Optimal. Leaf size=97 \[ \frac {(a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)}-\frac {(a \sec (c+d x)+a)^{n+3} \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)}{a^3 d (n+3)}-\frac {3 (a \sec (c+d x)+a)^{n+3}}{a^3 d (n+3)} \]
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Rubi [A] time = 0.08, antiderivative size = 97, 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} \[ -\frac {(a \sec (c+d x)+a)^{n+3} \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)}{a^3 d (n+3)}-\frac {3 (a \sec (c+d x)+a)^{n+3}}{a^3 d (n+3)}+\frac {(a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)} \]
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 ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{2+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^2 (a+a x)^{2+n}+\frac {a^2 (a+a x)^{2+n}}{x}+a (a+a x)^{3+n}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {3 (a+a \sec (c+d x))^{3+n}}{a^3 d (3+n)}+\frac {(a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{2+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {3 (a+a \sec (c+d x))^{3+n}}{a^3 d (3+n)}-\frac {\, _2F_1(1,3+n;4+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{3+n}}{a^3 d (3+n)}+\frac {(a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 72, normalized size = 0.74 \[ \frac {(\sec (c+d x)+1)^3 (a (\sec (c+d x)+1))^n (-(n+4) \, _2F_1(1,n+3;n+4;\sec (c+d x)+1)+(n+3) \sec (c+d x)-2 n-9)}{d (n+3) (n+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{5}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.19, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\tan ^{5}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{5}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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