Optimal. Leaf size=15 \[ -\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (3 x+7)\right ) \]
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Rubi [A] time = 0.029909, antiderivative size = 15, 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 = {3186, 216} \[ -\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (3 x+7)\right ) \]
Antiderivative was successfully verified.
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Rule 3186
Rule 216
Rubi steps
\begin{align*} \int \frac{\sin (7+3 x)}{\sqrt{3+\sin ^2(7+3 x)}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4-x^2}} \, dx,x,\cos (7+3 x)\right )\right )\\ &=-\frac{1}{3} \sin ^{-1}\left (\frac{1}{2} \cos (7+3 x)\right )\\ \end{align*}
Mathematica [C] time = 0.0933488, size = 39, normalized size = 2.6 \[ \frac{1}{3} i \log \left (\sqrt{7-\cos (2 (3 x+7))}+i \sqrt{2} \cos (3 x+7)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.797, size = 57, normalized size = 3.8 \begin{align*} -{\frac{1}{6\,\cos \left ( 7+3\,x \right ) }\sqrt{ \left ( 3+ \left ( \sin \left ( 7+3\,x \right ) \right ) ^{2} \right ) \left ( \cos \left ( 7+3\,x \right ) \right ) ^{2}}\arcsin \left ( -1+{\frac{ \left ( \cos \left ( 7+3\,x \right ) \right ) ^{2}}{2}} \right ){\frac{1}{\sqrt{3+ \left ( \sin \left ( 7+3\,x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41911, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \cos \left (3 \, x + 7\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81811, size = 251, normalized size = 16.73 \begin{align*} \frac{1}{6} \, \arctan \left (-\frac{4 \, \cos \left (3 \, x + 7\right ) \sin \left (3 \, x + 7\right ) -{\left (\cos \left (3 \, x + 7\right )^{3} - 2 \, \cos \left (3 \, x + 7\right )\right )} \sqrt{-\cos \left (3 \, x + 7\right )^{2} + 4}}{\cos \left (3 \, x + 7\right )^{4} - 8 \, \cos \left (3 \, x + 7\right )^{2} + 4}\right ) - \frac{1}{6} \, \arctan \left (\frac{\sin \left (3 \, x + 7\right )}{\cos \left (3 \, x + 7\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.30528, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \, \arcsin \left (\frac{1}{2} \, \cos \left (3 \, x + 7\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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