Optimal. Leaf size=28 \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac{x}{a} \]
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Rubi [A] time = 0.0380084, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2735, 2648} \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{x}{a}-\int \frac{1}{a+a \sin (c+d x)} \, dx\\ &=\frac{x}{a}+\frac{\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.108201, size = 72, normalized size = 2.57 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left ((c+d x-2) \sin \left (\frac{1}{2} (c+d x)\right )+(c+d x) \cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 41, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}+2\,{\frac{1}{da \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 = 1.42433, 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 = 1.74005, 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 [A] time = 1.54983, size = 90, normalized size = 3.21 \begin{align*} \begin{cases} \frac{d x \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{d x}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12511, 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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