Optimal. Leaf size=25 \[ -\frac{i a}{3 f (c-i c \tan (e+f x))^3} \]
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Rubi [A] time = 0.0738695, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ -\frac{i a}{3 f (c-i c \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^3} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^4} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{i a}{3 f (c-i c \tan (e+f x))^3}\\ \end{align*}
Mathematica [B] time = 0.445941, size = 56, normalized size = 2.24 \[ \frac{a (-2 i \sin (2 (e+f x))+4 \cos (2 (e+f x))+3) (\sin (4 (e+f x))-i \cos (4 (e+f x)))}{24 c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 21, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,f{c}^{3} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}} \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 = 1.35444, size = 131, normalized size = 5.24 \begin{align*} \frac{-i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, a e^{\left (2 i \, f x + 2 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 = 0.509905, size = 131, normalized size = 5.24 \begin{align*} \begin{cases} \frac{- 64 i a c^{6} f^{2} e^{6 i e} e^{6 i f x} - 192 i a c^{6} f^{2} e^{4 i e} e^{4 i f x} - 192 i a c^{6} f^{2} e^{2 i e} e^{2 i f x}}{1536 c^{9} f^{3}} & \text{for}\: 1536 c^{9} f^{3} \neq 0 \\\frac{x \left (a e^{6 i e} + 2 a e^{4 i e} + a e^{2 i e}\right )}{4 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 = 1.35298, size = 130, normalized size = 5.2 \begin{align*} -\frac{2 \,{\left (3 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 6 i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 10 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 6 i \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a \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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