Optimal. Leaf size=40 \[ -\frac{(a \sec (c+d x)+a)^n \text{Hypergeometric2F1}\left (1,n,n+1,\frac{1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
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Rubi [A] time = 0.0459735, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3873, 68} \[ -\frac{(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
Antiderivative was successfully verified.
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Rule 3873
Rule 68
Rubi steps
\begin{align*} \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{(a-a x)^{-1+n}}{-a-a x} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac{\, _2F_1\left (1,n;1+n;\frac{1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n}\\ \end{align*}
Mathematica [B] time = 0.768554, size = 92, normalized size = 2.3 \[ \frac{2^{n-1} (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left (\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x)\right )^{n-1} \text{Hypergeometric2F1}\left (1,1-n,2-n,\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )}{d (n-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.254, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( dx+c \right ) \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n}\, 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 (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{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 (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right ), 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 (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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