Optimal. Leaf size=49 \[ -\frac{(b \csc (e+f x))^{n+3} \text{Hypergeometric2F1}\left (2,\frac{n+3}{2},\frac{n+5}{2},\csc ^2(e+f x)\right )}{b^3 f (n+3)} \]
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Rubi [A] time = 0.0510699, antiderivative size = 49, 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 = {2621, 364} \[ -\frac{(b \csc (e+f x))^{n+3} \, _2F_1\left (2,\frac{n+3}{2};\frac{n+5}{2};\csc ^2(e+f x)\right )}{b^3 f (n+3)} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 364
Rubi steps
\begin{align*} \int (b \csc (e+f x))^n \sec ^3(e+f x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^{2+n}}{\left (-1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \csc (e+f x)\right )}{b^3 f}\\ &=-\frac{(b \csc (e+f x))^{3+n} \, _2F_1\left (2,\frac{3+n}{2};\frac{5+n}{2};\csc ^2(e+f x)\right )}{b^3 f (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0430347, size = 51, normalized size = 1.04 \[ -\frac{b (b \csc (e+f x))^{n-1} \text{Hypergeometric2F1}\left (2,\frac{1-n}{2},\frac{3-n}{2},\sin ^2(e+f x)\right )}{f (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.323, size = 0, normalized size = 0. \begin{align*} \int \left ( b\csc \left ( fx+e \right ) \right ) ^{n} \left ( \sec \left ( fx+e \right ) \right ) ^{3}\, 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} \sec \left (f x + e\right )^{3}\,{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} \sec \left (f x + e\right )^{3}, 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 )^{n} \sec \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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