Optimal. Leaf size=123 \[ \frac{\left (n^2-13 n+32\right ) (a \sec (c+d x)+a)^{n+3} \text{Hypergeometric2F1}(4,n+3,n+4,\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac{\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac{(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d} \]
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Rubi [A] time = 0.108252, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3873, 89, 78, 65} \[ \frac{\left (n^2-13 n+32\right ) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac{\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac{(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3873
Rule 89
Rule 78
Rule 65
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(-a-a x)^2 (a-a x)^{2+n}}{x^6} \, dx,x,-\sec (c+d x)\right )}{a^4 d}\\ &=-\frac{\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^{2+n} \left (a^3 (12-n)+5 a^3 x\right )}{x^5} \, dx,x,-\sec (c+d x)\right )}{5 a^5 d}\\ &=\frac{(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac{\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac{\left (32-13 n+n^2\right ) \operatorname{Subst}\left (\int \frac{(a-a x)^{2+n}}{x^4} \, dx,x,-\sec (c+d x)\right )}{20 a^2 d}\\ &=\frac{(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac{\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}+\frac{\left (32-13 n+n^2\right ) \, _2F_1(4,3+n;4+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{3+n}}{20 a^3 d (3+n)}\\ \end{align*}
Mathematica [A] time = 0.497712, size = 84, normalized size = 0.68 \[ -\frac{(\sec (c+d x)+1)^3 (a (\sec (c+d x)+1))^n \left ((n+3) \cos ^4(c+d x) (4 \cos (c+d x)+n-12)-\left (n^2-13 n+32\right ) \text{Hypergeometric2F1}(4,n+3,n+4,\sec (c+d x)+1)\right )}{20 d (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.658, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n} \left ( \sin \left ( dx+c \right ) \right ) ^{5}\, 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} \sin \left (d x + c\right )^{5}\,{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 (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )}{\left (a \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(-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} \sin \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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