Optimal. Leaf size=33 \[ \frac{(c-i c \tan (e+f x))^n}{f (-\tan (e+f x)+i)^2} \]
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Rubi [A] time = 0.105814, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {3588, 74} \[ \frac{(c-i c \tan (e+f x))^n}{f (-\tan (e+f x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 74
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^n (-i (2+n)+(-2+n) \tan (e+f x))}{(-i+\tan (e+f x))^2} \, dx &=-\frac{(i c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{-1+n} (-i (2+n)+(-2+n) x)}{(-i+x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(c-i c \tan (e+f x))^n}{f (i-\tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 3.70189, size = 56, normalized size = 1.7 \[ \frac{(c \sec (e+f x))^n \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.713, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{n} \left ( -i \left ( 2+n \right ) + \left ( n-2 \right ) \tan \left ( fx+e \right ) \right ) }{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}\, 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 [B] time = 1.37158, size = 153, normalized size = 4.64 \begin{align*} -\frac{\left (\frac{2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}{\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, f} \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 ({\left (n - 2\right )} \tan \left (f x + e\right ) - i \, n - 2 i\right )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}}{{\left (\tan \left (f x + e\right ) - i\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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