Optimal. Leaf size=120 \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^{m+1} \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right )}{d f (m+1)} \]
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Rubi [A] time = 0.121811, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3197, 511, 510} \[ \frac{\cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^{m+1} \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right )}{d f (m+1)} \]
Antiderivative was successfully verified.
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Rule 3197
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \left (a+b \sin ^2(e+f x)\right )^p (d \tan (e+f x))^m \, dx &=\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 ) \operatorname{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 ) \operatorname{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{F_1\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*}
Mathematica [A] time = 0.501859, size = 121, normalized size = 1.01 \[ \frac{\tan (e+f x) \cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^m \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right )}{f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.605, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{p} \left ( d\tan \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \left (d \tan \left (f x + e\right )\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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