Optimal. Leaf size=63 \[ \frac {\tan ^5(e+f x) \cos ^2(e+f x)^{\frac {m+5}{2}} (b \sec (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \[ \frac {\tan ^5(e+f x) \cos ^2(e+f x)^{\frac {m+5}{2}} (b \sec (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int (b \sec (e+f x))^m \tan ^4(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{\frac {5+m}{2}} \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7}{2};\sin ^2(e+f x)\right ) (b \sec (e+f x))^m \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 110, normalized size = 1.75 \[ \frac {\sin (2 (e+f x)) \cos ^2(e+f x)^{\frac {m-1}{2}} (b \sec (e+f x))^m \left (\, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {3}{2};\sin ^2(e+f x)\right )-2 \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {3}{2};\sin ^2(e+f x)\right )+\, _2F_1\left (\frac {1}{2},\frac {m+5}{2};\frac {3}{2};\sin ^2(e+f x)\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x +e \right )\right )^{m} \left (\tan ^{4}\left (f x +e \right )\right )\, dx \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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 \[ \int \left (b \sec {\left (e + f x \right )}\right )^{m} \tan ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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