Optimal. Leaf size=79 \[ \frac {2 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {1}{4} (3+2 m);\frac {1}{4} (7+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \tan (e+f x))^{3/2}}{b f (3+2 m)} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2682, 2657}
\begin {gather*} \frac {2 \cos ^2(e+f x)^{3/4} (b \tan (e+f x))^{3/2} (a \sin (e+f x))^m \, _2F_1\left (\frac {3}{4},\frac {1}{4} (2 m+3);\frac {1}{4} (2 m+7);\sin ^2(e+f x)\right )}{b f (2 m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int (a \sin (e+f x))^m \sqrt {b \tan (e+f x)} \, dx &=\frac {\left (a \cos ^{\frac {3}{2}}(e+f x) (b \tan (e+f x))^{3/2}\right ) \int \frac {(a \sin (e+f x))^{\frac {1}{2}+m}}{\sqrt {\cos (e+f x)}} \, dx}{b (a \sin (e+f x))^{3/2}}\\ &=\frac {2 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {1}{4} (3+2 m);\frac {1}{4} (7+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \tan (e+f x))^{3/2}}{b f (3+2 m)}\\ \end {align*}
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Mathematica [A]
time = 3.44, size = 87, normalized size = 1.10 \begin {gather*} \frac {2 \, _2F_1\left (\frac {2+m}{2},\frac {1}{4} (3+2 m);\frac {1}{4} (7+2 m);-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m (b \tan (e+f x))^{3/2}}{b f (3+2 m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (a \sin \left (f x +e \right )\right )^{m} \sqrt {b \tan \left (f x +e \right )}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin {\left (e + f x \right )}\right )^{m} \sqrt {b \tan {\left (e + f x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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