Optimal. Leaf size=54 \[ \frac{2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d}-\frac{2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.0277695, 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 = {3361, 3296, 2637} \[ \frac{2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d}-\frac{2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 3361
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \sin \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=-\frac{2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d}+\frac{2 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d}\\ &=-\frac{2 \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b d}+\frac{2 \sin \left (a+b \sqrt{c+d x}\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.0738688, size = 50, normalized size = 0.93 \[ \frac{2 \sin \left (a+b \sqrt{c+d x}\right )-2 b \sqrt{c+d x} \cos \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 61, normalized size = 1.1 \begin{align*} 2\,{\frac{\sin \left ( a+b\sqrt{dx+c} \right ) - \left ( a+b\sqrt{dx+c} \right ) \cos \left ( a+b\sqrt{dx+c} \right ) +a\cos \left ( a+b\sqrt{dx+c} \right ) }{d{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950191, size = 84, normalized size = 1.56 \begin{align*} -\frac{2 \,{\left ({\left (\sqrt{d x + c} b + a\right )} \cos \left (\sqrt{d x + c} b + a\right ) - a \cos \left (\sqrt{d x + c} b + a\right ) - \sin \left (\sqrt{d x + c} b + a\right )\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64205, size = 111, normalized size = 2.06 \begin{align*} -\frac{2 \,{\left (\sqrt{d x + c} b \cos \left (\sqrt{d x + c} b + a\right ) - \sin \left (\sqrt{d x + c} b + a\right )\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.4912, size = 66, normalized size = 1.22 \begin{align*} \begin{cases} x \sin{\left (a \right )} & \text{for}\: b = 0 \wedge d = 0 \\x \sin{\left (a + b \sqrt{c} \right )} & \text{for}\: d = 0 \\x \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{2 \sqrt{c + d x} \cos{\left (a + b \sqrt{c + d x} \right )}}{b d} + \frac{2 \sin{\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.
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Giac [B] time = 1.20166, size = 225, normalized size = 4.17 \begin{align*} -\frac{2 \,{\left ({\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )} \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 ) + \frac{b \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 )}{\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.
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