Optimal. Leaf size=62 \[ \frac {6 \sqrt {b \sin (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{12};\frac {13}{12};\sin ^2(e+f x)\right )}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2602, 2577} \[ \frac {6 \sqrt {b \sin (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{12};\frac {13}{12};\sin ^2(e+f x)\right )}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sin (e+f x)}}{(d \tan (e+f x))^{4/3}} \, dx &=\frac {\left (b \sqrt [3]{b \sin (e+f x)}\right ) \int \frac {\cos ^{\frac {4}{3}}(e+f x)}{(b \sin (e+f x))^{5/6}} \, dx}{d \sqrt [3]{\cos (e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ &=\frac {6 \, _2F_1\left (-\frac {1}{6},\frac {1}{12};\frac {13}{12};\sin ^2(e+f x)\right ) \sqrt {b \sin (e+f x)}}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 64, normalized size = 1.03 \[ \frac {6 \sqrt [4]{\sec ^2(e+f x)} \sqrt {b \sin (e+f x)} \, _2F_1\left (\frac {1}{12},\frac {5}{4};\frac {13}{12};-\tan ^2(e+f x)\right )}{d f \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {2}{3}}}{d^{2} \tan \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sin \left (f x +e \right )}}{\left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {\sqrt {b\,\sin \left (e+f\,x\right )}}{{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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