Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{5/3} \, _2F_1\left (\frac {3}{2},\frac {5}{3};\frac {5}{2};\sin ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 f} \]
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Rubi [A]
time = 0.03, 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 = {2697}
\begin {gather*} \frac {\cos ^2(e+f x)^{5/3} \tan ^3(e+f x) \sqrt [3]{d \sec (e+f x)} \, _2F_1\left (\frac {3}{2},\frac {5}{3};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \sqrt [3]{d \sec (e+f x)} \tan ^2(e+f x) \, dx &=\frac {\cos ^2(e+f x)^{5/3} \, _2F_1\left (\frac {3}{2},\frac {5}{3};\frac {5}{2};\sin ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 80, normalized size = 1.40 \begin {gather*} \frac {3 \sqrt [3]{d \sec (e+f x)} \left (-\, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\sin ^2(e+f x)\right ) \sin (2 (e+f x))+2 \sqrt [3]{\cos ^2(e+f x)} \tan (e+f x)\right )}{8 f \sqrt [3]{\cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (\tan ^{2}\left (f x +e \right )\right )\, dx\]
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 \sqrt [3]{d \sec {\left (e + f x \right )}} \tan ^{2}{\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 {d}{\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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