Optimal. Leaf size=23 \[ -\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d} \]
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Rubi [A] time = 0.0941638, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4397, 2669, 3767, 8} \[ -\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx &=\int (1+\cos (c+d x)) \csc ^2(c+d x) \, dx\\ &=-\frac{\csc (c+d x)}{d}+\int \csc ^2(c+d x) \, dx\\ &=-\frac{\csc (c+d x)}{d}-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0142455, size = 15, normalized size = 0.65 \[ -\frac{\cot \left (\frac{1}{2} (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 24, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( - \left ( \sin \left ( dx+c \right ) \right ) ^{-1}-\cot \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04373, size = 30, normalized size = 1.3 \begin{align*} -\frac{\frac{1}{\sin \left (d x + c\right )} + \frac{1}{\tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.457527, size = 51, normalized size = 2.22 \begin{align*} -\frac{\cos \left (d x + c\right ) + 1}{d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.24435, size = 27, normalized size = 1.17 \begin{align*} \begin{cases} \frac{- \cot{\left (c + d x \right )} - \csc{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (\cot{\left (c \right )} + \csc{\left (c \right )}\right ) \csc{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12489, size = 22, normalized size = 0.96 \begin{align*} -\frac{1}{d \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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