Optimal. Leaf size=64 \[ \frac{6 \cos ^2(e+f x)^{7/6} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac{7}{6},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right )}{17 d f} \]
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Rubi [A] time = 0.0926452, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2602, 2577} \[ \frac{6 \cos ^2(e+f x)^{7/6} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac{7}{6},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right )}{17 d f} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx &=\frac{\left (b \cos ^{\frac{7}{3}}(e+f x) (d \tan (e+f x))^{7/3}\right ) \int \frac{(b \sin (e+f x))^{11/6}}{\cos ^{\frac{4}{3}}(e+f x)} \, dx}{d (b \sin (e+f x))^{7/3}}\\ &=\frac{6 \cos ^2(e+f x)^{7/6} \, _2F_1\left (\frac{7}{6},\frac{17}{12};\frac{29}{12};\sin ^2(e+f x)\right ) \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{7/3}}{17 d f}\\ \end{align*}
Mathematica [A] time = 0.423323, size = 65, normalized size = 1.02 \[ -\frac{3 d \sqrt{b \sin (e+f x)} \sqrt [3]{d \tan (e+f x)} \left (\sqrt [4]{\sec ^2(e+f x)} \, _2F_1\left (\frac{5}{12},\frac{5}{4};\frac{17}{12};-\tan ^2(e+f x)\right )-1\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.136, size = 0, normalized size = 0. \begin{align*} \int \sqrt{b\sin \left ( fx+e \right ) } \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 \sqrt{b \sin \left (f x + e\right )} \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 \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac{1}{3}} d \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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