Optimal. Leaf size=42 \[ \frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0586445, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3310} \[ \frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3310
Rubi steps
\begin{align*} \int \left (-\frac{x}{3 \sqrt{\cos (a+b x)}}+x \cos ^{\frac{3}{2}}(a+b x)\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\cos (a+b x)}} \, dx\right )+\int x \cos ^{\frac{3}{2}}(a+b x) \, dx\\ &=\frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sqrt{\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.4149, size = 40, normalized size = 0.95 \[ \frac{\sqrt{\cos (a+b x)} \left (4 x \sin (a+b x)+\frac{8 \cos (a+b x)}{3 b}\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.275, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cos \left ( bx+a \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}} - \frac{x}{3 \, \sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}} - \frac{x}{3 \, \sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]