Optimal. Leaf size=33 \[ \frac{\sin \left (a+b x^2\right )}{2 b}-\frac{\sin ^3\left (a+b x^2\right )}{6 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0283865, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3380, 2633} \[ \frac{\sin \left (a+b x^2\right )}{2 b}-\frac{\sin ^3\left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 2633
Rubi steps
\begin{align*} \int x \cos ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cos ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (a+b x^2\right )\right )}{2 b}\\ &=\frac{\sin \left (a+b x^2\right )}{2 b}-\frac{\sin ^3\left (a+b x^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0170718, size = 33, normalized size = 1. \[ \frac{\sin \left (a+b x^2\right )}{2 b}-\frac{\sin ^3\left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.029, size = 26, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( b{x}^{2}+a \right ) \right ) ^{2} \right ) \sin \left ( b{x}^{2}+a \right ) }{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17494, size = 36, normalized size = 1.09 \begin{align*} \frac{\sin \left (3 \, b x^{2} + 3 \, a\right ) + 9 \, \sin \left (b x^{2} + a\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56497, size = 61, normalized size = 1.85 \begin{align*} \frac{{\left (\cos \left (b x^{2} + a\right )^{2} + 2\right )} \sin \left (b x^{2} + a\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.970186, size = 44, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sin ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac{\sin{\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \cos ^{3}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1619, size = 35, normalized size = 1.06 \begin{align*} -\frac{\sin \left (b x^{2} + a\right )^{3} - 3 \, \sin \left (b x^{2} + a\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]