Optimal. Leaf size=64 \[ \frac {2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac {7}{12},\frac {5}{4};\frac {9}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{5 d f (b \sec (e+f x))^{4/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697}
\begin {gather*} \frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac {7}{12},\frac {5}{4};\frac {9}{4};\sin ^2(e+f x)\right )}{5 d f (b \sec (e+f x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx &=\frac {2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac {7}{12},\frac {5}{4};\frac {9}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{5 d f (b \sec (e+f x))^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 71, normalized size = 1.11 \begin {gather*} \frac {3 \cot ^3(e+f x) \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {1}{3};\sec ^2(e+f x)\right ) (d \tan (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{3/4}}{4 f (b \sec (e+f x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx\]
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 {\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {4}{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 {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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