Optimal. Leaf size=74 \[ \frac {(a \sec (c+d x)+a)^n \, _2F_1(1,n;n+1;\sec (c+d x)+1)}{d n}-\frac {(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
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Rubi [A] time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3880, 86, 65, 68} \[ \frac {(a \sec (c+d x)+a)^n \, _2F_1(1,n;n+1;\sec (c+d x)+1)}{d n}-\frac {(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 86
Rule 3880
Rubi steps
\begin {align*} \int \cot (c+d x) (a+a \sec (c+d x))^n \, dx &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a+a x)^{-1+n}}{x (-a+a x)} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {(a+a x)^{-1+n}}{x} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {(a+a x)^{-1+n}}{-a+a x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n}+\frac {\, _2F_1(1,n;1+n;1+\sec (c+d x)) (a+a \sec (c+d x))^n}{d n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.77 \[ -\frac {(a (\sec (c+d x)+1))^n \left (\, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )-2 \, _2F_1(1,n;n+1;\sec (c+d x)+1)\right )}{2 d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \cot \left (d x +c \right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )\,{d x} \]
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 \[ \int \mathrm {cot}\left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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