Optimal. Leaf size=28 \[ \frac{a^2 \tan (e+f x)}{f (c-i c \tan (e+f x))^2} \]
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Rubi [A] time = 0.101718, antiderivative size = 28, 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, 34} \[ \frac{a^2 \tan (e+f x)}{f (c-i c \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 34
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^2} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(c-i c \tan (e+f x))^4} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{c-x}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{a^2 \tan (e+f x)}{f (c-i c \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 0.236265, size = 34, normalized size = 1.21 \[ \frac{a^2 (\sin (4 (e+f x))-i \cos (4 (e+f x)))}{4 c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 39, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}}{f{c}^{2}} \left ( - \left ( \tan \left ( fx+e \right ) +i \right ) ^{-1}+{\frac{i}{ \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \right ) } \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 [A] time = 1.48172, size = 54, normalized size = 1.93 \begin{align*} -\frac{i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{4 \, c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.559509, size = 48, normalized size = 1.71 \begin{align*} \begin{cases} - \frac{i a^{2} e^{4 i e} e^{4 i f x}}{4 c^{2} f} & \text{for}\: 4 c^{2} f \neq 0 \\\frac{a^{2} x e^{4 i e}}{c^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31115, size = 73, normalized size = 2.61 \begin{align*} -\frac{2 \,{\left (a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{c^{2} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + i\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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