Optimal. Leaf size=11 \[ -\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0083176, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ -\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (a+b x))}{b}\\ &=-\frac{\cot (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0082027, size = 11, normalized size = 1. \[ -\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 12, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01663, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \tan \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.448256, size = 43, normalized size = 3.91 \begin{align*} -\frac{\cos \left (b x + a\right )}{b \sin \left (b x + a\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 \csc ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28183, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \tan \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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