Integrand size = 23, antiderivative size = 434 \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=-\frac {1}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i f \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \cos \left (4 e-\frac {4 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {f \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2} \]
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Time = 0.85 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3809, 3378, 3384, 3380, 3383, 3394, 12} \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=-\frac {f \operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {i f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {i f \operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {1}{4 a^2 d (c+d x)} \]
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Rule 12
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3394
Rule 3809
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 a^2 (c+d x)^2}-\frac {\cos (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac {\cos ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}-\frac {i \sin (2 e+2 f x)}{2 a^2 (c+d x)^2}-\frac {\sin ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}+\frac {i \sin (4 e+4 f x)}{4 a^2 (c+d x)^2}\right ) \, dx \\ & = -\frac {1}{4 a^2 d (c+d x)}+\frac {i \int \frac {\sin (4 e+4 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac {i \int \frac {\sin (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2}+\frac {\int \frac {\cos ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac {\int \frac {\sin ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac {\int \frac {\cos (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2} \\ & = -\frac {1}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {(i f) \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{a^2 d}+\frac {(i f) \int \frac {\cos (4 e+4 f x)}{c+d x} \, dx}{a^2 d}+\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{a^2 d}+\frac {f \int -\frac {\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d}-\frac {f \int \frac {\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d} \\ & = -\frac {1}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}-2 \frac {f \int \frac {\sin (4 e+4 f x)}{c+d x} \, dx}{2 a^2 d}+\frac {\left (i f \cos \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 d}-\frac {\left (i f \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a^2 d}+\frac {\left (f \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a^2 d}-\frac {\left (i f \sin \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 d}+\frac {\left (i f \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a^2 d}+\frac {\left (f \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a^2 d} \\ & = -\frac {1}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i f \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \cos \left (4 e-\frac {4 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+\frac {f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-2 \left (\frac {\left (f \cos \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}+\frac {\left (f \sin \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}\right ) \\ & = -\frac {1}{4 a^2 d (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cos ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i f \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \cos \left (4 e-\frac {4 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+\frac {f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {i \sin (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sin ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-2 \left (\frac {f \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{2 a^2 d^2}+\frac {f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^2 d^2}\right ) \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=\frac {-d+2 d (\cos (2 (e+f x))+i \sin (2 (e+f x)))-d (\cos (4 (e+f x))+i \sin (4 (e+f x)))+4 f (c+d x) \left (-i \cos \left (2 e-\frac {2 c f}{d}\right )+\sin \left (2 e-\frac {2 c f}{d}\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )+(c+d x) \left (4 i f \cos \left (4 e-\frac {4 c f}{d}\right )-4 f \sin \left (4 e-\frac {4 c f}{d}\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {4 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d^2 (c+d x)} \]
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Time = 0.82 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {1}{4 a^{2} d \left (d x +c \right )}+\frac {i f \,{\mathrm e}^{2 i \left (f x +e \right )}}{2 a^{2} d^{2} \left (i f x +\frac {i c f}{d}\right )}+\frac {i f \,{\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{a^{2} d^{2}}-\frac {i f \,{\mathrm e}^{4 i \left (f x +e \right )}}{4 a^{2} d^{2} \left (i f x +\frac {i c f}{d}\right )}-\frac {i f \,{\mathrm e}^{-\frac {4 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-4 i f x -4 i e -\frac {4 \left (i c f -i d e \right )}{d}\right )}{a^{2} d^{2}}\) | \(193\) |
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Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.30 \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=-\frac {4 \, {\left (i \, d f x + i \, c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} + 4 \, {\left (-i \, d f x - i \, c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} + d e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, d e^{\left (2 i \, f x + 2 i \, e\right )} + d}{4 \, {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )}} \]
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\[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=- \frac {\int \frac {1}{c^{2} \cot ^{2}{\left (e + f x \right )} - 2 i c^{2} \cot {\left (e + f x \right )} - c^{2} + 2 c d x \cot ^{2}{\left (e + f x \right )} - 4 i c d x \cot {\left (e + f x \right )} - 2 c d x + d^{2} x^{2} \cot ^{2}{\left (e + f x \right )} - 2 i d^{2} x^{2} \cot {\left (e + f x \right )} - d^{2} x^{2}}\, dx}{a^{2}} \]
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Time = 0.44 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=-\frac {f^{2} \cos \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - 2 \, f^{2} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + 2 i \, f^{2} E_{2}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - i \, f^{2} E_{2}\left (\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + f^{2}}{4 \, {\left ({\left (f x + e\right )} a^{2} d^{2} - a^{2} d^{2} e + a^{2} c d f\right )} f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2249 vs. \(2 (410) = 820\).
Time = 1.44 (sec) , antiderivative size = 2249, normalized size of antiderivative = 5.18 \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]
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