Optimal. Leaf size=90 \[ \frac{i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac{i a^3 c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )^4}+\frac{4 i c^3}{5 f (a+i a \tan (e+f x))^5} \]
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Rubi [A] time = 0.117911, antiderivative size = 90, 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} \[ \frac{i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac{i a^3 c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )^4}+\frac{4 i c^3}{5 f (a+i a \tan (e+f x))^5} \]
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))^5} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(a+i a \tan (e+f x))^8} \, dx\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^6} \, 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)^6}-\frac{4 a}{(a+x)^5}+\frac{1}{(a+x)^4}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac{4 i c^3}{5 f (a+i a \tan (e+f x))^5}-\frac{i c^3}{a f (a+i a \tan (e+f x))^4}+\frac{i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 2.59558, size = 58, normalized size = 0.64 \[ \frac{c^3 (4 i \sin (2 (e+f x))+16 \cos (2 (e+f x))+15) (\sin (8 (e+f x))+i \cos (8 (e+f x)))}{240 a^5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 52, normalized size = 0.6 \begin{align*}{\frac{{c}^{3}}{f{a}^{5}} \left ( -{\frac{1}{3\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{i}{ \left ( \tan \left ( fx+e \right ) -i \right ) ^{4}}}+{\frac{4}{5\, \left ( \tan \left ( fx+e \right ) -i \right ) ^{5}}} \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 = 1.28632, size = 149, normalized size = 1.66 \begin{align*} \frac{{\left (10 i \, c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, c^{3}\right )} e^{\left (-10 i \, f x - 10 i \, e\right )}}{240 \, a^{5} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3829, size = 153, normalized size = 1.7 \begin{align*} \begin{cases} \frac{\left (640 i a^{10} c^{3} f^{2} e^{18 i e} e^{- 6 i f x} + 960 i a^{10} c^{3} f^{2} e^{16 i e} e^{- 8 i f x} + 384 i a^{10} c^{3} f^{2} e^{14 i e} e^{- 10 i f x}\right ) e^{- 24 i e}}{15360 a^{15} f^{3}} & \text{for}\: 15360 a^{15} f^{3} e^{24 i e} \neq 0 \\\frac{x \left (c^{3} e^{4 i e} + 2 c^{3} e^{2 i e} + c^{3}\right ) e^{- 10 i e}}{4 a^{5}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5311, size = 235, normalized size = 2.61 \begin{align*} -\frac{2 \,{\left (15 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} - 30 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 140 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 170 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 282 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 170 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 140 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 30 i \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 15 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{15 \, a^{5} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - i\right )}^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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