Optimal. Leaf size=123 \[ \frac {(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)}-\frac {5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac {(a \sec (c+d x)+a)^{n+4} \, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{a^4 d (n+4)}+\frac {7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)} \]
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Rubi [A] time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 65
Rule 88
Rule 3880
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*}
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Mathematica [A] time = 0.46, size = 87, normalized size = 0.71 \[ \frac {(\sec (c+d x)+1)^4 (a (\sec (c+d x)+1))^n \left (\frac {\, _2F_1(1,n+4;n+5;\sec (c+d x)+1)}{n+4}+\frac {(\sec (c+d x)+1)^2}{n+6}-\frac {5 (\sec (c+d x)+1)}{n+5}+\frac {7}{n+4}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.39, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\tan ^{7}\left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^7\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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