Optimal. Leaf size=64 \[ \frac{3 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac{7}{6},\frac{23}{12};\frac{13}{6};\sin ^2(e+f x)\right )}{7 d f} \]
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Rubi [A] time = 0.0579168, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2617} \[ \frac{3 \cos ^2(e+f x)^{23/12} (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac{7}{6},\frac{23}{12};\frac{13}{6};\sin ^2(e+f x)\right )}{7 d f} \]
Antiderivative was successfully verified.
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Rule 2617
Rubi steps
\begin{align*} \int (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{4/3} \, dx &=\frac{3 \cos ^2(e+f x)^{23/12} \, _2F_1\left (\frac{7}{6},\frac{23}{12};\frac{13}{6};\sin ^2(e+f x)\right ) (b \sec (e+f x))^{3/2} (d \tan (e+f x))^{7/3}}{7 d f}\\ \end{align*}
Mathematica [A] time = 0.145581, size = 64, normalized size = 1. \[ \frac{2 d (b \sec (e+f x))^{3/2} \sqrt [3]{d \tan (e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{3}{4};\frac{7}{4};\sec ^2(e+f x)\right )}{3 f \sqrt [6]{-\tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (d \tan \left (f x + e\right )\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac{1}{3}} b d \sec \left (f x + e\right ) \tan \left (f x + e\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (d \tan \left (f x + e\right )\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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