Integrand size = 23, antiderivative size = 124 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {(a+b (3+2 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{b f (3+2 p)} \]
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Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 396, 252, 251} \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {\left (\frac {a}{2 b p+3 b}+1\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right )}{f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
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Rule 251
Rule 252
Rule 396
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (1+\frac {a}{3 b+2 b p}\right ) \text {Subst}\left (\int \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (\left (1+\frac {a}{3 b+2 b p}\right ) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {\left (1+\frac {a}{3 b+2 b p}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p}}{f} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97 \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \left (-\left ((a+b (3+2 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \sin ^2(e+f x)}{a}\right )\right )+\left (a+b \sin ^2(e+f x)\right ) \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^p\right )}{b f (3+2 p)} \]
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\[\int \left (\cos ^{3}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p}d x\]
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\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
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\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \cos \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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