Optimal. Leaf size=52 \[ \frac{i (c-i c \tan (e+f x))^n \, _2F_1\left (2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right )}{4 a f n} \]
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Rubi [A] time = 0.124293, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 68} \[ \frac{i (c-i c \tan (e+f x))^n \, _2F_1\left (2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right )}{4 a f n} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 68
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx &=\frac{\int \cos ^2(e+f x) (c-i c \tan (e+f x))^{1+n} \, dx}{a c}\\ &=\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \frac{(c+x)^{-1+n}}{(c-x)^2} \, dx,x,-i c \tan (e+f x)\right )}{a f}\\ &=\frac{i \, _2F_1\left (2,n;1+n;\frac{1}{2} (1-i \tan (e+f x))\right ) (c-i c \tan (e+f x))^n}{4 a f n}\\ \end{align*}
Mathematica [A] time = 57.6802, size = 79, normalized size = 1.52 \[ \frac{i 2^{n-2} \left (1+e^{2 i (e+f x)}\right )^2 \left (\frac{c}{1+e^{2 i (e+f x)}}\right )^n \, _2F_1\left (2,2-n;3-n;1+e^{2 i (e+f x)}\right )}{a f (n-2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.527, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{n}}{a+ia\tan \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{\left (\frac{2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}{\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 \frac{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}}{i \, a \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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