Integrand size = 13, antiderivative size = 39 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {c+d x}{2 d \left (1-(c+d x)^2\right )}+\frac {\text {arctanh}(c+d x)}{2 d} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 205, 212} \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {\text {arctanh}(c+d x)}{2 d}+\frac {c+d x}{2 d \left (1-(c+d x)^2\right )} \]
[In]
[Out]
Rule 205
Rule 212
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 {\tanh ^{-1}(c+d x)}{2 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {-\frac {2 (c+d x)}{-1+(c+d x)^2}-\log (1-c-d x)+\log (1+c+d x)}{4 d} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.33
method | result | size |
default | \(-\frac {1}{4 d \left (d x +c -1\right )}-\frac {\ln \left (d x +c -1\right )}{4 d}-\frac {1}{4 d \left (d x +c +1\right )}+\frac {\ln \left (d x +c +1\right )}{4 d}\) | \(52\) |
norman | \(\frac {-\frac {c}{2 d}-\frac {x}{2}}{d^{2} x^{2}+2 c d x +c^{2}-1}-\frac {\ln \left (d x +c -1\right )}{4 d}+\frac {\ln \left (d x +c +1\right )}{4 d}\) | \(56\) |
risch | \(\frac {-\frac {c}{2 d}-\frac {x}{2}}{d^{2} x^{2}+2 c d x +c^{2}-1}-\frac {\ln \left (d x +c -1\right )}{4 d}+\frac {\ln \left (-d x -c -1\right )}{4 d}\) | \(59\) |
parallelrisch | \(-\frac {\ln \left (d x +c -1\right ) x^{2} d^{3}-\ln \left (d x +c +1\right ) x^{2} d^{3}+2 \ln \left (d x +c -1\right ) x c \,d^{2}-2 \ln \left (d x +c +1\right ) x c \,d^{2}+\ln \left (d x +c -1\right ) c^{2} d -\ln \left (d x +c +1\right ) c^{2} d +2 d^{2} x -\ln \left (d x +c -1\right ) d +\ln \left (d x +c +1\right ) d +2 c d}{4 d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\) | \(137\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (33) = 66\).
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=-\frac {2 \, d x - {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c + 1\right ) + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c - 1\right ) + 2 \, c}{4 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} - 1\right )} d\right )}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {- c - d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} - 2 d} + \frac {- \frac {\log {\left (x + \frac {c - 1}{d} \right )}}{4} + \frac {\log {\left (x + \frac {c + 1}{d} \right )}}{4}}{d} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=-\frac {d x + c}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} - 1\right )} d\right )}} + \frac {\log \left (d x + c + 1\right )}{4 \, d} - \frac {\log \left (d x + c - 1\right )}{4 \, d} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {\log \left ({\left | d x + c + 1 \right |}\right )}{4 \, d} - \frac {\log \left ({\left | d x + c - 1 \right |}\right )}{4 \, d} - \frac {d x + c}{2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} d} \]
[In]
[Out]
Time = 10.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx=\frac {\mathrm {atanh}\left (c+d\,x\right )}{2\,d}-\frac {\frac {x}{2}+\frac {c}{2\,d}}{c^2+2\,c\,d\,x+d^2\,x^2-1} \]
[In]
[Out]