Integrand size = 23, antiderivative size = 90 \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-p,\frac {3}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \sqrt {\cos ^2(e+f x)} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x)}{f} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3271, 441, 440} \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-p,\frac {3}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right )}{f} \]
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Rule 440
Rule 441
Rule 3271
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{\left (1-x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (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}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{\left (1-x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-p,\frac {3}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \sqrt {\cos ^2(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} \tan (e+f x)}{f} \\ \end{align*}
\[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx \]
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\[\int \left (\sec ^{4}\left (f x +e \right )\right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p}d x\]
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\[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
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\[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right )^{4} \,d x } \]
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\[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sec \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p}{{\cos \left (e+f\,x\right )}^4} \,d x \]
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