Optimal. Leaf size=63 \[ \frac{\tan ^3(e+f x) \cos ^2(e+f x)^{\frac{m+3}{2}} (b \sec (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
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Rubi [A] time = 0.0376106, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2617} \[ \frac{\tan ^3(e+f x) \cos ^2(e+f x)^{\frac{m+3}{2}} (b \sec (e+f x))^m \, _2F_1\left (\frac{3}{2},\frac{m+3}{2};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (b \sec (e+f x))^m \tan ^2(e+f x) \, dx &=\frac{\cos ^2(e+f x)^{\frac{3+m}{2}} \, _2F_1\left (\frac{3}{2},\frac{3+m}{2};\frac{5}{2};\sin ^2(e+f x)\right ) (b \sec (e+f x))^m \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [C] time = 24.9202, size = 6612, normalized size = 104.95 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \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 (b \sec \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 (b \sec \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 (b \sec{\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 (b \sec \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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