Optimal. Leaf size=69 \[ \frac {(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac {(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3880, 80, 65} \[ \frac {(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac {(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 80
Rule 3880
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \tan ^3(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x) (a+a x)^{1+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {(a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}-\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{1+n}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {(a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}+\frac {\, _2F_1(1,2+n;3+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.71 \[ \frac {(\sec (c+d x)+1)^2 (a (\sec (c+d x)+1))^n (\, _2F_1(1,n+2;n+3;\sec (c+d x)+1)+1)}{d (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.16, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\tan ^{3}\left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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