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 = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 2648
Rule 2747
Rule 3770
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*}
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Mathematica [A] time = 0.07, 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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fricas [B] time = 0.45, size = 97, normalized size = 2.55 \[ -\frac {{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) - 2}{2 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.47, size = 38, normalized size = 1.00 \[ \frac {\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 40, normalized size = 1.05 \[ \frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 51, normalized size = 1.34 \[ \frac {\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {2}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 39, normalized size = 1.03 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {2}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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 \[ \frac {\int \frac {\csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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