Optimal. Leaf size=78 \[ -\frac {\sin ^2(e+f x)^{3/4} (b \csc (e+f x))^{3/2} (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {3}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 78, 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 = {2586, 2576} \[ -\frac {\sin ^2(e+f x)^{3/4} (b \csc (e+f x))^{3/2} (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {3}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2586
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m \sqrt {b \csc (e+f x)} \, dx &=\frac {\left ((b \csc (e+f x))^{3/2} (b \sin (e+f x))^{3/2}\right ) \int \frac {(a \cos (e+f x))^m}{\sqrt {b \sin (e+f x)}} \, dx}{b^2}\\ &=-\frac {(a \cos (e+f x))^{1+m} (b \csc (e+f x))^{3/2} \, _2F_1\left (\frac {3}{4},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{3/4}}{a b f (1+m)}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 96, normalized size = 1.23 \[ \frac {2 \tan (e+f x) \sqrt {b \csc (e+f x)} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (a \cos (e+f x))^m \, _2F_1\left (\frac {1}{4} (1-2 m),\frac {1-m}{2};\frac {1}{4} (5-2 m);\csc ^2(e+f x)\right )}{f (2 m-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int \sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \sqrt {b \csc \left (f x +e \right )}\, dx \]
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 \[ \int \sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,\sqrt {\frac {b}{\sin \left (e+f\,x\right )}} \,d x \]
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 \[ \int \left (a \cos {\left (e + f x \right )}\right )^{m} \sqrt {b \csc {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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