Optimal. Leaf size=234 \[ \frac{c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{x \left (-6 c^2 d^2+4 i c^3 d+c^4-4 i c d^3-7 d^4\right )}{8 a^3 (c-i d) (c+i d)^4}+\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac{-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 0.672629, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3559, 3596, 3531, 3530} \[ \frac{c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{x \left (-6 c^2 d^2+4 i c^3 d+c^4-4 i c d^3-7 d^4\right )}{8 a^3 (c-i d) (c+i d)^4}+\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac{-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac{\int \frac{-3 a (i c-2 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac{\int \frac{-6 a^2 \left (c^2+3 i c d-4 d^2\right )-6 a^2 (c+3 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{\int \frac{6 a^3 \left (i c^3-4 c^2 d-7 i c d^2+8 d^3\right )+6 a^3 d \left (i c^2-4 c d-7 i d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac{\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{d^4 \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (c+i d)^4 (i c+d)}\\ &=\frac{\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.81287, size = 435, normalized size = 1.86 \[ \frac{\sec ^3(e+f x) \left (42 i c^2 d^2 \sin (e+f x)+72 c^2 d^2 f x \sin (3 (e+f x))-3 \left (-18 c^2 d^2+28 i c^3 d+9 c^4+28 i c d^3-27 d^4\right ) \cos (e+f x)+2 \cos (3 (e+f x)) \left (-36 i c^2 d^2 f x-2 c^3 d (12 f x+i)+c^4 (-1+6 i f x)+24 d^4 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+2 c d^3 (12 f x-i)+d^4 (1-42 i f x)\right )+36 c^3 d \sin (e+f x)-4 c^3 d \sin (3 (e+f x))-48 i c^3 d f x \sin (3 (e+f x))-9 i c^4 \sin (e+f x)-12 c^4 f x \sin (3 (e+f x))+2 i c^4 \sin (3 (e+f x))+36 c d^3 \sin (e+f x)-4 c d^3 \sin (3 (e+f x))+48 i c d^3 f x \sin (3 (e+f x))+48 i d^4 \sin (3 (e+f x)) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+51 i d^4 \sin (e+f x)-2 i d^4 \sin (3 (e+f x))+84 d^4 f x \sin (3 (e+f x))\right )}{96 a^3 f (c-i d) (c+i d)^4 (\tan (e+f x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 564, normalized size = 2.4 \begin{align*} \text{result too large to display} \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 = 1.73996, size = 667, normalized size = 2.85 \begin{align*} \frac{{\left (96 \, d^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - 2 \, c^{4} - 4 i \, c^{3} d - 4 i \, c d^{3} + 2 \, d^{4} +{\left (12 i \, c^{4} - 48 \, c^{3} d - 72 i \, c^{2} d^{2} + 48 \, c d^{3} - 180 i \, d^{4}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} -{\left (18 \, c^{4} + 60 i \, c^{3} d - 48 \, c^{2} d^{2} + 60 i \, c d^{3} - 66 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} -{\left (9 \, c^{4} + 24 i \, c^{3} d - 6 \, c^{2} d^{2} + 24 i \, c d^{3} - 15 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{{\left (96 i \, a^{3} c^{5} - 288 \, a^{3} c^{4} d - 192 i \, a^{3} c^{3} d^{2} - 192 \, a^{3} c^{2} d^{3} - 288 i \, a^{3} c d^{4} + 96 \, a^{3} d^{5}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50288, size = 603, normalized size = 2.58 \begin{align*} \frac{2 \,{\left (-\frac{i \, d^{5} \log \left (-i \, d \tan \left (f x + e\right ) - i \, c\right )}{2 \, a^{3} c^{5} d + 6 i \, a^{3} c^{4} d^{2} - 4 \, a^{3} c^{3} d^{3} + 4 i \, a^{3} c^{2} d^{4} - 6 \, a^{3} c d^{5} - 2 i \, a^{3} d^{6}} + \frac{{\left (-i \, c^{3} + 5 \, c^{2} d + 11 i \, c d^{2} - 15 \, d^{3}\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{32 \, a^{3} c^{4} + 128 i \, a^{3} c^{3} d - 192 \, a^{3} c^{2} d^{2} - 128 i \, a^{3} c d^{3} + 32 \, a^{3} d^{4}} + \frac{\log \left (\tan \left (f x + e\right ) + i\right )}{-32 i \, a^{3} c - 32 \, a^{3} d} + \frac{11 i \, c^{3} \tan \left (f x + e\right )^{3} - 55 \, c^{2} d \tan \left (f x + e\right )^{3} - 121 i \, c d^{2} \tan \left (f x + e\right )^{3} + 165 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 225 i \, c^{2} d \tan \left (f x + e\right )^{2} - 495 \, c d^{2} \tan \left (f x + e\right )^{2} - 579 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 345 \, c^{2} d \tan \left (f x + e\right ) + 711 i \, c d^{2} \tan \left (f x + e\right ) - 699 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 223 i \, c^{2} d + 385 \, c d^{2} + 301 i \, d^{3}}{{\left (192 \, a^{3} c^{4} + 768 i \, a^{3} c^{3} d - 1152 \, a^{3} c^{2} d^{2} - 768 i \, a^{3} c d^{3} + 192 \, a^{3} d^{4}\right )}{\left (\tan \left (f x + e\right ) - i\right )}^{3}}\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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