Optimal. Leaf size=50 \[ -\frac{i a^3 \left (c^2+i c^2 \tan (e+f x)\right )^3}{6 f \left (c^3-i c^3 \tan (e+f x)\right )^3} \]
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Rubi [A] time = 0.107305, 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} \[ -\frac{i a^3 \left (c^2+i c^2 \tan (e+f x)\right )^3}{6 f \left (c^3-i c^3 \tan (e+f x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac{\sec ^6(e+f x)}{(c-i c \tan (e+f x))^6} \, dx\\ &=\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{(c-x)^2}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac{i a^3 (c+i c \tan (e+f x))^3}{6 f \left (c^2-i c^2 \tan (e+f x)\right )^3}\\ \end{align*}
Mathematica [A] time = 0.219371, size = 34, normalized size = 0.68 \[ \frac{a^3 (\sin (6 (e+f x))-i \cos (6 (e+f x)))}{6 c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 50, normalized size = 1. \begin{align*}{\frac{{a}^{3}}{f{c}^{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 = 1.26545, size = 54, normalized size = 1.08 \begin{align*} -\frac{i \, a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{6 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.474164, size = 48, normalized size = 0.96 \begin{align*} \begin{cases} - \frac{i a^{3} e^{6 i e} e^{6 i f x}}{6 c^{3} f} & \text{for}\: 6 c^{3} f \neq 0 \\\frac{a^{3} x e^{6 i e}}{c^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40102, size = 97, normalized size = 1.94 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 10 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 3 \, a^{3} \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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