Integrand size = 21, antiderivative size = 68 \[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{f} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4219, 372, 371} \[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=-\frac {\cos (e+f x) \left (a+b \sec ^2(e+f x)\right )^p \left (\frac {b \sec ^2(e+f x)}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a}\right )}{f} \]
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Rule 371
Rule 372
Rule 4219
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\left (\left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{f} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b \sec ^2(e+f x)}{a}\right ) \left (a+b \sec ^2(e+f x)\right )^p \left (1+\frac {b \sec ^2(e+f x)}{a}\right )^{-p}}{f} \]
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\[\int \left (a +b \sec \left (f x +e \right )^{2}\right )^{p} \sin \left (f x +e \right )d x\]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{p} \sin {\left (e + f x \right )}\, dx \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right ) \,d x } \]
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Time = 19.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.16 \[ \int \left (a+b \sec ^2(e+f x)\right )^p \sin (e+f x) \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2}-p,-p;\ \frac {3}{2}-p;\ -\frac {a\,{\cos \left (e+f\,x\right )}^2}{b}\right )}{f\,\left (2\,p-1\right )\,{\left (\frac {a\,{\cos \left (e+f\,x\right )}^2}{b}+1\right )}^p} \]
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