Optimal. Leaf size=76 \[ -\frac {b \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {5}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a f (m+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 76, 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 = {2587, 2576} \[ -\frac {b \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \, _2F_1\left (\frac {5}{4},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{a f (m+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2587
Rubi steps
\begin {align*} \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx &=\left (b^2 \sqrt {b \csc (e+f x)} \sqrt {b \sin (e+f x)}\right ) \int \frac {(a \cos (e+f x))^m}{(b \sin (e+f x))^{3/2}} \, dx\\ &=-\frac {b (a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \, _2F_1\left (\frac {5}{4},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a f (1+m)}\\ \end {align*}
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Mathematica [A] time = 1.24, size = 94, normalized size = 1.24 \[ \frac {2 a b \sqrt {b \csc (e+f x)} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} (a \cos (e+f x))^{m-1} \, _2F_1\left (\frac {1}{4} (3-2 m),\frac {1-m}{2};\frac {1}{4} (7-2 m);\csc ^2(e+f x)\right )}{f (2 m-3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m} b \csc \left (f x + e\right ), x\right ) \]
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 \[ \int \left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \csc \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
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 \[ \int \left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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