Optimal. Leaf size=123 \[ \frac{(a \sec (c+d x)+a)^{n+4} \text{Hypergeometric2F1}(1,n+4,n+5,\sec (c+d x)+1)}{a^4 d (n+4)}+\frac{7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac{5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac{(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)} \]
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Rubi [A] time = 0.100469, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3880, 88, 65} \[ \frac{(a \sec (c+d x)+a)^{n+4} \, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{a^4 d (n+4)}+\frac{7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac{5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac{(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 88
Rule 65
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x)^3 (a+a x)^{3+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (7 a^3 (a+a x)^{3+n}-\frac{a^3 (a+a x)^{3+n}}{x}-5 a^2 (a+a x)^{4+n}+a (a+a x)^{5+n}\right ) \, dx,x,\sec (c+d x)\right )}{a^6 d}\\ &=\frac{7 (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}-\frac{5 (a+a \sec (c+d x))^{5+n}}{a^5 d (5+n)}+\frac{(a+a \sec (c+d x))^{6+n}}{a^6 d (6+n)}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{3+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{7 (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}+\frac{\, _2F_1(1,4+n;5+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}-\frac{5 (a+a \sec (c+d x))^{5+n}}{a^5 d (5+n)}+\frac{(a+a \sec (c+d x))^{6+n}}{a^6 d (6+n)}\\ \end{align*}
Mathematica [A] time = 0.41777, size = 87, normalized size = 0.71 \[ \frac{(\sec (c+d x)+1)^4 (a (\sec (c+d x)+1))^n \left (\frac{\text{Hypergeometric2F1}(1,n+4,n+5,\sec (c+d x)+1)}{n+4}+\frac{(\sec (c+d x)+1)^2}{n+6}-\frac{5 (\sec (c+d x)+1)}{n+5}+\frac{7}{n+4}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.439, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n} \left ( \tan \left ( dx+c \right ) \right ) ^{7}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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} \tan \left (d x + c\right )^{7}, 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} \tan \left (d x + c\right )^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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