Optimal. Leaf size=62 \[ \frac{i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac{i a^2 c^2}{3 f \left (a^2+i a^2 \tan (e+f x)\right )^3} \]
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Rubi [A] time = 0.108742, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 43} \[ \frac{i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac{i a^2 c^2}{3 f \left (a^2+i a^2 \tan (e+f x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \left (\frac{2 a}{(a+x)^5}-\frac{1}{(a+x)^4}\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=\frac{i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac{i c^2}{3 a f (a+i a \tan (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 1.81146, size = 58, normalized size = 0.94 \[ \frac{c^2 (3 i \sin (2 (e+f x))+9 \cos (2 (e+f x))+8) (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{96 a^4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 39, normalized size = 0.6 \begin{align*}{\frac{{c}^{2}}{f{a}^{4}} \left ({\frac{{\frac{i}{2}}}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{4}}}+{\frac{1}{3\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}} \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 = 1.37984, size = 142, normalized size = 2.29 \begin{align*} \frac{{\left (6 i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c^{2}\right )} e^{\left (-8 i \, f x - 8 i \, e\right )}}{96 \, a^{4} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07346, size = 153, normalized size = 2.47 \begin{align*} \begin{cases} \frac{\left (384 i a^{8} c^{2} f^{2} e^{14 i e} e^{- 4 i f x} + 512 i a^{8} c^{2} f^{2} e^{12 i e} e^{- 6 i f x} + 192 i a^{8} c^{2} f^{2} e^{10 i e} e^{- 8 i f x}\right ) e^{- 18 i e}}{6144 a^{12} f^{3}} & \text{for}\: 6144 a^{12} f^{3} e^{18 i e} \neq 0 \\\frac{x \left (c^{2} e^{4 i e} + 2 c^{2} e^{2 i e} + c^{2}\right ) e^{- 8 i e}}{4 a^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46167, size = 189, normalized size = 3.05 \begin{align*} -\frac{2 \,{\left (3 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 6 i \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 17 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 16 i \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 17 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 6 i \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, a^{4} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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