Optimal. Leaf size=151 \[ -\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {x (c+d x)}{4 a^2}+\frac {3 d}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {3 i d x}{16 a^2 f}-\frac {d x^2}{8 a^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2} \]
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Rubi [A] time = 0.14, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3479, 8, 3730} \[ -\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {x (c+d x)}{4 a^2}+\frac {3 d}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {3 i d x}{16 a^2 f}-\frac {d x^2}{8 a^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2} \]
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))^2} \, dx &=\frac {x (c+d x)}{4 a^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}-d \int \left (\frac {x}{4 a^2}-\frac {i}{4 f (a+i a \cot (e+f x))^2}-\frac {i}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}\right ) \, dx\\ &=-\frac {d x^2}{8 a^2}+\frac {x (c+d x)}{4 a^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {(i d) \int \frac {1}{(a+i a \cot (e+f x))^2} \, dx}{4 f}+\frac {(i d) \int \frac {1}{a^2+i a^2 \cot (e+f x)} \, dx}{4 f}\\ &=-\frac {d x^2}{8 a^2}+\frac {x (c+d x)}{4 a^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {d}{8 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {(i d) \int 1 \, dx}{8 a^2 f}+\frac {(i d) \int \frac {1}{a+i a \cot (e+f x)} \, dx}{8 a f}\\ &=\frac {i d x}{8 a^2 f}-\frac {d x^2}{8 a^2}+\frac {x (c+d x)}{4 a^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {3 d}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}+\frac {(i d) \int 1 \, dx}{16 a^2 f}\\ &=\frac {3 i d x}{16 a^2 f}-\frac {d x^2}{8 a^2}+\frac {x (c+d x)}{4 a^2}+\frac {d}{16 f^2 (a+i a \cot (e+f x))^2}-\frac {i (c+d x)}{4 f (a+i a \cot (e+f x))^2}+\frac {3 d}{16 f^2 \left (a^2+i a^2 \cot (e+f x)\right )}-\frac {i (c+d x)}{4 f \left (a^2+i a^2 \cot (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 165, normalized size = 1.09 \[ \frac {8 i (2 c f+d (2 f x+i)) \cos (2 (e+f x))+(-4 i c f-4 i d f x+d) \cos (4 (e+f x))-16 c f \sin (2 (e+f x))+4 c f \sin (4 (e+f x))+16 c e f+16 c f^2 x-8 d e^2-8 i d \sin (2 (e+f x))-16 d f x \sin (2 (e+f x))+i d \sin (4 (e+f x))+4 d f x \sin (4 (e+f x))+8 d f^2 x^2}{64 a^2 f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 69, normalized size = 0.46 \[ \frac {8 \, d f^{2} x^{2} + 16 \, c f^{2} x + {\left (-4 i \, d f x - 4 i \, c f + d\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (16 i \, d f x + 16 i \, c f - 8 \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.73, size = 108, normalized size = 0.72 \[ \frac {8 \, d f^{2} x^{2} + 16 \, c f^{2} x - 4 i \, d f x e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} - 4 i \, c f e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} + d e^{\left (4 i \, f x + 4 i \, e\right )} - 8 \, d e^{\left (2 i \, f x + 2 i \, e\right )}}{64 \, a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.66, size = 390, normalized size = 2.58 \[ -\frac {\frac {2 i d \left (\frac {\left (f x +e \right ) \left (\sin ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{16}-\frac {3 f x}{32}-\frac {3 e}{32}\right )}{f}+\frac {i c \left (\sin ^{4}\left (f x +e \right )\right )}{2}-\frac {i d e \left (\sin ^{4}\left (f x +e \right )\right )}{2 f}+\frac {2 d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{16}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{16}-\left (f x +e \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {\left (\sin ^{4}\left (f x +e \right )\right )}{16}\right )}{f}+2 c \left (-\frac {\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{8}+\frac {f x}{8}+\frac {e}{8}\right )-\frac {2 e d \left (-\frac {\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{4}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{8}+\frac {f x}{8}+\frac {e}{8}\right )}{f}-\frac {d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}-c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {e d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}}{a^{2} f} \]
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 = 0.34, size = 102, normalized size = 0.68 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,a^2\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{4\,a^2\,f}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^2\,f^2}+\frac {d\,x\,1{}\mathrm {i}}{16\,a^2\,f}\right )+\frac {d\,x^2}{8\,a^2}+\frac {c\,x}{4\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 214, normalized size = 1.42 \[ \begin {cases} \frac {\left (128 i a^{2} c f^{3} e^{2 i e} + 128 i a^{2} d f^{3} x e^{2 i e} - 64 a^{2} d f^{2} e^{2 i e}\right ) e^{2 i f x} + \left (- 32 i a^{2} c f^{3} e^{4 i e} - 32 i a^{2} d f^{3} x e^{4 i e} + 8 a^{2} d f^{2} e^{4 i e}\right ) e^{4 i f x}}{512 a^{4} f^{4}} & \text {for}\: 512 a^{4} f^{4} \neq 0 \\\frac {x^{2} \left (d e^{4 i e} - 2 d e^{2 i e}\right )}{8 a^{2}} + \frac {x \left (c e^{4 i e} - 2 c e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c x}{4 a^{2}} + \frac {d x^{2}}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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