Optimal. Leaf size=85 \[ -\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac {1-m}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2576} \[ -\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac {1-m}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin {align*} \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx &=-\frac {(d \cos (e+f x))^{1+n} \, _2F_1\left (\frac {1-m}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{d f (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.96 \[ \frac {d \sin ^{m+1}(e+f x) \cos ^2(e+f x)^{\frac {1-n}{2}} (d \cos (e+f x))^{n-1} \, _2F_1\left (\frac {m+1}{2},\frac {1-n}{2};\frac {m+3}{2};\sin ^2(e+f x)\right )}{f (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}, x\right ) \]
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 \[ \int \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x +e \right )\right )^{n} \left (\sin ^{m}\left (f x +e \right )\right )\, dx \]
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 \[ \int \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^m\,{\left (d\,\cos \left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (d \cos {\left (e + f x \right )}\right )^{n} \sin ^{m}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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