Optimal. Leaf size=58 \[ \frac{2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac{i c^2}{2 a f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.107872, antiderivative size = 58, 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} \[ \frac{2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac{i c^2}{2 a f (a+i a \tan (e+f x))^2} \]
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))^3} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(a+i a \tan (e+f x))^5} \, dx\\ &=-\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^4} \, 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)^4}-\frac{1}{(a+x)^3}\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=\frac{2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac{i c^2}{2 a f (a+i a \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 1.5956, size = 53, normalized size = 0.91 \[ \frac{c^2 (5 \cos (e+f x)+i \sin (e+f x)) (\sin (5 (e+f x))+i \cos (5 (e+f x)))}{24 a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 39, normalized size = 0.7 \begin{align*}{\frac{{c}^{2}}{f{a}^{3}} \left ({\frac{{\frac{i}{2}}}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{2}{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 = 1.33407, size = 101, normalized size = 1.74 \begin{align*} \frac{{\left (3 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, c^{2}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{24 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.879648, size = 109, normalized size = 1.88 \begin{align*} \begin{cases} \frac{\left (12 i a^{3} c^{2} f e^{6 i e} e^{- 4 i f x} + 8 i a^{3} c^{2} f e^{4 i e} e^{- 6 i f x}\right ) e^{- 10 i e}}{96 a^{6} f^{2}} & \text{for}\: 96 a^{6} f^{2} e^{10 i e} \neq 0 \\\frac{x \left (c^{2} e^{2 i e} + c^{2}\right ) e^{- 6 i e}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41555, size = 143, normalized size = 2.47 \begin{align*} -\frac{2 \,{\left (3 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 3 i \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 8 \, c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 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^{3} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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