Optimal. Leaf size=11 \[ \frac{\tan ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0253427, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4338, 203} \[ \frac{\tan ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4338
Rule 203
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\tan ^{-1}(\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0362957, size = 11, normalized size = 1. \[ \frac{\tan ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 13, normalized size = 1.2 \begin{align*} -{\frac{\arctan \left ( \csc \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66518, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\sin \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.486142, size = 32, normalized size = 2.91 \begin{align*} \frac{\arctan \left (\sin \left (d x + c\right )\right )}{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*} \int \frac{\cot{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + \csc{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13988, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\sin \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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