Optimal. Leaf size=40 \[ -\frac {(b \sec (e+f x))^m \, _2F_1\left (1,\frac {m}{2};\frac {m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
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Rubi [A] time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2606, 364} \[ -\frac {(b \sec (e+f x))^m \, _2F_1\left (1,\frac {m}{2};\frac {m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2606
Rubi steps
\begin {align*} \int \cot (e+f x) (b \sec (e+f x))^m \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(b x)^{-1+m}}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\, _2F_1\left (1,\frac {m}{2};\frac {2+m}{2};\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m}\\ \end {align*}
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Mathematica [B] time = 0.82, size = 124, normalized size = 3.10 \[ \frac {b \sec ^2\left (\frac {1}{2} (e+f x)\right ) (b \sec (e+f x))^{m-1} \left ((\cos (e+f x)+1) \, _2F_1(1,1-m;2-m;\cos (e+f x))-2^m \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m} \, _2F_1\left (1-m,1-m;2-m;\frac {1}{2} \cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )}{4 f (m-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.11, size = 0, normalized size = 0.00 \[ \int \cot \left (f x +e \right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {cot}\left (e+f\,x\right )\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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