Optimal. Leaf size=89 \[ \frac {2 \cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (3+2 n);\frac {1}{4} (7+2 n);\sin ^2(e+f x)\right ) \sqrt {a \sin (e+f x)} (b \tan (e+f x))^{1+n}}{b f (3+2 n)} \]
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Rubi [A]
time = 0.07, 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 = {2682, 2657}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rule 2682
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 = 11.77, size = 91, normalized size = 1.02 \begin {gather*} \frac {\cos ^2(e+f x)^{\frac {1}{2} (-1+n)} \, _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)} \sin (2 (e+f x)) (b \tan (e+f x))^n}{f (3+2 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, 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 \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 \sqrt {a \sin {\left (e + f x \right )}} \left (b \tan {\left (e + f x \right )}\right )^{n}\, 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 \sqrt {a\,\sin \left (e+f\,x\right )}\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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