Optimal. Leaf size=90 \[ -\frac {(b \cot (e+f x))^{1+n} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {1}{2} (3-m+n);\cos ^2(e+f x)\right ) (a \sec (e+f x))^m \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1-m+n)} \]
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Rubi [A]
time = 0.12, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2698, 2682,
2656} \begin {gather*} -\frac {\sin ^2(e+f x)^{\frac {n+1}{2}} (a \sec (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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rule 2682
Rule 2698
Rubi steps
\begin {align*} \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx &=\left (\left (\frac {\cos (e+f x)}{a}\right )^m (a \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{a}\right )^{-m} (b \cot (e+f x))^n \, dx\\ &=-\frac {\left (\left (\frac {\cos (e+f x)}{a}\right )^{-1+m-n} (b \cot (e+f x))^{1+n} (a \sec (e+f x))^m (-\sin (e+f x))^{1+n}\right ) \int \left (\frac {\cos (e+f x)}{a}\right )^{-m+n} (-\sin (e+f x))^{-n} \, dx}{a b}\\ &=-\frac {(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 ) (a \sec (e+f x))^m \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1-m+n)}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 83, normalized size = 0.92 \begin {gather*} -\frac {b (b \cot (e+f x))^{-1+n} \, _2F_1\left (1-\frac {m}{2},\frac {1-n}{2};\frac {3-n}{2};-\tan ^2(e+f x)\right ) (a \sec (e+f x))^m \sec ^2(e+f x)^{-m/2}}{f (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \sec \left (f x +e \right )\right )^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a \sec {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\cos \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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