Optimal. Leaf size=78 \[ -\frac {(a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac {1}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1) \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)}} \]
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Rubi [A] time = 0.11, 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} \[ -\frac {(a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac {1}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1) \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2586
Rubi steps
\begin {align*} \int \frac {(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx &=\frac {\int (a \cos (e+f x))^m (b \sin (e+f x))^{3/2} \, dx}{b^2 \sqrt {b \csc (e+f x)} \sqrt {b \sin (e+f x)}}\\ &=-\frac {(a \cos (e+f x))^{1+m} \, _2F_1\left (-\frac {1}{4},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right )}{a b f (1+m) \sqrt {b \csc (e+f x)} \sqrt [4]{\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.45, size = 116, normalized size = 1.49 \[ \frac {2 a \cos (2 (e+f x)) \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (a \cos (e+f x))^{m-1} \, _2F_1\left (\frac {1}{4} (-2 m-3),\frac {1-m}{2};\frac {1}{4} (1-2 m);\csc ^2(e+f x)\right )}{b f (2 m+3) \left (\csc ^2(e+f x)-2\right ) \sqrt {b \csc (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}}{b^{2} \csc \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \cos \left (f x +e \right )\right )^{m}}{\left (b \csc \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a\,\cos \left (e+f\,x\right )\right )}^m}{{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a \cos {\left (e + f x \right )}\right )^{m}}{\left (b \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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