Optimal. Leaf size=43 \[ \frac{a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{d f (n+1)} \]
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Rubi [A] time = 0.0488593, 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} \[ \frac{a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 3537
Rule 64
Rubi steps
\begin{align*} \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx &=-\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{i d x}{a}\right )^n}{-a^2+a x} \, dx,x,-i a \tan (e+f x)\right )}{f}\\ &=\frac{a \, _2F_1(1,1+n;2+n;-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0615825, size = 44, normalized size = 1.02 \[ \frac{a \tan (e+f x) (d \tan (e+f x))^n \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{f (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \left ( d\tan \left ( fx+e \right ) \right ) ^{n} \left ( a-ia\tan \left ( fx+e \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 (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\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{2 \, a \left (\frac{-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} - a \left (\int - \left (d \tan{\left (e + f x \right )}\right )^{n}\, dx + \int i \left (d \tan{\left (e + f x \right )}\right )^{n} \tan{\left (e + f 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 (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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