Optimal. Leaf size=84 \[ -\frac {\sin ^2(e+f x)^{\frac {n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\cos ^2(e+f x)\right )}{b f (m+n+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2602, 2576} \[ -\frac {\sin ^2(e+f x)^{\frac {n+1}{2}} (a \cos (e+f x))^m (b \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\cos ^2(e+f x)\right )}{b f (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2602
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m (b \cot (e+f x))^n \, dx &=-\frac {\left (a (a \cos (e+f x))^{-1-n} (b \cot (e+f x))^{1+n} (-\sin (e+f x))^{1+n}\right ) \int (a \cos (e+f x))^{m+n} (-\sin (e+f x))^{-n} \, dx}{b}\\ &=-\frac {(a \cos (e+f x))^m (b \cot (e+f x))^{1+n} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1+m+n);\frac {1}{2} (3+m+n);\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1+m+n)}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 83, normalized size = 0.99 \[ -\frac {b \sec ^2(e+f x)^{m/2} (a \cos (e+f x))^m (b \cot (e+f x))^{n-1} \, _2F_1\left (\frac {m+2}{2},\frac {1-n}{2};\frac {3-n}{2};-\tan ^2(e+f x)\right )}{f (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.64, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \cot \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \cos \left (f x + e\right )\right )^{m} \left (b \cot \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a \cos {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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