Optimal. Leaf size=115 \[ \frac{(a+b \sec (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \sec (c+d x)}{a-b}\right )}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \sec (c+d x)}{a+b}\right )}{2 d (n+1) (a+b)} \]
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Rubi [A] time = 0.118902, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3874, 73, 712, 68} \[ \frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right )}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right )}{2 d (n+1) (a+b)} \]
Antiderivative was successfully verified.
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Rule 3874
Rule 73
Rule 712
Rule 68
Rubi steps
\begin{align*} \int \csc (c+d x) (a+b \sec (c+d x))^n \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-b x)^n}{(-1+x) (1+x)} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a-b x)^n}{-1+x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{(a-b x)^n}{2 (1-x)}-\frac{(a-b x)^n}{2 (1+x)}\right ) \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-b x)^n}{1-x} \, dx,x,-\sec (c+d x)\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{(a-b x)^n}{1+x} \, dx,x,-\sec (c+d x)\right )}{2 d}\\ &=\frac{\, _2F_1\left (1,1+n;2+n;\frac{a+b \sec (c+d x)}{a-b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a-b) d (1+n)}-\frac{\, _2F_1\left (1,1+n;2+n;\frac{a+b \sec (c+d x)}{a+b}\right ) (a+b \sec (c+d x))^{1+n}}{2 (a+b) d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.945822, size = 132, normalized size = 1.15 \[ \frac{(a+b \sec (c+d x))^n \left (\text{Hypergeometric2F1}\left (1,-n,1-n,\frac{(a+b) \cos (c+d x)}{a \cos (c+d x)+b}\right )-2^n \left (\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{b}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-n,1-n,\frac{(b-a) \cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{2 b}\right )\right )}{2 d n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.241, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( dx+c \right ) \left ( a+b\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 (b \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 (b \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 (b \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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