Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d} \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4397, 3186, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3186
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx &=\int \frac {\sin (c+d x)}{1+\sin ^2(c+d x)} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 61, normalized size = 2.65 \[ -\frac {\tanh ^{-1}\left (\frac {\cos (c)-(\sin (c)-i) \tan \left (\frac {d x}{2}\right )}{\sqrt {2}}\right )+\tanh ^{-1}\left (\frac {\cos (c)-(\sin (c)+i) \tan \left (\frac {d x}{2}\right )}{\sqrt {2}}\right )}{\sqrt {2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 44, normalized size = 1.91 \[ \frac {\sqrt {2} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - 2}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 68, normalized size = 2.96 \[ \frac {\sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 6 \right |}}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 21, normalized size = 0.91 \[ -\frac {\arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 176, normalized size = 7.65 \[ \frac {\sqrt {2} \log \left (-\frac {2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) - \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} - 3}{2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} + 3}\right ) + \sqrt {2} \log \left (-\frac {2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) - \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} - 3}{2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} + 3}\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 42, normalized size = 1.83 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2\,\sqrt {2}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1}\right )}{2\,d} \]
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 \[ \int \frac {1}{\sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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