Optimal. Leaf size=53 \[ \frac {3 \, _2F_1\left (-\frac {5}{6},-\frac {1}{2};\frac {1}{6};\cos ^2(e+f x)\right ) \sec ^{\frac {5}{3}}(e+f x) \sin (e+f x)}{5 f \sqrt {\sin ^2(e+f x)}} \]
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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 {5}{3}}(e+f x) \, _2F_1\left (-\frac {5}{6},-\frac {1}{2};\frac {1}{6};\cos ^2(e+f x)\right )}{5 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 {8}{3}}(e+f x) \sin ^2(e+f x) \, dx &=\left (\cos ^{\frac {2}{3}}(e+f x) \sec ^{\frac {2}{3}}(e+f x)\right ) \int \frac {\sin ^2(e+f x)}{\cos ^{\frac {8}{3}}(e+f x)} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {5}{6},-\frac {1}{2};\frac {1}{6};\cos ^2(e+f x)\right ) \sec ^{\frac {5}{3}}(e+f x) \sin (e+f x)}{5 f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 56, normalized size = 1.06 \begin {gather*} -\frac {3 \left (-1+\cos ^2(e+f x)^{5/6} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\sin ^2(e+f x)\right )\right ) \sec ^{\frac {5}{3}}(e+f x) \sin (e+f x)}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (\sec ^{\frac {2}{3}}\left (f x +e \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )\, dx\]
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 \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 \tan ^{2}{\left (e + f x \right )} \sec ^{\frac {2}{3}}{\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 {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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