Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2617} \[ \frac {\cos ^2(e+f x)^{19/6} \tan ^5(e+f x) (d \sec (e+f x))^{4/3} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{4/3} \tan ^4(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{19/6} \, _2F_1\left (\frac {5}{2},\frac {19}{6};\frac {7}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{4/3} \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 92, normalized size = 1.61 \[ \frac {3 d \sqrt [3]{d \sec (e+f x)} \left (-18 \sin (e+f x) \sqrt [6]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right )+27 \sin (e+f x)+\tan (e+f x) \sec (e+f x) \left (7 \sec ^2(e+f x)-16\right )\right )}{91 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} d \sec \left (f x + e\right ) \tan \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {4}{3}} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {4}{3}} \left (\tan ^{4}\left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{4/3} \,d x \]
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 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{\frac {4}{3}} \tan ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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