Optimal. Leaf size=48 \[ \frac{(a \sin (e+f x))^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right )}{a^2 f (m+2)} \]
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Rubi [A] time = 0.0348985, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2592, 364} \[ \frac{(a \sin (e+f x))^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(e+f x)\right )}{a^2 f (m+2)} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 364
Rubi steps
\begin{align*} \int (a \sin (e+f x))^m \tan (e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^{1+m}}{a^2-x^2} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac{\, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};\sin ^2(e+f x)\right ) (a \sin (e+f x))^{2+m}}{a^2 f (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0325041, size = 53, normalized size = 1.1 \[ \frac{\sin ^2(e+f x) (a \sin (e+f x))^m \, _2F_1\left (1,\frac{m+2}{2};\frac{m+2}{2}+1;\sin ^2(e+f x)\right )}{f (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.546, size = 0, normalized size = 0. \begin{align*} \int \left ( a\sin \left ( fx+e \right ) \right ) ^{m}\tan \left ( fx+e \right ) \, 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 (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )\,{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 (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin{\left (e + f x \right )}\right )^{m} \tan{\left (e + f x \right )}\, dx \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 (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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