Optimal. Leaf size=76 \[ -\frac {b (a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \, _2F_1\left (\frac {5}{4},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a f (1+m)} \]
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Rubi [A]
time = 0.08, 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 = {2667, 2656}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rule 2667
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 = 5.43, size = 94, normalized size = 1.24 \begin {gather*} \frac {2 a b (a \cos (e+f x))^{-1+m} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} \sqrt {b \csc (e+f x)} \, _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 (-3+2 m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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