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 = 0.11, antiderivative size = 48, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.51, size = 46, normalized size = 0.96 \[ -\frac {\sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 72, normalized size = 1.50 \[ \frac {\sqrt {2} {\left (d x + c + \arctan \left (-\frac {\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sin \left (2 \, d x + 2 \, c\right )}{\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 2}\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 20, normalized size = 0.42 \[ \frac {\sqrt {2}\, \arctan \left (\tan \left (d x +c \right ) \sqrt {2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 245, normalized size = 5.10 \[ \frac {\sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sin \left (d x + c\right )}{2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} + 3}, \frac {\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) - 1}{2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} + 3}\right ) - \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sin \left (d x + c\right )}{2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} + 3}, \frac {\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 1}{2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} + 3}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 56, normalized size = 1.17 \[ \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {7\,\sqrt {2}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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