Optimal. Leaf size=92 \[ -\frac {a b \sin ^2(e+f x)^{\frac {1-m}{2}} (a \sin (e+f x))^{m-1} (b \sec (e+f x))^{n-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.10, antiderivative size = 92, 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 = {2587, 2576} \[ -\frac {a b \sin ^2(e+f x)^{\frac {1-m}{2}} (a \sin (e+f x))^{m-1} (b \sec (e+f x))^{n-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2587
Rubi steps
\begin {align*} \int (b \sec (e+f x))^n (a \sin (e+f x))^m \, dx &=\left (b^2 (b \cos (e+f x))^{-1+n} (b \sec (e+f x))^{-1+n}\right ) \int (b \cos (e+f x))^{-n} (a \sin (e+f x))^m \, dx\\ &=-\frac {a b \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) (b \sec (e+f x))^{-1+n} (a \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 289, normalized size = 3.14 \[ \frac {4 (m+3) \sin \left (\frac {1}{2} (e+f x)\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (a \sin (e+f x))^m (b \sec (e+f x))^n F_1\left (\frac {m+1}{2};n,m-n+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m+1) \left ((m+3) (\cos (e+f x)+1) F_1\left (\frac {m+1}{2};n,m-n+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 \sin ^2\left (\frac {1}{2} (e+f x)\right ) \left ((m-n+1) F_1\left (\frac {m+3}{2};n,m-n+2;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n F_1\left (\frac {m+3}{2};n+1,m-n+1;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x +e \right )\right )^{n} \left (a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \sec {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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