Optimal. Leaf size=58 \[ \frac{i a^2}{2 c f (c-i c \tan (e+f x))^2}-\frac{2 i a^2}{3 f (c-i c \tan (e+f x))^3} \]
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Rubi [A] time = 0.113296, 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} \[ \frac{i a^2}{2 c f (c-i c \tan (e+f x))^2}-\frac{2 i a^2}{3 f (c-i c \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^3} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(c-i c \tan (e+f x))^5} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{c-x}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (\frac{2 c}{(c+x)^4}-\frac{1}{(c+x)^3}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=-\frac{2 i a^2}{3 f (c-i c \tan (e+f x))^3}+\frac{i a^2}{2 c f (c-i c \tan (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 1.3485, size = 53, normalized size = 0.91 \[ \frac{a^2 (5 \cos (e+f x)-i \sin (e+f x)) (\sin (5 (e+f x))-i \cos (5 (e+f x)))}{24 c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 39, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}}{f{c}^{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 = 1.26518, size = 101, normalized size = 1.74 \begin{align*} \frac{-2 i \, a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{24 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.447423, size = 97, normalized size = 1.67 \begin{align*} \begin{cases} \frac{- 8 i a^{2} c^{3} f e^{6 i e} e^{6 i f x} - 12 i a^{2} c^{3} f e^{4 i e} e^{4 i f x}}{96 c^{6} f^{2}} & \text{for}\: 96 c^{6} f^{2} \neq 0 \\\frac{x \left (a^{2} e^{6 i e} + a^{2} e^{4 i e}\right )}{2 c^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.48326, size = 143, normalized size = 2.47 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3 i \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 8 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 i \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{3 \, c^{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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