Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{7/6} \, _2F_1\left (\frac {7}{6},\frac {3}{2};\frac {5}{2};\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f (d \sec (e+f x))^{2/3}} \]
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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)^{7/6} \tan ^3(e+f x) \, _2F_1\left (\frac {7}{6},\frac {3}{2};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f (d \sec (e+f x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{(d \sec (e+f x))^{2/3}} \, dx &=\frac {\cos ^2(e+f x)^{7/6} \, _2F_1\left (\frac {7}{6},\frac {3}{2};\frac {5}{2};\sin ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f (d \sec (e+f x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 79, normalized size = 1.39 \begin {gather*} \frac {-\frac {3}{2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right ) \sin (2 (e+f x))+3 \cos ^2(e+f x)^{5/6} \tan (e+f x)}{f \cos ^2(e+f x)^{5/6} (d \sec (e+f x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{2}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {2}{3}}}\, dx\]
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 ^{2}{\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}}}\, 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 )}^2}{{\left (\frac {d}{\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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