Optimal. Leaf size=95 \[ -\frac{a^2 (c+d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{2 a^2 (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)} \]
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Rubi [A] time = 0.127351, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3543, 3537, 68} \[ -\frac{a^2 (c+d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{2 a^2 (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 3537
Rule 68
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^n \, dx &=-\frac{a^2 (c+d \tan (e+f x))^{1+n}}{d f (1+n)}+\int \left (2 a^2+2 i a^2 \tan (e+f x)\right ) (c+d \tan (e+f x))^n \, dx\\ &=-\frac{a^2 (c+d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac{\left (4 i a^4\right ) \operatorname{Subst}\left (\int \frac{\left (c-\frac{i d x}{2 a^2}\right )^n}{-4 a^4+2 a^2 x} \, dx,x,2 i a^2 \tan (e+f x)\right )}{f}\\ &=-\frac{a^2 (c+d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac{2 a^2 \, _2F_1\left (1,1+n;2+n;\frac{c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)}\\ \end{align*}
Mathematica [F] time = 4.51852, size = 0, normalized size = 0. \[ \int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.275, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{2} \left ( c+d\tan \left ( fx+e \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 (f x + e\right ) + a\right )}^{2}{\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{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 \, a^{2} \left (\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (4 i \, f x + 4 i \, e\right )}}{e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\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 (f x + e\right ) + a\right )}^{2}{\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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