Optimal. Leaf size=53 \[ \frac {3 \sin (e+f x) \sec ^{\frac {4}{3}}(e+f x) \, _2F_1\left (-\frac {2}{3},-\frac {1}{2};\frac {1}{3};\cos ^2(e+f x)\right )}{4 f \sqrt {\sin ^2(e+f x)}} \]
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Rubi [A] time = 0.06, 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 = {2632, 2576} \[ \frac {3 \sin (e+f x) \sec ^{\frac {4}{3}}(e+f x) \, _2F_1\left (-\frac {2}{3},-\frac {1}{2};\frac {1}{3};\cos ^2(e+f x)\right )}{4 f \sqrt {\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2632
Rubi steps
\begin {align*} \int \sec ^{\frac {7}{3}}(e+f x) \sin ^2(e+f x) \, dx &=\left (\sqrt [3]{\cos (e+f x)} \sqrt [3]{\sec (e+f x)}\right ) \int \frac {\sin ^2(e+f x)}{\cos ^{\frac {7}{3}}(e+f x)} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{2};\frac {1}{3};\cos ^2(e+f x)\right ) \sec ^{\frac {4}{3}}(e+f x) \sin (e+f x)}{4 f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 56, normalized size = 1.06 \[ -\frac {3 \sin (e+f x) \sec ^{\frac {4}{3}}(e+f x) \left (\cos ^2(e+f x)^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\sin ^2(e+f x)\right )-1\right )}{4 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2}, x\right ) \]
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 \[ \int \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{\frac {1}{3}}\left (f x +e \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )\, dx \]
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 \[ \int \sec \left (f x + e\right )^{\frac {1}{3}} \tan \left (f x + e\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^{1/3} \,d x \]
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 \[ \int \tan ^{2}{\left (e + f x \right )} \sqrt [3]{\sec {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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