Optimal. Leaf size=143 \[ \frac{A \sin (c+d x) (b \sec (c+d x))^{n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-n-1),\frac{1-n}{2},\cos ^2(c+d x)\right )}{b d (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^{n+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (-n-2),-\frac{n}{2},\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.126553, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {16, 3787, 3772, 2643} \[ \frac{A \sin (c+d x) (b \sec (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\cos ^2(c+d x)\right )}{b d (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx &=\frac{\int (b \sec (c+d x))^{2+n} (A+B \sec (c+d x)) \, dx}{b^2}\\ &=\frac{A \int (b \sec (c+d x))^{2+n} \, dx}{b^2}+\frac{B \int (b \sec (c+d x))^{3+n} \, dx}{b^3}\\ &=\frac{\left (A \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 )^{-2-n} \, dx}{b^2}+\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 )^{-3-n} \, dx}{b^3}\\ &=\frac{A \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-1-n);\frac{1-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{1+n} \sin (c+d x)}{b d (1+n) \sqrt{\sin ^2(c+d x)}}+\frac{B \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-2-n);-\frac{n}{2};\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^n \tan (c+d x)}{d (2+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.220685, size = 119, normalized size = 0.83 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) (b \sec (c+d x))^n \left (A (n+3) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+2}{2},\frac{n+4}{2},\sec ^2(c+d x)\right )+B (n+2) \sec (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n+3}{2},\frac{n+5}{2},\sec ^2(c+d x)\right )\right )}{d (n+2) (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.924, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{2} \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} \sec \left (d x + c\right )^{2}\,{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 \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )^{2}\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 ) \sec ^{2}{\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} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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