Optimal. Leaf size=62 \[ \frac {6 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {1}{12},\frac {3}{4};\frac {13}{12};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{d f (b \sin (e+f x))^{4/3}} \]
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Rubi [A]
time = 0.07, 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 = {2682, 2657}
\begin {gather*} \frac {6 \cos ^2(e+f x)^{3/4} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac {1}{12},\frac {3}{4};\frac {13}{12};\sin ^2(e+f x)\right )}{d f (b \sin (e+f x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int \frac {\sqrt {d \tan (e+f x)}}{(b \sin (e+f x))^{4/3}} \, dx &=\frac {\left (b \cos ^{\frac {3}{2}}(e+f x) (d \tan (e+f x))^{3/2}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} (b \sin (e+f x))^{5/6}} \, dx}{d (b \sin (e+f x))^{3/2}}\\ &=\frac {6 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {1}{12},\frac {3}{4};\frac {13}{12};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{d f (b \sin (e+f x))^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 10.37, size = 67, normalized size = 1.08 \begin {gather*} \frac {3 \, _2F_1\left (\frac {1}{12},\frac {3}{4};\frac {13}{12};\sin ^2(e+f x)\right ) \sin (2 (e+f x)) \sqrt {d \tan (e+f x)}}{f \sqrt [4]{\cos ^2(e+f x)} (b \sin (e+f x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \tan \left (f x +e \right )}}{\left (b \sin \left (f x +e \right )\right )^{\frac {4}{3}}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (b\,\sin \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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