Optimal. Leaf size=87 \[ -\frac {(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 87, 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 = {2603, 2617} \[ -\frac {(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \, _2F_1\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1);\frac {n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2603
Rule 2617
Rubi steps
\begin {align*} \int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx &=\left (\left (\frac {\csc (e+f x)}{a}\right )^m (a \sin (e+f x))^m\right ) \int (b \cot (e+f x))^n \left (\frac {\csc (e+f x)}{a}\right )^{-m} \, dx\\ &=-\frac {(b \cot (e+f x))^{1+n} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) (a \sin (e+f x))^m \sin ^2(e+f x)^{\frac {1}{2} (1-m+n)}}{b f (1+n)}\\ \end {align*}
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Mathematica [C] time = 1.80, size = 289, normalized size = 3.32 \[ \frac {(m-n+3) \sin (e+f x) (a \sin (e+f x))^m (b \cot (e+f x))^n F_1\left (\frac {1}{2} (m-n+1);-n,m+1;\frac {1}{2} (m-n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m-n+1) \left ((m-n+3) F_1\left (\frac {1}{2} (m-n+1);-n,m+1;\frac {1}{2} (m-n+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (n F_1\left (\frac {1}{2} (m-n+3);1-n,m+1;\frac {1}{2} (m-n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m+1) F_1\left (\frac {1}{2} (m-n+3);-n,m+2;\frac {1}{2} (m-n+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.96, size = 0, normalized size = 0.00 \[ \int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sin \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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