Optimal. Leaf size=50 \[ \frac{i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]
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Rubi [A] time = 0.105004, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 37} \[ \frac{i c^3 \left (a^2-i a^2 \tan (e+f x)\right )^3}{6 f \left (a^3+i a^3 \tan (e+f x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 37
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(a+i a \tan (e+f x))^6} \, dx\\ &=-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{(a-x)^2}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a^2 f}\\ &=\frac{i c^3 (1-i \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 0.243045, size = 34, normalized size = 0.68 \[ \frac{c^3 (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{6 a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 50, normalized size = 1. \begin{align*}{\frac{{c}^{3}}{f{a}^{3}} \left ( \left ( \tan \left ( fx+e \right ) -i \right ) ^{-1}+{\frac{2\,i}{ \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 = 1.42533, size = 54, normalized size = 1.08 \begin{align*} \frac{i \, c^{3} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.640469, size = 53, normalized size = 1.06 \begin{align*} \begin{cases} \frac{i c^{3} e^{- 6 i e} e^{- 6 i f x}}{6 a^{3} f} & \text{for}\: 6 a^{3} f e^{6 i e} \neq 0 \\\frac{c^{3} x e^{- 6 i e}}{a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.52313, size = 97, normalized size = 1.94 \begin{align*} -\frac{2 \,{\left (3 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 10 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 \, c^{3} \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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