Optimal. Leaf size=31 \[ -\frac {\tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{2 d}-\frac {x}{4} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2658} \[ -\frac {\tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{2 d}-\frac {x}{4} \]
Antiderivative was successfully verified.
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Rule 2658
Rubi steps
\begin {align*} \int \frac {1}{-5-3 \sin (c+d x)} \, dx &=-\frac {x}{4}-\frac {\tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.81 \[ -\frac {\tan ^{-1}\left (\frac {2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.84 \[ -\frac {\arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 49, normalized size = 1.58 \[ -\frac {d x + c + 2 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 0.65 \[ -\frac {\arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 26, normalized size = 0.84 \[ -\frac {\arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.51, size = 40, normalized size = 1.29 \[ \frac {\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}}{2\,d}-\frac {\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 49, normalized size = 1.58 \[ \begin {cases} - \frac {\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} + \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text {for}\: d \neq 0 \\\frac {x}{- 3 \sin {\relax (c )} - 5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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