Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0067261, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2637} \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \, dx &=\frac{\sqrt{b \cos (c+d x)} \int \cos (c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{\sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0281434, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.269, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d}\sqrt{b\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.7803, size = 18, normalized size = 0.56 \begin{align*} \frac{\sqrt{b} \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61199, size = 78, normalized size = 2.44 \begin{align*} \frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \sqrt{\cos \left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 20.8331, size = 29, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\sqrt{b} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sqrt{b \cos{\left (c \right )}} \sqrt{\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 = 3.37655, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \, \sqrt{b} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{d \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]