 b}-\frac {\cos (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = 0.01, 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]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 22, normalized size = 0.81 \[ \frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.19, size = 25, normalized size = 0.93 \[ \frac {\cos \left (b x + a\right )^{3}}{3 \, b} - \frac {\cos \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 22, normalized size = 0.81 \[ -\frac {\left (\sin ^{2}\left (b x +a \right )+2\right ) \cos \left (b x +a \right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 22, normalized size = 0.81 \[ \frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 24, normalized size = 0.89 \[ -\frac {3\,\cos \left (a+b\,x\right )-{\cos \left (a+b\,x\right )}^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.47, size = 37, normalized size = 1.37 \[ \begin {cases} - \frac {\sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 \cos ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________