Optimal. Leaf size=31 \[ -\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{6 d}-\frac {x}{12} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3783, 2657} \[ -\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{6 d}-\frac {x}{12} \]
Antiderivative was successfully verified.
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Rule 2657
Rule 3783
Rubi steps
\begin {align*} \int \frac {1}{3+5 \csc (c+d x)} \, dx &=\frac {x}{3}-\frac {1}{3} \int \frac {1}{1+\frac {3}{5} \sin (c+d x)} \, dx\\ &=-\frac {x}{12}-\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{6 d}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 66, normalized size = 2.13 \[ \frac {2 (c+d x)-5 \tan ^{-1}\left (\frac {2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 33, normalized size = 1.06 \[ \frac {4 \, d x - 5 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 49, normalized size = 1.58 \[ -\frac {d x + c + 10 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 36, normalized size = 1.16 \[ -\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{6 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 49, normalized size = 1.58 \[ -\frac {5 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right ) - 4 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 39, normalized size = 1.26 \[ \frac {x}{3}-\frac {5\,\mathrm {atan}\left (\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-15}{24\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+20}\right )}{6\,d} \]
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 \[ \int \frac {1}{5 \csc {\left (c + d x \right )} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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