Optimal. Leaf size=74 \[ -\frac {\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n}+\frac {\, _2F_1(1,n;1+n;1+\sec (c+d x)) (a+a \sec (c+d x))^n}{d n} \]
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Rubi [A]
time = 0.04, 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 = {3965, 88, 67,
70} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 88
Rule 3965
Rubi steps
\begin {align*} \int \cot (c+d x) (a+a \sec (c+d x))^n \, dx &=\frac {a^2 \text {Subst}\left (\int \frac {(a+a x)^{-1+n}}{x (-a+a x)} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a \text {Subst}\left (\int \frac {(a+a x)^{-1+n}}{x} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a^2 \text {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 = 0.05, size = 57, normalized size = 0.77 \begin {gather*} -\frac {\left (\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right )-2 \, _2F_1(1,n;1+n;1+\sec (c+d x))\right ) (a (1+\sec (c+d x)))^n}{2 d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \cot {\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {cot}\left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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