Optimal. Leaf size=209 \[ -\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}+\frac {d}{36 f^2 (a+i a \cot (e+f x))^3} \]
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Rubi [A] time = 0.22, antiderivative size = 209, normalized size of antiderivative = 1.00, 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} \[ -\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {x (c+d x)}{8 a^3}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {11 i d x}{96 a^3 f}-\frac {d x^2}{16 a^3}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}+\frac {d}{36 f^2 (a+i a \cot (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rule 3730
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+i a \cot (e+f x))^3} \, dx &=\frac {x (c+d x)}{8 a^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}-d \int \left (\frac {x}{8 a^3}-\frac {i}{6 f (a+i a \cot (e+f x))^3}-\frac {i}{8 a f (a+i a \cot (e+f x))^2}-\frac {i}{8 f \left (a^3+i a^3 \cot (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 \cot (e+f x))^3}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {(i d) \int \frac {1}{a^3+i a^3 \cot (e+f x)} \, dx}{8 f}+\frac {(i d) \int \frac {1}{(a+i a \cot (e+f x))^3} \, dx}{6 f}+\frac {(i d) \int \frac {1}{(a+i a \cot (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 \cot (e+f x))^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {d}{32 a f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac {d}{16 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {(i d) \int 1 \, dx}{16 a^3 f}+\frac {(i d) \int \frac {1}{a+i a \cot (e+f x)} \, dx}{16 a^2 f}+\frac {(i d) \int \frac {1}{(a+i a \cot (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 \cot (e+f x))^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac {3 d}{32 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}+\frac {(i d) \int 1 \, dx}{32 a^3 f}+\frac {(i d) \int \frac {1}{a+i a \cot (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 \cot (e+f x))^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (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 \cot (e+f x))^3}-\frac {i (c+d x)}{6 f (a+i a \cot (e+f x))^3}+\frac {5 d}{96 a f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{8 a f (a+i a \cot (e+f x))^2}+\frac {11 d}{96 f^2 \left (a^3+i a^3 \cot (e+f x)\right )}-\frac {i (c+d x)}{8 f \left (a^3+i a^3 \cot (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 244, normalized size = 1.17 \[ \frac {108 i (2 c f+d (2 f x+i)) \cos (2 (e+f x))+27 (-4 i c f-4 i d f x+d) \cos (4 (e+f x))-216 c f \sin (2 (e+f x))+108 c f \sin (4 (e+f x))-24 c f \sin (6 (e+f x))+24 i c f \cos (6 (e+f x))+144 c e f+144 c f^2 x-72 d e^2-108 i d \sin (2 (e+f x))-216 d f x \sin (2 (e+f x))+27 i d \sin (4 (e+f x))+108 d f x \sin (4 (e+f x))-4 i d \sin (6 (e+f x))-24 d f x \sin (6 (e+f x))-4 d \cos (6 (e+f x))+24 i d f x \cos (6 (e+f x))+72 d f^2 x^2}{1152 a^3 f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 94, normalized size = 0.45 \[ \frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x + {\left (24 i \, d f x + 24 i \, c f - 4 \, d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-108 i \, d f x - 108 i \, c f + 27 \, d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (216 i \, d f x + 216 i \, c f - 108 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.08, size = 151, normalized size = 0.72 \[ \frac {72 \, d f^{2} x^{2} + 144 \, c f^{2} x + 24 i \, d f x e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c f e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 216 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, d e^{\left (6 i \, f x + 6 i \, e\right )} + 27 \, d e^{\left (4 i \, f x + 4 i \, e\right )} - 108 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.06, size = 653, normalized size = 3.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 144, normalized size = 0.69 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{16\,a^3\,f}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (12\,c\,f+d\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{128\,a^3\,f^2}+\frac {d\,x\,3{}\mathrm {i}}{32\,a^3\,f}\right )+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{288\,a^3\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{48\,a^3\,f}\right )+\frac {d\,x^2}{16\,a^3}+\frac {c\,x}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 301, normalized size = 1.44 \[ \begin {cases} - \frac {\left (- 221184 i a^{6} c f^{5} e^{2 i e} - 221184 i a^{6} d f^{5} x e^{2 i e} + 110592 a^{6} d f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (110592 i a^{6} c f^{5} e^{4 i e} + 110592 i a^{6} d f^{5} x e^{4 i e} - 27648 a^{6} d f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (- 24576 i a^{6} c f^{5} e^{6 i e} - 24576 i a^{6} d f^{5} x e^{6 i e} + 4096 a^{6} d f^{4} e^{6 i e}\right ) e^{6 i f x}}{1179648 a^{9} f^{6}} & \text {for}\: 1179648 a^{9} f^{6} \neq 0 \\\frac {x^{2} \left (- d e^{6 i e} + 3 d e^{4 i e} - 3 d e^{2 i e}\right )}{16 a^{3}} + \frac {x \left (- c e^{6 i e} + 3 c e^{4 i e} - 3 c e^{2 i e}\right )}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c x}{8 a^{3}} + \frac {d x^{2}}{16 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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