Optimal. Leaf size=84 \[ -\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}+\frac {i d x}{4 a f} \]
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Rubi [A] time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3723, 3479, 8} \[ -\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}+\frac {i d x}{4 a f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rule 3723
Rubi steps
\begin {align*} \int \frac {c+d x}{a+i a \cot (e+f x)} \, dx &=\frac {(c+d x)^2}{4 a d}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(i d) \int \frac {1}{a+i a \cot (e+f x)} \, dx}{2 f}\\ &=\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}+\frac {(i d) \int 1 \, dx}{4 a f}\\ &=\frac {i d x}{4 a f}+\frac {(c+d x)^2}{4 a d}+\frac {d}{4 f^2 (a+i a \cot (e+f x))}-\frac {i (c+d x)}{2 f (a+i a \cot (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 107, normalized size = 1.27 \[ \frac {(\cos (e+f x)+i \sin (e+f x)) \left (\left (2 c f (2 f x+i)+d \left (2 f^2 x^2+2 i f x-1\right )\right ) \cos (e+f x)-i \left (2 c f (2 f x-i)+d \left (2 f^2 x^2-2 i f x+1\right )\right ) \sin (e+f x)\right )}{8 a f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 48, normalized size = 0.57 \[ \frac {2 \, d f^{2} x^{2} + 4 \, c f^{2} x + {\left (2 i \, d f x + 2 i \, c f - d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, a f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 67, normalized size = 0.80 \[ \frac {2 \, d f^{2} x^{2} + 4 \, c f^{2} x + 2 i \, d f x e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, c f e^{\left (2 i \, f x + 2 i \, e\right )} - d e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, a f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 177, normalized size = 2.11 \[ \frac {d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )+c f \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-i d \left (-\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}+\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )+\frac {i c f \left (\cos ^{2}\left (f x +e \right )\right )}{2}-\frac {i d e \left (\cos ^{2}\left (f x +e \right )\right )}{2}}{f^{2} a} \]
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 = 0.38, size = 105, normalized size = 1.25 \[ -\frac {d\,\cos \left (2\,e+2\,f\,x\right )-2\,d\,f^2\,x^2+2\,c\,f\,\sin \left (2\,e+2\,f\,x\right )-4\,c\,f^2\,x+2\,d\,f\,x\,\sin \left (2\,e+2\,f\,x\right )+d\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-c\,f\,\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}-d\,f\,x\,\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}}{8\,a\,f^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 104, normalized size = 1.24 \[ \begin {cases} - \frac {\left (- 2 i c f e^{2 i e} - 2 i d f x e^{2 i e} + d e^{2 i e}\right ) e^{2 i f x}}{8 a f^{2}} & \text {for}\: 8 a f^{2} \neq 0 \\- \frac {c x e^{2 i e}}{2 a} - \frac {d x^{2} e^{2 i e}}{4 a} & \text {otherwise} \end {cases} + \frac {c x}{2 a} + \frac {d x^{2}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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