Integrand size = 25, antiderivative size = 120 \[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1+m}{2},-p,\frac {3+m}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \cos ^2(e+f x)^{\frac {1+m}{2}} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} (d \tan (e+f x))^{1+m}}{d f (1+m)} \]
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Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3276, 525, 524} \[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\frac {\cos ^2(e+f x)^{\frac {m+1}{2}} (d \tan (e+f x))^{m+1} \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {b \sin ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2},\frac {m+1}{2},-p,\frac {m+3}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right )}{d f (m+1)} \]
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Rule 524
Rule 525
Rule 3276
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\cos ^2(e+f x)^{\frac {1+m}{2}} \sin ^{-1-m}(e+f x) (d \tan (e+f x))^{1+m}\right ) \text {Subst}\left (\int x^m \left (1-x^2\right )^{\frac {1}{2} (-1-m)} \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{d f} \\ & = \frac {\left (\cos ^2(e+f x)^{\frac {1+m}{2}} \sin ^{-1-m}(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} (d \tan (e+f x))^{1+m}\right ) \text {Subst}\left (\int x^m \left (1-x^2\right )^{\frac {1}{2} (-1-m)} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{d f} \\ & = \frac {\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1+m}{2},-p,\frac {3+m}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \cos ^2(e+f x)^{\frac {1+m}{2}} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} (d \tan (e+f x))^{1+m}}{d f (1+m)} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1+m}{2},-p,\frac {3+m}{2},\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \cos ^2(e+f x)^{\frac {1+m}{2}} \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac {b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x) (d \tan (e+f x))^m}{f (1+m)} \]
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\[\int {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{p} \left (d \tan \left (f x +e \right )\right )^{m}d x\]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\text {Timed out} \]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \left (d \tan \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \]
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