Optimal. Leaf size=164 \[ \frac{(A n+B n+B) \sin (c+d x) \sec ^{1-n}(c+d x) \left (\frac{\sec (c+d x)+1}{1-\sec (c+d x)}\right )^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n \text{Hypergeometric2F1}\left (\frac{1}{2}-n,-n,1-n,-\frac{2 \sec (c+d x)}{1-\sec (c+d x)}\right )}{d n (n+1) (\sec (c+d x)+1)}+\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)} \]
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Rubi [A] time = 0.255129, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {4013, 3828, 3825, 132} \[ \frac{(A n+B n+B) \sin (c+d x) \sec ^{1-n}(c+d x) \left (\frac{\sec (c+d x)+1}{1-\sec (c+d x)}\right )^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n \, _2F_1\left (\frac{1}{2}-n,-n;1-n;-\frac{2 \sec (c+d x)}{1-\sec (c+d x)}\right )}{d n (n+1) (\sec (c+d x)+1)}+\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 4013
Rule 3828
Rule 3825
Rule 132
Rubi steps
\begin{align*} \int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n (A+B \sec (c+d x)) \, dx &=\frac{A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac{(B+A n+B n) \int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \, dx}{1+n}\\ &=\frac{A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac{\left ((B+A n+B n) (1+\sec (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int \sec ^{-n}(c+d x) (1+\sec (c+d x))^n \, dx}{1+n}\\ &=\frac{A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac{\left ((B+A n+B n) (1+\sec (c+d x))^{-\frac{1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{(1-x)^{-1-n} (2-x)^{-\frac{1}{2}+n}}{\sqrt{x}} \, dx,x,1-\sec (c+d x)\right )}{d (1+n) \sqrt{1-\sec (c+d x)}}\\ &=\frac{A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}+\frac{(B+A n+B n) \, _2F_1\left (\frac{1}{2}-n,-n;1-n;-\frac{2 \sec (c+d x)}{1-\sec (c+d x)}\right ) \sec ^{1-n}(c+d x) \left (\frac{1+\sec (c+d x)}{1-\sec (c+d x)}\right )^{\frac{1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)}{d n (1+n) (1+\sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.07748, size = 111, normalized size = 0.68 \[ \frac{\sin (c+d x) \sec ^{-n}(c+d x) (a (\sec (c+d x)+1))^n \left (\frac{(A n+B n+B) \left (-\cot ^2\left (\frac{1}{2} (c+d x)\right )\right )^{\frac{1}{2}-n} \text{Hypergeometric2F1}\left (\frac{1}{2}-n,-n,1-n,\csc ^2\left (\frac{1}{2} (c+d x)\right )\right )}{n (\cos (c+d x)+1)}+A\right )}{d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.14, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{-1-n} \left ( a+a\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 (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1}\,{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 ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1}, 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 (B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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