Optimal. Leaf size=56 \[ \frac {i a (a+i a \tan (c+d x))^{n-1} \, _2F_1\left (2,n-1;n;\frac {1}{2} (i \tan (c+d x)+1)\right )}{4 d (1-n)} \]
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Rubi [A] time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 68} \[ \frac {i a (a+i a \tan (c+d x))^{n-1} \text {Hypergeometric2F1}\left (2,n-1,n,\frac {1}{2} (1+i \tan (c+d x))\right )}{4 d (1-n)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3487
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^n \, dx &=-\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-2+n}}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {i a \, _2F_1\left (2,-1+n;n;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-1+n}}{4 d (1-n)}\\ \end {align*}
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Mathematica [B] time = 13.29, size = 141, normalized size = 2.52 \[ -\frac {i 2^{n-3} e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \, _2F_1\left (1,2;n;-e^{2 i (c+d x)}\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{4} \, \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} {\left (e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.39, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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