Optimal. Leaf size=10 \[ \frac {\sec (c+d x)}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 8
Rule 2606
Rule 4397
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*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {\sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 1.20 \[ \frac {1}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 28, normalized size = 2.80 \[ \frac {2}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 13, normalized size = 1.30 \[ \frac {1}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 28, normalized size = 2.80 \[ -\frac {2}{d {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 20, normalized size = 2.00 \[ -\frac {2}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \frac {1}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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