Optimal. Leaf size=24 \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0327388, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{2 (a+a \sin (c+d x))^{3/2}}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0809359, size = 44, normalized size = 1.83 \[ \frac{2 \sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 21, normalized size = 0.9 \begin{align*}{\frac{2}{3\,da} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.958411, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58152, size = 69, normalized size = 2.88 \begin{align*} \frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sin \left (d x + c\right ) + 1\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.484342, size = 58, normalized size = 2.42 \begin{align*} \begin{cases} \frac{2 \sqrt{a \sin{\left (c + d x \right )} + a} \sin{\left (c + d x \right )}}{3 d} + \frac{2 \sqrt{a \sin{\left (c + d x \right )} + a}}{3 d} & \text{for}\: d \neq 0 \\x \sqrt{a \sin{\left (c \right )} + a} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.52082, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]