Optimal. Leaf size=43 \[ \frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d e (m+1)} \]
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Rubi [A] time = 0.0462262, antiderivative size = 43, 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 = {3537, 64} \[ \frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 3537
Rule 64
Rubi steps
\begin{align*} \int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i e x}{a}\right )^m}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{a \, _2F_1(1,1+m;2+m;i \tan (c+d x)) (e \tan (c+d x))^{1+m}}{d e (1+m)}\\ \end{align*}
Mathematica [B] time = 0.961127, size = 159, normalized size = 3.7 \[ \frac{a e^{-i c} 2^{-m-1} \left (-\frac{i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}\right )^{m+1} \left (1+e^{2 i (c+d x)}\right )^{m+1} \cos (c+d x) (1+i \tan (c+d x)) \, _2F_1\left (m+1,m+1;m+2;\frac{1}{2} \left (1-e^{2 i (c+d x)}\right )\right ) \tan ^{-m}(c+d x) (e \tan (c+d x))^m}{d (m+1) (\cos (d x)+i \sin (d x))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.406, size = 0, normalized size = 0. \begin{align*} \int \left ( e\tan \left ( dx+c \right ) \right ) ^{m} \left ( a+ia\tan \left ( dx+c \right ) \right ) \, 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 )} \left (e \tan \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{2 \, a \left (\frac{-i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} e^{\left (2 i \, d x + 2 i \, c\right )}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \left (e \tan{\left (c + d x \right )}\right )^{m}\, dx + \int i \left (e \tan{\left (c + d x \right )}\right )^{m} \tan{\left (c + d x \right )}\, dx\right ) \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 )} \left (e \tan \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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