Optimal. Leaf size=209 \[ \frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{x (c+d x)}{8 a^3}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 0.263967, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3479, 8, 3730} \[ \frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{x (c+d x)}{8 a^3}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rule 3730
Rubi steps
\begin{align*} \int \frac{c+d x}{(a+i a \tan (e+f x))^3} \, dx &=\frac{x (c+d x)}{8 a^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-d \int \left (\frac{x}{8 a^3}+\frac{i}{6 f (a+i a \tan (e+f x))^3}+\frac{i}{8 a f (a+i a \tan (e+f x))^2}+\frac{i}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\right ) \, dx\\ &=-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{(i d) \int \frac{1}{a^3+i a^3 \tan (e+f x)} \, dx}{8 f}-\frac{(i d) \int \frac{1}{(a+i a \tan (e+f x))^3} \, dx}{6 f}-\frac{(i d) \int \frac{1}{(a+i a \tan (e+f x))^2} \, dx}{8 a f}\\ &=-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{d}{32 a f^2 (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{d}{16 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{(i d) \int 1 \, dx}{16 a^3 f}-\frac{(i d) \int \frac{1}{a+i a \tan (e+f x)} \, dx}{16 a^2 f}-\frac{(i d) \int \frac{1}{(a+i a \tan (e+f x))^2} \, dx}{12 a f}\\ &=-\frac{i d x}{16 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{3 d}{32 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{(i d) \int 1 \, dx}{32 a^3 f}-\frac{(i d) \int \frac{1}{a+i a \tan (e+f x)} \, dx}{24 a^2 f}\\ &=-\frac{3 i d x}{32 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{(i d) \int 1 \, dx}{48 a^3 f}\\ &=-\frac{11 i d x}{96 a^3 f}-\frac{d x^2}{16 a^3}+\frac{x (c+d x)}{8 a^3}+\frac{d}{36 f^2 (a+i a \tan (e+f x))^3}+\frac{i (c+d x)}{6 f (a+i a \tan (e+f x))^3}+\frac{5 d}{96 a f^2 (a+i a \tan (e+f x))^2}+\frac{i (c+d x)}{8 a f (a+i a \tan (e+f x))^2}+\frac{11 d}{96 f^2 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i (c+d x)}{8 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.671978, size = 205, normalized size = 0.98 \[ \frac{i \sec ^3(e+f x) \left (4 \left (6 c f (6 f x+i)+d \left (18 f^2 x^2+6 i f x+1\right )\right ) \cos (3 (e+f x))+27 (12 i c f+d (5+12 i f x)) \cos (e+f x)+144 i c f^2 x \sin (3 (e+f x))-108 c f \sin (e+f x)+24 c f \sin (3 (e+f x))+72 i d f^2 x^2 \sin (3 (e+f x))-108 d f x \sin (e+f x)+24 d f x \sin (3 (e+f x))+81 i d \sin (e+f x)-4 i d \sin (3 (e+f x))\right )}{1152 a^3 f^2 (\tan (e+f x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.275, size = 114, normalized size = 0.6 \begin{align*}{\frac{d{x}^{2}}{16\,{a}^{3}}}+{\frac{cx}{8\,{a}^{3}}}+{\frac{{\frac{3\,i}{32}} \left ( 2\,dfx-id+2\,cf \right ){{\rm e}^{-2\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{2}}}+{\frac{{\frac{3\,i}{128}} \left ( 4\,dfx-id+4\,cf \right ){{\rm e}^{-4\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{2}}}+{\frac{{\frac{i}{288}} \left ( 6\,dfx-id+6\,cf \right ){{\rm e}^{-6\,i \left ( fx+e \right ) }}}{{a}^{3}{f}^{2}}} \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 = 1.57334, size = 309, normalized size = 1.48 \begin{align*} \frac{{\left (24 i \, d f x + 24 i \, c f + 72 \,{\left (d f^{2} x^{2} + 2 \, c f^{2} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (216 i \, d f x + 216 i \, c f + 108 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (108 i \, d f x + 108 i \, c f + 27 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{1152 \, a^{3} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.29434, size = 313, normalized size = 1.5 \begin{align*} \begin{cases} \frac{\left (\left (24576 i a^{15} c f^{8} e^{18 i e} + 24576 i a^{15} d f^{8} x e^{18 i e} + 4096 a^{15} d f^{7} e^{18 i e}\right ) e^{- 6 i f x} + \left (110592 i a^{15} c f^{8} e^{20 i e} + 110592 i a^{15} d f^{8} x e^{20 i e} + 27648 a^{15} d f^{7} e^{20 i e}\right ) e^{- 4 i f x} + \left (221184 i a^{15} c f^{8} e^{22 i e} + 221184 i a^{15} d f^{8} x e^{22 i e} + 110592 a^{15} d f^{7} e^{22 i e}\right ) e^{- 2 i f x}\right ) e^{- 24 i e}}{1179648 a^{18} f^{9}} & \text{for}\: 1179648 a^{18} f^{9} e^{24 i e} \neq 0 \\\frac{x^{2} \left (3 d e^{4 i e} + 3 d e^{2 i e} + d\right ) e^{- 6 i e}}{16 a^{3}} + \frac{x \left (3 c e^{4 i e} + 3 c e^{2 i e} + c\right ) e^{- 6 i e}}{8 a^{3}} & \text{otherwise} \end{cases} + \frac{c x}{8 a^{3}} + \frac{d x^{2}}{16 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18024, size = 204, normalized size = 0.98 \begin{align*} \frac{{\left (72 \, d f^{2} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 144 \, c f^{2} x e^{\left (6 i \, f x + 6 i \, e\right )} + 216 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 108 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, d f x + 216 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 108 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f + 108 \, d e^{\left (4 i \, f x + 4 i \, e\right )} + 27 \, d e^{\left (2 i \, f x + 2 i \, e\right )} + 4 \, d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{1152 \, a^{3} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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