Optimal. Leaf size=10 \[ \frac{\sec (c+d x)}{d} \]
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Rubi [A] time = 0.0250854, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4397, 2606, 8} \[ \frac{\sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\csc (c+d x)-\sin (c+d x)} \, dx &=\int \sec (c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=\frac{\sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0091558, size = 10, normalized size = 1. \[ \frac{\sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 13, normalized size = 1.3 \begin{align*}{\frac{1}{d\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04778, size = 38, normalized size = 3.8 \begin{align*} -\frac{2}{d{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.449018, size = 27, normalized size = 2.7 \begin{align*} \frac{1}{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{1}{- \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 [B] time = 1.10124, size = 38, normalized size = 3.8 \begin{align*} \frac{2}{d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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