Optimal. Leaf size=28 \[ \frac {\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac {x}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 28, 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 = {3777, 8} \[ \frac {\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3777
Rubi steps
\begin {align*} \int \frac {1}{a+a \csc (c+d x)} \, dx &=\frac {\cot (c+d x)}{d (a+a \csc (c+d x))}+\frac {\int a \, dx}{a^2}\\ &=\frac {x}{a}+\frac {\cot (c+d x)}{d (a+a \csc (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 47, normalized size = 1.68 \[ \frac {-\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+c+d x}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 54, normalized size = 1.93 \[ \frac {d x + {\left (d x + 1\right )} \cos \left (d x + c\right ) + {\left (d x - 1\right )} \sin \left (d x + c\right ) + 1}{a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 32, normalized size = 1.14 \[ \frac {\frac {d x + c}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 41, normalized size = 1.46 \[ \frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 50, normalized size = 1.79 \[ \frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 27, normalized size = 0.96 \[ \frac {x}{a}+\frac {2}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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 \[ \frac {\int \frac {1}{\csc {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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