Optimal. Leaf size=38 \[ \frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)} \]
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Rubi [A] time = 0.927861, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 88, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {4023, 3828, 3825, 132, 133, 4087} \[ \frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)} \]
Antiderivative was successfully verified.
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Rule 4023
Rule 3828
Rule 3825
Rule 132
Rule 133
Rule 4087
Rubi steps
\begin{align*} \int \left (\frac{\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a A n-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )\right ) \, dx &=\frac{\int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a A n-a C (1+n) \sec (c+d x)) \, dx}{a (1+n)}+\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1+n)}-\frac{C \int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^{1+n} \, dx}{a}+\frac{\int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (a A n+a C (1+n) \sec (c+d x)) \, dx}{a (1+n)}+\frac{(C-A n+C 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{C \int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^{1+n} \, dx}{a}+\left (-C+\frac{A n}{1+n}\right ) \int \sec ^{-n}(c+d x) (a+a \sec (c+d x))^n \, dx-\left (C (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))^{1+n} \, dx+\frac{\left ((C-A n+C 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)}+\left (C (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))^{1+n} \, dx+\left (\left (-C+\frac{A n}{1+n}\right ) (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-\frac{\left (C (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 \sqrt{1-\sec (c+d x)}}+\frac{\left ((C-A n+C 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{(C-A n+C 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))}-\frac{2^{\frac{3}{2}+n} C F_1\left (\frac{1}{2};1+n,-\frac{1}{2}-n;\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac{1}{2}-n} (a+a \sec (c+d x))^n \tan (c+d x)}{d}+\frac{\left (C (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 \sqrt{1-\sec (c+d x)}}+\frac{\left (\left (-C+\frac{A n}{1+n}\right ) (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 \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{(C-A n+C 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))}-\frac{\left (C-\frac{A n}{1+n}\right ) \, _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+\sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.162311, size = 38, normalized size = 1. \[ \frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a (\sec (c+d x)+1))^n}{d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.27, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n} \left ( -aAn-aC \left ( 1+n \right ) \sec \left ( dx+c \right ) \right ) }{ \left ( 1+n \right ) a \left ( \sec \left ( dx+c \right ) \right ) ^{n}}}+ \left ( \sec \left ( dx+c \right ) \right ) ^{-1-n} \left ( a+a\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 [B] time = 11.1309, size = 419, normalized size = 11.03 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (-{\left (d n + d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) - c\right ) \sin \left (c n\right ) -{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (-{\left (d n - d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) + c\right ) \sin \left (c n\right ) -{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (c n\right ) \sin \left (-{\left (d n + d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) - c\right ) +{\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )}^{n} A a^{n} \cos \left (c n\right ) \sin \left (-{\left (d n - d\right )} x + 2 \, n \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) + c\right )}{2 \,{\left ({\left (d n + d\right )} 2^{n} \cos \left (c n\right )^{2} +{\left (d n + d\right )} 2^{n} \sin \left (c n\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.908504, size = 142, normalized size = 3.74 \begin{align*} \frac{A \left (\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}\right )^{n} \frac{1}{\cos \left (d x + c\right )}^{-n - 1} \sin \left (d x + c\right )}{{\left (d n + d\right )} \cos \left (d x + c\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} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{-n - 1} - \frac{{\left (C a{\left (n + 1\right )} \sec \left (d x + c\right ) + A a n\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{a{\left (n + 1\right )} \sec \left (d x + c\right )^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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