Optimal. Leaf size=32 \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.0359275, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2707, 36, 29, 31} \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a \sin (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\log (\sin (c+d x))}{a d}-\frac{\log (1+\sin (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0155676, size = 32, normalized size = 1. \[ \frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 33, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06519, size = 42, normalized size = 1.31 \begin{align*} -\frac{\frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{\log \left (\sin \left (d x + c\right )\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46932, size = 74, normalized size = 2.31 \begin{align*} \frac{\log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - \log \left (\sin \left (d x + c\right ) + 1\right )}{a 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*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27787, size = 45, normalized size = 1.41 \begin{align*} -\frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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