Optimal. Leaf size=53 \[ \frac {3 \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};\cos ^2(e+f x)\right ) \sec ^{\frac {8}{3}}(e+f x) \sin (e+f x)}{8 f \sqrt {\sin ^2(e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 53, 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 = {2712, 2656}
\begin {gather*} \frac {3 \sin (e+f x) \sec ^{\frac {8}{3}}(e+f x) \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};\cos ^2(e+f x)\right )}{8 f \sqrt {\sin ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rule 2712
Rubi steps
\begin {align*} \int \sec ^{\frac {11}{3}}(e+f x) \sin ^4(e+f x) \, dx &=\left (\cos ^{\frac {2}{3}}(e+f x) \sec ^{\frac {2}{3}}(e+f x)\right ) \int \frac {\sin ^4(e+f x)}{\cos ^{\frac {11}{3}}(e+f x)} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{3};\cos ^2(e+f x)\right ) \sec ^{\frac {8}{3}}(e+f x) \sin (e+f x)}{8 f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 78, normalized size = 1.47 \begin {gather*} \frac {3 \sec ^{\frac {2}{3}}(e+f x) \left (-11 \sin (e+f x)+9 \sqrt [3]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right ) \sin (e+f x)+2 \sec (e+f x) \tan (e+f x)\right )}{16 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{4}\left (f x +e \right )}{\sec \left (f x +e \right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{\sec {\left (e + f x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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