Optimal. Leaf size=48 \[ \frac{\tan ^{-1}\left (\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right )}{\sqrt{2} d}+\frac{x}{\sqrt{2}} \]
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Rubi [A] time = 0.10813, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {203} \[ \frac{\tan ^{-1}\left (\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right )}{\sqrt{2} d}+\frac{x}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 203
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\cos (c+d x) \sin (c+d x)}{1+\sqrt{2}+\sin ^2(c+d x)}\right )}{\sqrt{2} d}\\ \end{align*}
Mathematica [A] time = 0.0249839, size = 22, normalized size = 0.46 \[ \frac{\tan ^{-1}\left (\sqrt{2} \tan (c+d x)\right )}{\sqrt{2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 20, normalized size = 0.4 \begin{align*}{\frac{\sqrt{2}\arctan \left ( \sqrt{2}\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.83484, size = 331, normalized size = 6.9 \begin{align*} \frac{\sqrt{2} \arctan \left (\frac{2 \, \sqrt{2} \sin \left (d x + c\right )}{2 \,{\left (\sqrt{2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt{2} + 3}, \frac{\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) - 1}{2 \,{\left (\sqrt{2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt{2} + 3}\right ) - \sqrt{2} \arctan \left (\frac{2 \, \sqrt{2} \sin \left (d x + c\right )}{2 \,{\left (\sqrt{2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt{2} + 3}, \frac{\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 1}{2 \,{\left (\sqrt{2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt{2} + 3}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.485324, size = 128, normalized size = 2.67 \begin{align*} -\frac{\sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2}}{4 \, \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{4 \, 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{\csc{\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 = 1.18044, size = 97, normalized size = 2.02 \begin{align*} \frac{\sqrt{2}{\left (d x + c + \arctan \left (-\frac{\sqrt{2} \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sin \left (2 \, d x + 2 \, c\right )}{\sqrt{2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 2}\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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