Optimal. Leaf size=113 \[ \frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0831302, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4046, 3772, 2643} \[ \frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right )}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \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{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{C (b \sec (c+d x))^n \tan (c+d x)}{d (1+n)}\\ \end{align*}
Mathematica [C] time = 5.61037, size = 273, normalized size = 2.42 \[ -\frac{i 2^{n+1} e^{-i (n+1) (c+d x)} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+1} \sec ^{-n-2}(c+d x) \left (A+C \sec ^2(c+d x)\right ) (b \sec (c+d x))^n \left (n e^{i (n+2) (c+d x)} \left (2 (n+4) (A+2 C) \text{Hypergeometric2F1}\left (1,-\frac{n}{2},\frac{n+4}{2},-e^{2 i (c+d x)}\right )+A (n+2) e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},\frac{n+6}{2},-e^{2 i (c+d x)}\right )\right )+A \left (n^2+6 n+8\right ) e^{i n (c+d x)} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-1,\frac{n+2}{2},-e^{2 i (c+d x)}\right )\right )}{d n (n+2) (n+4) (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.687, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sec \left ( dx+c \right ) \right ) ^{n} \left ( A+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} + 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} + 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 + 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} + 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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