Optimal. Leaf size=60 \[ -\frac{\sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0302036, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {16, 2643} \[ -\frac{\sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \sec (c+d x) \, dx &=b \int (b \cos (c+d x))^{-1+n} \, dx\\ &=-\frac{(b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{2+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0558222, size = 63, normalized size = 1.05 \[ -\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.711, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{n}\sec \left ( dx+c \right ) \, 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 \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )\,{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 \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ), 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 \cos{\left (c + d x \right )}\right )^{n} \sec{\left (c + d 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 \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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