Optimal. Leaf size=175 \[ -\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{n}{2},\frac{2-n}{2},\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)} \]
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Rubi [A] time = 0.142766, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {4047, 3772, 2643, 4046} \[ -\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{B \int (b \sec (c+d x))^{1+n} \, dx}{b}+\int (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}+\frac{(A+A n+C n) \int (b \sec (c+d x))^n \, dx}{1+n}+\frac{\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 )^{-1-n} \, dx}{b}\\ &=\frac{B \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n \sqrt{\sin ^2(c+d x)}}+\frac{C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}+\frac{\left ((A+A n+C 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 )^{-n} \, dx}{1+n}\\ &=-\frac{(A+A n+C n) \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 \left (1-n^2\right ) \sqrt{\sin ^2(c+d x)}}+\frac{B \, _2F_1\left (\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n \sqrt{\sin ^2(c+d x)}}+\frac{C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}\\ \end{align*}
Mathematica [C] time = 4.8301, size = 401, normalized size = 2.29 \[ -\frac{i 2^{n+1} e^{-i d n x} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \left (1+e^{2 i (c+d x)}\right )^n \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (e^{2 i c} \left (\frac{2 (A+2 C) e^{i d (n+2) x} \text{Hypergeometric2F1}\left (\frac{n+2}{2},n+2,\frac{n+4}{2},-e^{2 i (c+d x)}\right )}{n+2}+\frac{A e^{i (2 c+d (n+4) x)} \text{Hypergeometric2F1}\left (n+2,\frac{n+4}{2},\frac{n+6}{2},-e^{2 i (c+d x)}\right )}{n+4}+\frac{2 B e^{i (c+d (n+3) x)} \text{Hypergeometric2F1}\left (n+2,\frac{n+3}{2},\frac{n+5}{2},-e^{2 i (c+d x)}\right )}{n+3}\right )+\frac{A e^{i d n x} \text{Hypergeometric2F1}\left (\frac{n}{2},n+2,\frac{n+2}{2},-e^{2 i (c+d x)}\right )}{n}+\frac{2 B e^{i (c+d (n+1) x)} \text{Hypergeometric2F1}\left (\frac{n+1}{2},n+2,\frac{n+3}{2},-e^{2 i (c+d x)}\right )}{n+1}\right )}{d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.612, size = 0, normalized size = 0. \begin{align*} \int \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}\,{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 (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\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 )} + C \sec ^{2}{\left (c + d x \right )}\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 (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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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