Optimal. Leaf size=83 \[ \frac {\cos ^2(e+f x)^{\frac {n+1}{2}} (a \sin (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sin ^2(e+f x)\right )}{b f (m+n+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2602, 2577} \[ \frac {\cos ^2(e+f x)^{\frac {n+1}{2}} (a \sin (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sin ^2(e+f x)\right )}{b f (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rubi steps
\begin {align*} \int (a \sin (e+f x))^m (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))^{m+n} \, dx}{b}\\ &=\frac {\cos ^2(e+f x)^{\frac {1+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1+m+n);\frac {1}{2} (3+m+n);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \tan (e+f x))^{1+n}}{b f (1+m+n)}\\ \end {align*}
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Mathematica [C] time = 1.96, size = 260, normalized size = 3.13 \[ \frac {(m+n+3) \sin (e+f x) (a \sin (e+f x))^m (b \tan (e+f x))^n F_1\left (\frac {1}{2} (m+n+1);n,m+1;\frac {1}{2} (m+n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m+n+1) \left ((m+n+3) F_1\left (\frac {1}{2} (m+n+1);n,m+1;\frac {1}{2} (m+n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left ((m+1) F_1\left (\frac {1}{2} (m+n+3);n,m+2;\frac {1}{2} (m+n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n F_1\left (\frac {1}{2} (m+n+3);n+1,m+1;\frac {1}{2} (m+n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \left (b \tan \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \tan {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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