Optimal. Leaf size=54 \[ \frac{2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d}+\frac{2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0274606, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3362, 3296, 2638} \[ \frac{2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d}+\frac{2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3362
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \cos \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d}-\frac{2 \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d}\\ &=\frac{2 \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d}+\frac{2 \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )}{b d}\\ \end{align*}
Mathematica [A] time = 0.0734567, size = 48, normalized size = 0.89 \[ \frac{2 \left (b \sqrt{c+d x} \sin \left (a+b \sqrt{c+d x}\right )+\cos \left (a+b \sqrt{c+d x}\right )\right )}{b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 61, normalized size = 1.1 \begin{align*} 2\,{\frac{\cos \left ( a+b\sqrt{dx+c} \right ) + \left ( a+b\sqrt{dx+c} \right ) \sin \left ( a+b\sqrt{dx+c} \right ) -a\sin \left ( a+b\sqrt{dx+c} \right ) }{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15409, size = 81, normalized size = 1.5 \begin{align*} \frac{2 \,{\left ({\left (\sqrt{d x + c} b + a\right )} \sin \left (\sqrt{d x + c} b + a\right ) - a \sin \left (\sqrt{d x + c} b + a\right ) + \cos \left (\sqrt{d x + c} b + a\right )\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57493, size = 109, normalized size = 2.02 \begin{align*} \frac{2 \,{\left (\sqrt{d x + c} b \sin \left (\sqrt{d x + c} b + a\right ) + \cos \left (\sqrt{d x + c} b + a\right )\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.535975, size = 66, normalized size = 1.22 \begin{align*} \begin{cases} x \cos{\left (a \right )} & \text{for}\: b = 0 \wedge d = 0 \\x \cos{\left (a + b \sqrt{c} \right )} & \text{for}\: d = 0 \\x \cos{\left (a \right )} & \text{for}\: b = 0 \\\frac{2 \sqrt{c + d x} \sin{\left (a + b \sqrt{c + d x} \right )}}{b d} + \frac{2 \cos{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18209, size = 227, normalized size = 4.2 \begin{align*} -\frac{2 \,{\left ({\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )} \sin \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right ) - \frac{b \cos \left (-{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) + a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - a\right )}{\mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )}\right )}}{b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]