Optimal. Leaf size=73 \[ -\frac{c \sin (a+b x) (c \sec (a+b x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(a+b x)\right )}{b (1-n) \sqrt{\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.0311051, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ -\frac{c \sin (a+b x) (c \sec (a+b x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right )}{b (1-n) \sqrt{\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (c \sec (a+b x))^n \, dx &=\left (\frac{\cos (a+b x)}{c}\right )^n (c \sec (a+b x))^n \int \left (\frac{\cos (a+b x)}{c}\right )^{-n} \, dx\\ &=-\frac{\cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right ) (c \sec (a+b x))^n \sin (a+b x)}{b (1-n) \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0469835, size = 61, normalized size = 0.84 \[ \frac{\sqrt{-\tan ^2(a+b x)} \cot (a+b x) (c \sec (a+b x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\sec ^2(a+b x)\right )}{b n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.265, size = 0, normalized size = 0. \begin{align*} \int \left ( c\sec \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 \left (c \sec \left (b x + a\right )\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 (\left (c \sec \left (b x + a\right )\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 \left (c \sec{\left (a + b x \right )}\right )^{n}\, 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 (c \sec \left (b x + a\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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