Integrand size = 18, antiderivative size = 53 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=-\frac {1}{2} i b d x-\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}-\frac {b d \log (i+c x)}{c} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4972, 641, 45} \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=-\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}-\frac {b d \log (c x+i)}{c}-\frac {1}{2} i b d x \]
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Rule 45
Rule 641
Rule 4972
Rubi steps \begin{align*} \text {integral}& = -\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}+\frac {(i b) \int \frac {(d+i c d x)^2}{1+c^2 x^2} \, dx}{2 d} \\ & = -\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}+\frac {(i b) \int \frac {d+i c d x}{\frac {1}{d}-\frac {i c x}{d}} \, dx}{2 d} \\ & = -\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}+\frac {(i b) \int \left (-d^2+\frac {2 i d^2}{i+c x}\right ) \, dx}{2 d} \\ & = -\frac {1}{2} i b d x-\frac {i d (1+i c x)^2 (a+b \arctan (c x))}{2 c}-\frac {b d \log (i+c x)}{c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=a d x-\frac {1}{2} i b d x+\frac {1}{2} i a c d x^2+\frac {i b d \arctan (c x)}{2 c}+b d x \arctan (c x)+\frac {1}{2} i b c d x^2 \arctan (c x)-\frac {b d \log \left (1+c^2 x^2\right )}{2 c} \]
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Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32
method | result | size |
parts | \(-i a d \left (-\frac {1}{2} c \,x^{2}+i x \right )+\frac {b d \left (\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )+\frac {i \left (-c x +i \ln \left (c^{2} x^{2}+1\right )+\arctan \left (c x \right )\right )}{2}\right )}{c}\) | \(70\) |
derivativedivides | \(\frac {-i a d \left (-\frac {1}{2} c^{2} x^{2}+i c x \right )+b d \left (\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )+\frac {i \left (-c x +i \ln \left (c^{2} x^{2}+1\right )+\arctan \left (c x \right )\right )}{2}\right )}{c}\) | \(74\) |
default | \(\frac {-i a d \left (-\frac {1}{2} c^{2} x^{2}+i c x \right )+b d \left (\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )+\frac {i \left (-c x +i \ln \left (c^{2} x^{2}+1\right )+\arctan \left (c x \right )\right )}{2}\right )}{c}\) | \(74\) |
parallelrisch | \(\frac {i b d \arctan \left (c x \right ) x^{2} c^{2}+i a \,c^{2} d \,x^{2}-i b d x c +2 b c d x \arctan \left (c x \right )+i b d \arctan \left (c x \right )+2 a c d x -b d \ln \left (c^{2} x^{2}+1\right )}{2 c}\) | \(79\) |
risch | \(\frac {b d \left (c \,x^{2}-2 i x \right ) \ln \left (i c x +1\right )}{4}+\frac {i a c d \,x^{2}}{2}-\frac {d c \,x^{2} b \ln \left (-i c x +1\right )}{4}+\frac {i b d x \ln \left (-i c x +1\right )}{2}-\frac {i b d x}{2}+\frac {i d b \arctan \left (c x \right )}{2 c}+a d x -\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}\) | \(102\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (41) = 82\).
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.68 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=\frac {2 i \, a c^{2} d x^{2} + 2 \, {\left (2 \, a - i \, b\right )} c d x - 3 \, b d \log \left (\frac {c x + i}{c}\right ) - b d \log \left (\frac {c x - i}{c}\right ) - {\left (b c^{2} d x^{2} - 2 i \, b c d x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{4 \, c} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (44) = 88\).
Time = 1.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.42 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=\frac {i a c d x^{2}}{2} + \frac {b d \left (- \frac {\log {\left (b c d x - i b d \right )}}{4} - \frac {5 \log {\left (b c d x + i b d \right )}}{12}\right )}{c} + x \left (a d - \frac {i b d}{2}\right ) + \left (\frac {b c d x^{2}}{4} - \frac {i b d x}{2}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (- 3 b c^{2} d x^{2} + 6 i b c d x - 4 b d\right ) \log {\left (- i c x + 1 \right )}}{12 c} \]
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none
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=\frac {1}{2} i \, a c d x^{2} + \frac {1}{2} i \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b c d + a d x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \]
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\[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int (d+i c d x) (a+b \arctan (c x)) \, dx=\frac {d\,\left (2\,a\,x+2\,b\,x\,\mathrm {atan}\left (c\,x\right )-b\,x\,1{}\mathrm {i}\right )}{2}+\frac {c\,d\,\left (a\,x^2\,1{}\mathrm {i}+b\,x^2\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}\right )}{2}+\frac {d\,\left (-b\,\ln \left (c^2\,x^2+1\right )+b\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}\right )}{2\,c} \]
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