Optimal. Leaf size=64 \[ \frac{6 \cos ^2(e+f x)^{5/4} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac{13}{12},\frac{5}{4};\frac{25}{12};\sin ^2(e+f x)\right )}{13 d f \sqrt [3]{b \sin (e+f x)}} \]
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Rubi [A] time = 0.0960796, 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} \[ \frac{6 \cos ^2(e+f x)^{5/4} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac{13}{12},\frac{5}{4};\frac{25}{12};\sin ^2(e+f x)\right )}{13 d f \sqrt [3]{b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^{3/2}}{\sqrt [3]{b \sin (e+f x)}} \, dx &=\frac{\left (b \cos ^{\frac{5}{2}}(e+f x) (d \tan (e+f x))^{5/2}\right ) \int \frac{(b \sin (e+f x))^{7/6}}{\cos ^{\frac{3}{2}}(e+f x)} \, dx}{d (b \sin (e+f x))^{5/2}}\\ &=\frac{6 \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac{13}{12},\frac{5}{4};\frac{25}{12};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{13 d f \sqrt [3]{b \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.436655, size = 63, normalized size = 0.98 \[ -\frac{2 d \sqrt{d \tan (e+f x)} \left (\sqrt [4]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{12},\frac{1}{4};\frac{13}{12};\sin ^2(e+f x)\right )-1\right )}{f \sqrt [3]{b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.114, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [3]{b\sin \left ( fx+e \right ) }}}}\, 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 \frac{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \sin \left (f x + e\right )\right )^{\frac{1}{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 (\frac{\left (b \sin \left (f x + e\right )\right )^{\frac{2}{3}} \sqrt{d \tan \left (f x + e\right )} d \tan \left (f x + e\right )}{b \sin \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 \frac{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (b \sin \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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