Optimal. Leaf size=78 \[ -\frac {\sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac {3}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b^3 f (m+1)} \]
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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} \[ -\frac {(a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac {3}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1) \sin ^2(e+f x)^{3/4} (b \csc (e+f x))^{3/2}} \]
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))^{5/2}} \, dx &=\frac {\int (a \cos (e+f x))^m (b \sin (e+f x))^{5/2} \, dx}{b^2 (b \csc (e+f x))^{3/2} (b \sin (e+f x))^{3/2}}\\ &=-\frac {(a \cos (e+f x))^{1+m} \, _2F_1\left (-\frac {3}{4},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right )}{a b f (1+m) (b \csc (e+f x))^{3/2} \sin ^2(e+f x)^{3/4}}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 125, normalized size = 1.60 \[ \frac {2 (2 \cos (2 (e+f x))+1) \tan (e+f x) \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (a \cos (e+f x))^m \, _2F_1\left (\frac {1}{4} (-2 m-5),\frac {1-m}{2};\frac {1}{4} (-2 m-1);\csc ^2(e+f x)\right )}{b^2 f (2 m+5) \left (3 \csc ^2(e+f x)-4\right ) \sqrt {b \csc (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}}{b^{3} \csc \left (f x + e\right )^{3}}, x\right ) \]
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 \[ \int \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \cos \left (f x +e \right )\right )^{m}}{\left (b \csc \left (f x +e \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a\,\cos \left (e+f\,x\right )\right )}^m}{{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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