Optimal. Leaf size=68 \[ \frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+3} \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(e+f x)\right )}{a^3 f (m+3)} \]
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Rubi [A] time = 0.0888997, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2600, 2577} \[ \frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+3} \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(e+f x)\right )}{a^3 f (m+3)} \]
Antiderivative was successfully verified.
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Rule 2600
Rule 2577
Rubi steps
\begin{align*} \int (a \sin (e+f x))^m \tan ^2(e+f x) \, dx &=\frac{\int \sec ^2(e+f x) (a \sin (e+f x))^{2+m} \, dx}{a^2}\\ &=\frac{\sqrt{\cos ^2(e+f x)} \, _2F_1\left (\frac{3}{2},\frac{3+m}{2};\frac{5+m}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{3+m}}{a^3 f (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0844246, size = 71, normalized size = 1.04 \[ \frac{\sin ^2(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(e+f x)\right )}{f (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \left ( a\sin \left ( fx+e \right ) \right ) ^{m} \left ( \tan \left ( fx+e \right ) \right ) ^{2}\, 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 )^{2}\,{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 )^{2}, 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 ^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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