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 = 0.112016, antiderivative size = 76, normalized size of antiderivative = 1., 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 2587
Rule 2576
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*}
Mathematica [A] time = 1.25679, 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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Maple [F] time = 0.33, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\csc \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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