Optimal. Leaf size=25 \[ \frac{i c}{3 f (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 0.0708741, 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} \[ \frac{i c}{3 f (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^3} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(a+i a \tan (e+f x))^4} \, dx\\ &=-\frac{(i c) \operatorname{Subst}\left (\int \frac{1}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{i c}{3 f (a+i a \tan (e+f x))^3}\\ \end{align*}
Mathematica [B] time = 0.678858, size = 56, normalized size = 2.24 \[ \frac{c (2 i \sin (2 (e+f x))+4 \cos (2 (e+f x))+3) (\sin (4 (e+f x))+i \cos (4 (e+f x)))}{24 a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 21, normalized size = 0.8 \begin{align*} -{\frac{c}{3\,f{a}^{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 = 1.13552, size = 131, normalized size = 5.24 \begin{align*} \frac{{\left (3 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 3 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\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 = 0.804346, size = 143, normalized size = 5.72 \begin{align*} \begin{cases} \frac{\left (192 i a^{6} c f^{2} e^{10 i e} e^{- 2 i f x} + 192 i a^{6} c f^{2} e^{8 i e} e^{- 4 i f x} + 64 i a^{6} c f^{2} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{1536 a^{9} f^{3}} & \text{for}\: 1536 a^{9} f^{3} e^{12 i e} \neq 0 \\\frac{x \left (c e^{4 i e} + 2 c e^{2 i e} + c\right ) e^{- 6 i e}}{4 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 = 1.49928, size = 130, normalized size = 5.2 \begin{align*} -\frac{2 \,{\left (3 \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 6 i \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 10 \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 6 i \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, c \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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