Optimal. Leaf size=86 \[ -\frac {\, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)} \]
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Rubi [A]
time = 0.06, antiderivative size = 86, 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 = {2667, 2656}
\begin {gather*} -\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} \sec ^{n-1}(e+f x) \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rule 2667
Rubi steps
\begin {align*} \int \sec ^n(e+f x) \sin ^m(e+f x) \, dx &=\left (\cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \sin ^m(e+f x) \, dx\\ &=-\frac {\, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 1.01, size = 285, normalized size = 3.31 \begin {gather*} \frac {4 (3+m) F_1\left (\frac {1+m}{2};n,1+m-n;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^n(e+f x) \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^m(e+f x)}{f (1+m) \left ((3+m) F_1\left (\frac {1+m}{2};n,1+m-n;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1+\cos (e+f x))-4 \left ((1+m-n) F_1\left (\frac {3+m}{2};n,2+m-n;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n F_1\left (\frac {3+m}{2};1+n,1+m-n;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (\sec ^{n}\left (f x +e \right )\right ) \left (\sin ^{m}\left (f x +e \right )\right )\, dx\]
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 \sin ^{m}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, 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 {\sin \left (e+f\,x\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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