Integrand size = 17, antiderivative size = 80 \[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=-\frac {\cos ^{1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2656} \[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=-\frac {\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac {1-m}{2}} \cos ^{n+1}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{f (n+1)} \]
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Rule 2656
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^{1+n}(e+f x) \operatorname {Hypergeometric2F1}\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}}}{f (1+n)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=\frac {\cos ^{-1+n}(e+f x) \cos ^2(e+f x)^{\frac {1-n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1-n}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) \sin ^{1+m}(e+f x)}{f (1+m)} \]
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\[\int \left (\cos ^{n}\left (f x +e \right )\right ) \left (\sin ^{m}\left (f x +e \right )\right )d x\]
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\[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=\int { \cos \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]
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\[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=\int \sin ^{m}{\left (e + f x \right )} \cos ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=\int { \cos \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]
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\[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=\int { \cos \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m} \,d x } \]
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Time = 2.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \cos ^n(e+f x) \sin ^m(e+f x) \, dx=-\frac {{\cos \left (e+f\,x\right )}^{n+1}\,{\sin \left (e+f\,x\right )}^{m+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2}-\frac {m}{2},\frac {n}{2}+\frac {1}{2};\ \frac {n}{2}+\frac {3}{2};\ {\cos \left (e+f\,x\right )}^2\right )}{f\,\left (n+1\right )\,{\left ({\sin \left (e+f\,x\right )}^2\right )}^{\frac {m}{2}+\frac {1}{2}}} \]
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