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 = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 216
Rule 3186
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*}
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Mathematica [C] time = 0.09, size = 39, normalized size = 2.60 \[ \frac {1}{3} i \log \left (\sqrt {7-\cos (2 (3 x+7))}+i \sqrt {2} \cos (3 x+7)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 94, normalized size = 6.27 \[ \frac {1}{6} \, \arctan \left (-\frac {4 \, \cos \left (3 \, x + 7\right ) \sin \left (3 \, x + 7\right ) - {\left (\cos \left (3 \, x + 7\right )^{3} - 2 \, \cos \left (3 \, x + 7\right )\right )} \sqrt {-\cos \left (3 \, x + 7\right )^{2} + 4}}{\cos \left (3 \, x + 7\right )^{4} - 8 \, \cos \left (3 \, x + 7\right )^{2} + 4}\right ) - \frac {1}{6} \, \arctan \left (\frac {\sin \left (3 \, x + 7\right )}{\cos \left (3 \, x + 7\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 11, normalized size = 0.73 \[ -\frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \cos \left (3 \, x + 7\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.16, size = 57, normalized size = 3.80 \[ -\frac {\sqrt {\left (3+\sin ^{2}\left (7+3 x \right )\right ) \left (\cos ^{2}\left (7+3 x \right )\right )}\, \arcsin \left (-1+\frac {\left (\cos ^{2}\left (7+3 x \right )\right )}{2}\right )}{6 \cos \left (7+3 x \right ) \sqrt {3+\sin ^{2}\left (7+3 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 11, normalized size = 0.73 \[ -\frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \cos \left (3 \, x + 7\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {\sin \left (3\,x+7\right )}{\sqrt {{\sin \left (3\,x+7\right )}^2+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (3 x + 7 \right )}}{\sqrt {\sin ^{2}{\left (3 x + 7 \right )} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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