Optimal. Leaf size=86 \[ -\frac{i a^2 2^{2-\frac{m}{2}} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left (\frac{m-2}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right )}{d m} \]
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Rubi [A] time = 0.219014, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3515, 3505, 3523, 70, 69} \[ -\frac{i a^2 2^{2-\frac{m}{2}} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \, _2F_1\left (\frac{m-2}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right )}{d m} \]
Antiderivative was successfully verified.
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Rule 3515
Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx &=\left ((e \cos (c+d x))^m (e \sec (c+d x))^m\right ) \int (e \sec (c+d x))^{-m} (a+i a \tan (c+d x))^2 \, dx\\ &=\left ((e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \int (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{2-\frac{m}{2}} \, dx\\ &=\frac{\left (a^2 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \operatorname{Subst}\left (\int (a-i a x)^{-1-\frac{m}{2}} (a+i a x)^{1-\frac{m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\left (2^{1-\frac{m}{2}} a^3 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} \left (\frac{a+i a \tan (c+d x)}{a}\right )^{m/2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{i x}{2}\right )^{1-\frac{m}{2}} (a-i a x)^{-1-\frac{m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{i 2^{2-\frac{m}{2}} a^2 (e \cos (c+d x))^m \, _2F_1\left (\frac{1}{2} (-2+m),-\frac{m}{2};1-\frac{m}{2};\frac{1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{d m}\\ \end{align*}
Mathematica [A] time = 1.58629, size = 125, normalized size = 1.45 \[ -\frac{i a^2 2^{2-m} e^{i (c+d x)} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{m-1} (\tan (c+d x)-i)^2 \, _2F_1\left (1,\frac{m+2}{2};3-\frac{m}{2};-e^{2 i (c+d x)}\right ) \cos ^{2-m}(c+d x) (e \cos (c+d x))^m}{d (m-4)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.419, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{m} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{2}\, 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 )}^{2} \left (e \cos \left (d x + c\right )\right )^{m}\,{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{4 \, \left (\frac{1}{2} \,{\left (e e^{\left (2 i \, d x + 2 i \, c\right )} + e\right )} e^{\left (-i \, d x - i \, c\right )}\right )^{m} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )}}{e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, 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 )}^{2} \left (e \cos \left (d x + c\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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