Integrand size = 14, antiderivative size = 78 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \]
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Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 441, 440} \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f} \]
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Rule 440
Rule 441
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (\left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1}{2},1,-p,\frac {3}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(78)=156\).
Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.46 \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {3 a \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sin (2 (e+f x)) \left (a+b \tan ^2(e+f x)\right )^p}{6 a f \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+4 f \left (b p \operatorname {AppellF1}\left (\frac {3}{2},1-p,1,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a \operatorname {AppellF1}\left (\frac {3}{2},-p,2,\frac {5}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)} \]
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\[\int \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
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\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, dx \]
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\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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