Optimal. Leaf size=69 \[ \frac{\cos (a+b x) \csc ^{n-1}(a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\sin ^2(a+b x)\right )}{b (1-n) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0259089, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3772, 2643} \[ \frac{\cos (a+b x) \csc ^{n-1}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \csc ^n(a+b x) \, dx &=\csc ^n(a+b x) \sin ^n(a+b x) \int \sin ^{-n}(a+b x) \, dx\\ &=\frac{\cos (a+b x) \csc ^{-1+n}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right )}{b (1-n) \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0921054, size = 59, normalized size = 0.86 \[ -\frac{\cos (a+b x) \sin ^2(a+b x)^{\frac{n-1}{2}} \csc ^{n-1}(a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{3}{2},\cos ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.641, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( bx+a \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 \csc \left (b x + a\right )^{n}\,{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 (\csc \left (b x + a\right )^{n}, 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 \csc ^{n}{\left (a + b 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 \csc \left (b x + a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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