Optimal. Leaf size=88 \[ \frac{b \cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rubi [A] time = 0.0921135, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2587, 2577} \[ \frac{b \cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
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Rule 2587
Rule 2577
Rubi steps
\begin{align*} \int \cos ^m(e+f x) (b \csc (e+f x))^n \, dx &=\left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int \cos ^m(e+f x) (b \sin (e+f x))^{-n} \, dx\\ &=\frac{b \cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)}\\ \end{align*}
Mathematica [C] time = 0.270629, size = 314, normalized size = 3.57 \[ -\frac{2 (n-3) \sin \left (\frac{1}{2} (e+f x)\right ) \cos ^3\left (\frac{1}{2} (e+f x)\right ) \cos ^m(e+f x) (b \csc (e+f x))^n F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (n-1) \left (2 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \left (m F_1\left (\frac{3}{2}-\frac{n}{2};1-m,m-n+1;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(m-n+1) F_1\left (\frac{3}{2}-\frac{n}{2};-m,m-n+2;\frac{5}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+(n-3) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2}-\frac{n}{2};-m,m-n+1;\frac{3}{2}-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.98, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{m} \left ( b\csc \left ( fx+e \right ) \right ) ^{n}\, 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 )^{n} \cos \left (f x + e\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 (\left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc{\left (e + f x \right )}\right )^{n} \cos ^{m}{\left (e + f x \right )}\, dx \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 )^{n} \cos \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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