Optimal. Leaf size=38 \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.0537851, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2747, 3770, 2648} \[ \frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2747
Rule 3770
Rule 2648
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \csc (c+d x) \, dx}{a}-\int \frac{1}{a+a \sin (c+d x)} \, dx\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\cos (c+d x)}{d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0660189, size = 48, normalized size = 1.26 \[ -\frac{\sec (c+d x) \left (\sin (c+d x)+\sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\cos ^2(c+d x)}\right )-1\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 40, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{da \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}+{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00464, size = 69, normalized size = 1.82 \begin{align*} \frac{\frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{2}{a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60375, size = 293, normalized size = 7.71 \begin{align*} -\frac{{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) - 2}{2 \,{\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \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{\csc{\left (c + d x \right )}}{\sin{\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 = 1.18004, size = 51, normalized size = 1.34 \begin{align*} \frac{\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{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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