Integrand size = 11, antiderivative size = 37 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {c+d x}{2 d \left (1+(c+d x)^2\right )}+\frac {\arctan (c+d x)}{2 d} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 205, 209} \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {\arctan (c+d x)}{2 d}+\frac {c+d x}{2 d \left ((c+d x)^2+1\right )} \]
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Rule 205
Rule 209
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {c+d x}{2 d \left (1+(c+d x)^2\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{2 d} \\ & = \frac {c+d x}{2 d \left (1+(c+d x)^2\right )}+\frac {\tan ^{-1}(c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {\frac {c+d x}{1+(c+d x)^2}+\arctan (c+d x)}{2 d} \]
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Time = 0.92 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {\frac {x}{2}+\frac {c}{2 d}}{d^{2} x^{2}+2 c d x +c^{2}+1}+\frac {\arctan \left (d x +c \right )}{2 d}\) | \(43\) |
default | \(\frac {2 d^{2} x +2 c d}{4 d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}+\frac {\arctan \left (\frac {2 d^{2} x +2 c d}{2 d}\right )}{2 d}\) | \(59\) |
parallelrisch | \(-\frac {i \ln \left (d x +c -i\right ) x^{2} d^{3}-i \ln \left (d x +c +i\right ) x^{2} d^{3}+2 i \ln \left (d x +c -i\right ) x c \,d^{2}-2 i \ln \left (d x +c +i\right ) x c \,d^{2}+i \ln \left (d x +c -i\right ) c^{2} d -i \ln \left (d x +c +i\right ) c^{2} d +i \ln \left (d x +c -i\right ) d -i \ln \left (d x +c +i\right ) d -2 d^{2} x -2 c d}{4 d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}\) | \(156\) |
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Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {d x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \arctan \left (d x + c\right ) + c}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} + 1\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {c + d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} + 2 d} + \frac {- \frac {i \log {\left (x + \frac {c - i}{d} \right )}}{4} + \frac {i \log {\left (x + \frac {c + i}{d} \right )}}{4}}{d} \]
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Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {d x + c}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} + 1\right )} d\right )}} + \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{2 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {\arctan \left (d x + c\right )}{2 \, d} + \frac {d x + c}{2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} d} \]
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Time = 10.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^2} \, dx=\frac {\frac {x}{2}+\frac {c}{2\,d}}{c^2+2\,c\,d\,x+d^2\,x^2+1}+\frac {\mathrm {atan}\left (c+d\,x\right )}{2\,d} \]
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