Optimal. Leaf size=87 \[ \frac{i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2}-\frac{4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac{i c^3}{f (a+i a \tan (e+f x))^4} \]
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Rubi [A] time = 0.119052, antiderivative size = 87, 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} \[ \frac{i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2}-\frac{4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac{i c^3}{f (a+i a \tan (e+f x))^4} \]
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))^3}{(a+i a \tan (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(a+i a \tan (e+f x))^7} \, dx\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^5} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \left (\frac{4 a^2}{(a+x)^5}-\frac{4 a}{(a+x)^4}+\frac{1}{(a+x)^3}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac{i c^3}{f (a+i a \tan (e+f x))^4}-\frac{4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac{i c^3}{2 f \left (a^2+i a^2 \tan (e+f x)\right )^2}\\ \end{align*}
Mathematica [A] time = 2.07449, size = 64, normalized size = 0.74 \[ -\frac{c^3 (\tan (e+f x)-7 i) \sec ^3(e+f x) (\cos (3 (e+f x))-i \sin (3 (e+f x)))}{48 a^4 f (\tan (e+f x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 53, normalized size = 0.6 \begin{align*}{\frac{{c}^{3}}{f{a}^{4}} \left ({\frac{i}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{4}}}-{\frac{{\frac{i}{2}}}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{4}{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 = 1.35966, size = 101, normalized size = 1.16 \begin{align*} \frac{{\left (4 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c^{3}\right )} e^{\left (-8 i \, f x - 8 i \, e\right )}}{48 \, a^{4} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.958526, size = 109, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\left (16 i a^{4} c^{3} f e^{8 i e} e^{- 6 i f x} + 12 i a^{4} c^{3} f e^{6 i e} e^{- 8 i f x}\right ) e^{- 14 i e}}{192 a^{8} f^{2}} & \text{for}\: 192 a^{8} f^{2} e^{14 i e} \neq 0 \\\frac{x \left (c^{3} e^{2 i e} + c^{3}\right ) e^{- 8 i e}}{2 a^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50508, size = 189, normalized size = 2.17 \begin{align*} -\frac{2 \,{\left (3 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 3 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 17 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 10 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 17 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 3 \, c^{3} \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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