Optimal. Leaf size=10 \[ \frac{\tan (c+d x)}{d} \]
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Rubi [A] time = 0.0862574, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {8} \[ \frac{\tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\tan (c+d x))}{d}\\ &=\frac{\tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0039757, size = 10, normalized size = 1. \[ \frac{\tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 11, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05784, size = 59, normalized size = 5.9 \begin{align*} -\frac{2 \, \sin \left (d x + c\right )}{d{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.447982, size = 42, normalized size = 4.2 \begin{align*} \frac{\sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \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*} \int \frac{\csc{\left (c + d x \right )}}{- \sin{\left (c + d x \right )} + \csc{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14395, size = 14, normalized size = 1.4 \begin{align*} \frac{\tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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