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 = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 2577
Rule 2600
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*}
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Mathematica [A] time = 0.09, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \left (\tan ^{2}\left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \tan ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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