Optimal. Leaf size=64 \[ \frac {6 \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac {13}{12},\frac {5}{4};\frac {25}{12};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{13 d f \sqrt [3]{b \sin (e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 64, 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)^{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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
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*}
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Mathematica [A]
time = 50.54, size = 63, normalized size = 0.98 \begin {gather*} -\frac {2 d \left (-1+\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 )\right ) \sqrt {d \tan (e+f x)}}{f \sqrt [3]{b \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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