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 = 0.0126391, antiderivative size = 28, normalized size of antiderivative = 1., 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 3777
Rule 8
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*}
Mathematica [A] time = 0.0894926, 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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Maple [A] time = 0.042, size = 41, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}+2\,{\frac{1}{ad \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47103, size = 68, normalized size = 2.43 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.467268, size = 142, normalized size = 5.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\csc{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2772, size = 43, normalized size = 1.54 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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