Optimal. Leaf size=191 \[ \frac{b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2-n}{2},\frac{4-n}{2},\cos ^2(c+d x)\right )}{d (2-n) n \sqrt{\sin ^2(c+d x)}}-\frac{b B \sin (c+d x) (b \sec (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(c+d x)\right )}{d (1-n) \sqrt{\sin ^2(c+d x)}}+\frac{b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n} \]
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Rubi [A] time = 0.193549, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {16, 4047, 3772, 2643, 4046} \[ \frac{b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right )}{d (2-n) n \sqrt{\sin ^2(c+d x)}}-\frac{b B \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt{\sin ^2(c+d x)}}+\frac{b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n} \]
Antiderivative was successfully verified.
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Rule 16
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \cos (c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=b \int (b \sec (c+d x))^{-1+n} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=b \int (b \sec (c+d x))^{-1+n} \left (A+C \sec ^2(c+d x)\right ) \, dx+B \int (b \sec (c+d x))^n \, dx\\ &=\frac{b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}+\frac{(b (C (-1+n)+A n)) \int (b \sec (c+d x))^{-1+n} \, dx}{n}+\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{b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}+\frac{\left (b (C (-1+n)+A n) \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}{n}\\ &=-\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{(C (1-n)-A n) \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) n \sqrt{\sin ^2(c+d x)}}+\frac{b C (b \sec (c+d x))^{-1+n} \tan (c+d x)}{d n}\\ \end{align*}
Mathematica [A] time = 0.340157, size = 161, normalized size = 0.84 \[ \frac{\sqrt{-\tan ^2(c+d x)} (b \sec (c+d x))^n \left (A n (n+1) \cos (c+d x) \cot (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n-1}{2},\frac{n+1}{2},\sec ^2(c+d x)\right )+(n-1) \csc (c+d x) \left (B (n+1) \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n}{2},\frac{n+2}{2},\sec ^2(c+d x)\right )+C n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+1}{2},\frac{n+3}{2},\sec ^2(c+d x)\right )\right )\right )}{d (n-1) n (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.822, 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 ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \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 (C \sec \left (d x + c\right )^{2} + 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 (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 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(-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 (C \sec \left (d x + c\right )^{2} + 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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