Optimal. Leaf size=10 \[ \frac{\cot (e+f x)}{f} \]
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Rubi [A] time = 0.0088738, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3767, 8} \[ \frac{\cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int -\csc ^2(e+f x) \, dx &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (e+f x))}{f}\\ &=\frac{\cot (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0161906, size = 10, normalized size = 1. \[ \frac{\cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 11, normalized size = 1.1 \begin{align*}{\frac{\cot \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962417, size = 16, normalized size = 1.6 \begin{align*} \frac{1}{f \tan \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48445, size = 42, normalized size = 4.2 \begin{align*} \frac{\cos \left (f x + e\right )}{f \sin \left (f x + e\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 (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12689, size = 18, normalized size = 1.8 \begin{align*} \frac{1}{f \tan \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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