Optimal. Leaf size=10 \[ \frac {\cot (e+f x)}{f} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3767, 8} \[ \frac {\cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
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*}
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Mathematica [A] time = 0.02, size = 10, normalized size = 1.00 \[ \frac {\cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 1.80 \[ \frac {\cos \left (f x + e\right )}{f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 13, normalized size = 1.30 \[ \frac {1}{f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 11, normalized size = 1.10 \[ \frac {\cot \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 12, normalized size = 1.20 \[ \frac {1}{f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.18, size = 10, normalized size = 1.00 \[ \frac {\mathrm {cot}\left (e+f\,x\right )}{f} \]
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 \[ - \int \csc ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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