Optimal. Leaf size=151 \[ -\frac{A b^2 \sin (c+d x) (b \sec (c+d x))^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2-n}{2},\frac{4-n}{2},\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b B \sin (c+d x) (b \sec (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(c+d x)\right )}{d (1-n) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.126733, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {16, 3787, 3772, 2643} \[ -\frac{A b^2 \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b B \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \cos (c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx &=b \int (b \sec (c+d x))^{-1+n} (A+B \sec (c+d x)) \, dx\\ &=(A b) \int (b \sec (c+d x))^{-1+n} \, dx+B \int (b \sec (c+d x))^n \, dx\\ &=\left (A b \left (\frac{\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{1-n} \, dx+\left (B \left (\frac{\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{-n} \, dx\\ &=-\frac{B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-n) \sqrt{\sin ^2(c+d x)}}-\frac{A \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (2-n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.157987, size = 107, normalized size = 0.71 \[ \frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left (A n \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-1}{2},\frac{n+1}{2},\sec ^2(c+d x)\right )+B (n-1) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\sec ^2(c+d x)\right )\right )}{d (n-1) n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.897, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+B\sec \left ( dx+c \right ) \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 \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \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 ) \sec \left (d x + c\right ) + A \cos \left (d x + c\right )\right )} \left (b \sec \left (d x + c\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 (b \sec{\left (c + d x \right )}\right )^{n} \left (A + B \sec{\left (c + d x \right )}\right ) \cos{\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 \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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