Optimal. Leaf size=89 \[ \frac {2 \sqrt {a \sin (e+f x)} \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\sin ^2(e+f x)\right )}{b f (2 n+3)} \]
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Rubi [A] time = 0.11, antiderivative size = 89, 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 = {2602, 2577} \[ \frac {2 \sqrt {a \sin (e+f x)} \cos ^2(e+f x)^{\frac {n+1}{2}} (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\sin ^2(e+f x)\right )}{b f (2 n+3)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rubi steps
\begin {align*} \int \sqrt {a \sin (e+f x)} (b \tan (e+f x))^n \, dx &=\frac {\left (a \cos ^{1+n}(e+f x) (a \sin (e+f x))^{-1-n} (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) (a \sin (e+f x))^{\frac {1}{2}+n} \, dx}{b}\\ &=\frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (3+2 n);\frac {1}{4} (7+2 n);\sin ^2(e+f x)\right ) \sqrt {a \sin (e+f x)} (b \tan (e+f x))^{1+n}}{b f (3+2 n)}\\ \end {align*}
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Mathematica [A] time = 1.62, size = 91, normalized size = 1.02 \[ \frac {\sin (2 (e+f x)) \sqrt {a \sin (e+f x)} \cos ^2(e+f x)^{\frac {n-1}{2}} (b \tan (e+f x))^n \, _2F_1\left (\frac {n+1}{2},\frac {1}{4} (2 n+3);\frac {1}{4} (2 n+7);\sin ^2(e+f x)\right )}{f (2 n+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x +e \right )}\, \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \sqrt {a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a\,\sin \left (e+f\,x\right )}\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \sqrt {a \sin {\left (e + f x \right )}} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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