Optimal. Leaf size=62 \[ -\frac {3 \cos ^2(e+f x)^{7/12} \, _2F_1\left (-\frac {1}{6},\frac {7}{12};\frac {5}{6};\sin ^2(e+f x)\right ) (b \sec (e+f x))^{3/2}}{d f \sqrt [3]{d \tan (e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 62, 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 {3 \cos ^2(e+f x)^{7/12} (b \sec (e+f x))^{3/2} \, _2F_1\left (-\frac {1}{6},\frac {7}{12};\frac {5}{6};\sin ^2(e+f x)\right )}{d f \sqrt [3]{d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2697
Rubi steps
\begin {align*} \int \frac {(b \sec (e+f x))^{3/2}}{(d \tan (e+f x))^{4/3}} \, dx &=-\frac {3 \cos ^2(e+f x)^{7/12} \, _2F_1\left (-\frac {1}{6},\frac {7}{12};\frac {5}{6};\sin ^2(e+f x)\right ) (b \sec (e+f x))^{3/2}}{d f \sqrt [3]{d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 64, normalized size = 1.03 \begin {gather*} \frac {2 d \, _2F_1\left (\frac {3}{4},\frac {7}{6};\frac {7}{4};\sec ^2(e+f x)\right ) (b \sec (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{7/6}}{3 f (d \tan (e+f x))^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx\]
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 (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (d \tan {\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 (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\left (d\,\mathrm {tan}\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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