Optimal. Leaf size=56 \[ \frac{i a (a+i a \tan (c+d x))^{n-1} \text{Hypergeometric2F1}\left (2,n-1,n,\frac{1}{2} (1+i \tan (c+d x))\right )}{4 d (1-n)} \]
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Rubi [A] time = 0.0572094, antiderivative size = 56, normalized size of antiderivative = 1., 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 3487
Rule 68
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*}
Mathematica [B] time = 13.1871, size = 141, normalized size = 2.52 \[ -\frac{i 2^{n-3} e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \left (e^{i d x}\right )^n \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} \text{Hypergeometric2F1}\left (1,2,n,-e^{2 i (c+d x)}\right ) (a+i a \tan (c+d x))^n}{d (n-1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.096, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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