Optimal. Leaf size=140 \[ \frac{(c+i d) (c-3 i d) (d+i c)}{8 a^3 f (1+i \tan (e+f x))}+\frac{x (c-i d)^3}{8 a^3}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c+i d)^2 (d+i c)}{8 a f (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.219159, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3546, 3540, 3526, 8} \[ \frac{(c+i d) (c-3 i d) (d+i c)}{8 a^3 f (1+i \tan (e+f x))}+\frac{x (c-i d)^3}{8 a^3}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c+i d)^2 (d+i c)}{8 a f (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 3546
Rule 3540
Rule 3526
Rule 8
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx &=\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c-i d) \int \frac{(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx}{2 a}\\ &=\frac{(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c-i d) \int \frac{a \left (c^2-2 i c d+d^2\right )-2 i a d^2 \tan (e+f x)}{a+i a \tan (e+f x)} \, dx}{4 a^3}\\ &=\frac{(c+i d) (c-3 i d) (i c+d)}{8 a^3 f (1+i \tan (e+f x))}+\frac{(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c-i d)^3 \int 1 \, dx}{8 a^3}\\ &=\frac{(c-i d)^3 x}{8 a^3}+\frac{(c+i d) (c-3 i d) (i c+d)}{8 a^3 f (1+i \tan (e+f x))}+\frac{(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 1.79059, size = 260, normalized size = 1.86 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (12 f x (c-i d)^3 (\cos (3 e)+i \sin (3 e))+18 i (c+i d) (c-i d)^2 (\cos (e)+i \sin (e)) \cos (2 f x)+18 (c+i d) (c-i d)^2 (\cos (e)+i \sin (e)) \sin (2 f x)+9 (c+i d)^2 (c-i d) (\cos (e)-i \sin (e)) \sin (4 f x)+9 (c+i d)^2 (d+i c) (\cos (e)-i \sin (e)) \cos (4 f x)+2 (c+i d)^3 (\sin (3 e)+i \cos (3 e)) \cos (6 f x)+2 (c+i d)^3 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)\right )}{96 f (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 454, normalized size = 3.2 \begin{align*}{\frac{{c}^{3}}{8\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{3\,c{d}^{2}}{8\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{8}}{c}^{3}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{9\,i}{8}}c{d}^{2}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( fx+e \right ) -i \right ){c}^{3}}{f{a}^{3}}}+{\frac{c{d}^{2}}{2\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( fx+e \right ) +i \right ){c}^{3}}{f{a}^{3}}}-{\frac{{c}^{3}}{6\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{{\frac{7\,i}{8}}{d}^{3}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{\ln \left ( \tan \left ( fx+e \right ) -i \right ){d}^{3}}{16\,f{a}^{3}}}+{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) c{d}^{2}}{f{a}^{3}}}-{\frac{3\,\ln \left ( \tan \left ( fx+e \right ) -i \right ){c}^{2}d}{16\,f{a}^{3}}}-{\frac{3\,{c}^{2}d}{8\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{5\,{d}^{3}}{8\,f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{\frac{i}{6}}{d}^{3}}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) c{d}^{2}}{f{a}^{3}}}+{\frac{3\,\ln \left ( \tan \left ( fx+e \right ) +i \right ){c}^{2}d}{16\,f{a}^{3}}}-{\frac{\ln \left ( \tan \left ( fx+e \right ) +i \right ){d}^{3}}{16\,f{a}^{3}}}-{\frac{{\frac{3\,i}{8}}{c}^{2}d}{f{a}^{3} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{2}}{c}^{2}d}{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 [A] time = 1.61577, size = 369, normalized size = 2.64 \begin{align*} \frac{{\left ({\left (12 \, c^{3} - 36 i \, c^{2} d - 36 \, c d^{2} + 12 i \, d^{3}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 2 i \, c^{3} - 6 \, c^{2} d - 6 i \, c d^{2} + 2 \, d^{3} +{\left (18 i \, c^{3} + 18 \, c^{2} d + 18 i \, c d^{2} + 18 \, d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (9 i \, c^{3} - 9 \, c^{2} d + 9 i \, c d^{2} - 9 \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61707, size = 554, normalized size = 3.96 \begin{align*} \begin{cases} \frac{\left (\left (512 i a^{6} c^{3} f^{2} e^{6 i e} - 1536 a^{6} c^{2} d f^{2} e^{6 i e} - 1536 i a^{6} c d^{2} f^{2} e^{6 i e} + 512 a^{6} d^{3} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{3} f^{2} e^{8 i e} - 2304 a^{6} c^{2} d f^{2} e^{8 i e} + 2304 i a^{6} c d^{2} f^{2} e^{8 i e} - 2304 a^{6} d^{3} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{3} f^{2} e^{10 i e} + 4608 a^{6} c^{2} d f^{2} e^{10 i e} + 4608 i a^{6} c d^{2} f^{2} e^{10 i e} + 4608 a^{6} d^{3} f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{24576 a^{9} f^{3}} & \text{for}\: 24576 a^{9} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac{c^{3} - 3 i c^{2} d - 3 c d^{2} + i d^{3}}{8 a^{3}} + \frac{\left (c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} - 3 i c^{2} d e^{6 i e} - 3 i c^{2} d e^{4 i e} + 3 i c^{2} d e^{2 i e} + 3 i c^{2} d - 3 c d^{2} e^{6 i e} + 3 c d^{2} e^{4 i e} + 3 c d^{2} e^{2 i e} - 3 c d^{2} + i d^{3} e^{6 i e} - 3 i d^{3} e^{4 i e} + 3 i d^{3} e^{2 i e} - i d^{3}\right ) e^{- 6 i e}}{8 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (c^{3} - 3 i c^{2} d - 3 c d^{2} + i d^{3}\right )}{8 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.99743, size = 386, normalized size = 2.76 \begin{align*} -\frac{\frac{6 \,{\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3}} + \frac{6 \,{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \log \left (i \, \tan \left (f x + e\right ) - 1\right )}{a^{3}} + \frac{-11 i \, c^{3} \tan \left (f x + e\right )^{3} - 33 \, c^{2} d \tan \left (f x + e\right )^{3} + 33 i \, c d^{2} \tan \left (f x + e\right )^{3} + 11 \, d^{3} \tan \left (f x + e\right )^{3} - 45 \, c^{3} \tan \left (f x + e\right )^{2} + 135 i \, c^{2} d \tan \left (f x + e\right )^{2} + 135 \, c d^{2} \tan \left (f x + e\right )^{2} + 51 i \, d^{3} \tan \left (f x + e\right )^{2} + 69 i \, c^{3} \tan \left (f x + e\right ) + 207 \, c^{2} d \tan \left (f x + e\right ) - 63 i \, c d^{2} \tan \left (f x + e\right ) + 75 \, d^{3} \tan \left (f x + e\right ) + 51 \, c^{3} - 57 i \, c^{2} d - 9 \, c d^{2} - 29 i \, d^{3}}{a^{3}{\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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