Optimal. Leaf size=48 \[ \frac{b (a+b \sec (c+d x))^{n+1} \text{Hypergeometric2F1}\left (2,n+1,n+2,\frac{b \sec (c+d x)}{a}+1\right )}{a^2 d (n+1)} \]
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Rubi [A] time = 0.0394676, antiderivative size = 48, 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 = {3874, 65} \[ \frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right )}{a^2 d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3874
Rule 65
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^n \sin (c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-b x)^n}{x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac{b \, _2F_1\left (2,1+n;2+n;1+\frac{b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^2 d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.48009, size = 72, normalized size = 1.5 \[ \frac{b \cos (c+d x) (a+b \sec (c+d x))^n \text{Hypergeometric2F1}\left (2,1-n,2-n,\frac{a \cos (c+d x)}{a \cos (c+d x)+b}\right )}{d (n-1) (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.319, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n}\sin \left ( dx+c \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 )}^{n} \sin \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} \sin \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{n} \sin{\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 )}^{n} \sin \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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