Integrand size = 13, antiderivative size = 64 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {3 \text {arctanh}(c+d x)}{8 d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 205, 212} \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=\frac {3 \text {arctanh}(c+d x)}{8 d}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2} \]
[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 )^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,c+d x\right )}{4 d} \\ & = \frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{8 d} \\ & = \frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {3 \tanh ^{-1}(c+d x)}{8 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=\frac {\frac {4 (c+d x)}{\left (-1+(c+d x)^2\right )^2}-\frac {6 (c+d x)}{-1+(c+d x)^2}-3 \log (1-c-d x)+3 \log (1+c+d x)}{16 d} \]
[In]
[Out]
Time = 0.84 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {1}{16 d \left (d x +c -1\right )^{2}}-\frac {3}{16 d \left (d x +c -1\right )}-\frac {3 \ln \left (d x +c -1\right )}{16 d}-\frac {1}{16 d \left (d x +c +1\right )^{2}}-\frac {3}{16 d \left (d x +c +1\right )}+\frac {3 \ln \left (d x +c +1\right )}{16 d}\) | \(78\) |
risch | \(\frac {-\frac {3 d^{2} x^{3}}{8}-\frac {9 x^{2} c d}{8}+\left (-\frac {9 c^{2}}{8}+\frac {5}{8}\right ) x -\frac {c \left (3 c^{2}-5\right )}{8 d}}{\left (d^{2} x^{2}+2 c d x +c^{2}-1\right )^{2}}-\frac {3 \ln \left (d x +c -1\right )}{16 d}+\frac {3 \ln \left (-d x -c -1\right )}{16 d}\) | \(87\) |
norman | \(\frac {\frac {-3 c^{3} d^{3}+5 d^{3} c}{8 d^{4}}+\frac {\left (-9 c^{2} d^{3}+5 d^{3}\right ) x}{8 d^{3}}-\frac {3 d^{2} x^{3}}{8}-\frac {9 x^{2} c d}{8}}{\left (d^{2} x^{2}+2 c d x +c^{2}-1\right )^{2}}-\frac {3 \ln \left (d x +c -1\right )}{16 d}+\frac {3 \ln \left (d x +c +1\right )}{16 d}\) | \(102\) |
parallelrisch | \(-\frac {3 \ln \left (d x +c -1\right ) c^{4} d^{3}+6 d^{6} x^{3}+18 x^{2} c \,d^{5}+18 x \,c^{2} d^{4}-6 \ln \left (d x +c -1\right ) c^{2} d^{3}+6 c^{3} d^{3}-10 d^{3} c +12 \ln \left (d x +c -1\right ) x^{3} c \,d^{6}-12 \ln \left (d x +c +1\right ) x^{3} c \,d^{6}+18 \ln \left (d x +c -1\right ) x^{2} c^{2} d^{5}-18 \ln \left (d x +c +1\right ) x^{2} c^{2} d^{5}+12 \ln \left (d x +c -1\right ) x \,c^{3} d^{4}-12 \ln \left (d x +c +1\right ) x \,c^{3} d^{4}-12 \ln \left (d x +c -1\right ) x c \,d^{4}+12 \ln \left (d x +c +1\right ) x c \,d^{4}+3 \ln \left (d x +c -1\right ) d^{3}-3 \ln \left (d x +c +1\right ) d^{3}-10 x \,d^{4}+6 \ln \left (d x +c +1\right ) c^{2} d^{3}+6 \ln \left (d x +c +1\right ) x^{2} d^{5}-3 \ln \left (d x +c +1\right ) c^{4} d^{3}+3 \ln \left (d x +c -1\right ) x^{4} d^{7}-6 \ln \left (d x +c -1\right ) x^{2} d^{5}-3 \ln \left (d x +c +1\right ) x^{4} d^{7}}{16 d^{4} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )^{2}}\) | \(344\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (54) = 108\).
Time = 0.26 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=-\frac {6 \, d^{3} x^{3} + 18 \, c d^{2} x^{2} + 6 \, c^{3} + 2 \, {\left (9 \, c^{2} - 5\right )} d x - 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c + 1\right ) + 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c - 1\right ) - 10 \, c}{16 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} - c\right )} d^{2} x + {\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (53) = 106\).
Time = 0.51 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=- \frac {3 c^{3} + 9 c d^{2} x^{2} - 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d - 5 d\right )}{8 c^{4} d - 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \cdot \left (48 c^{2} d^{3} - 16 d^{3}\right ) + x \left (32 c^{3} d^{2} - 32 c d^{2}\right )} - \frac {\frac {3 \log {\left (x + \frac {3 c - 3}{3 d} \right )}}{16} - \frac {3 \log {\left (x + \frac {3 c + 3}{3 d} \right )}}{16}}{d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (54) = 108\).
Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=-\frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} - 5\right )} d x - 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} - c\right )} d^{2} x + {\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} + \frac {3 \, \log \left (d x + c + 1\right )}{16 \, d} - \frac {3 \, \log \left (d x + c - 1\right )}{16 \, d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=\frac {3 \, \log \left ({\left | d x + c + 1 \right |}\right )}{16 \, d} - \frac {3 \, \log \left ({\left | d x + c - 1 \right |}\right )}{16 \, d} - \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} - 5 \, d x - 5 \, c}{8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )}^{2} d} \]
[In]
[Out]
Time = 10.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx=\frac {3\,\mathrm {atanh}\left (c+d\,x\right )}{8\,d}-\frac {x\,\left (\frac {9\,c^2}{8}-\frac {5}{8}\right )-\frac {5\,c-3\,c^3}{8\,d}+\frac {3\,d^2\,x^3}{8}+\frac {9\,c\,d\,x^2}{8}}{c^4-2\,c^2-x^2\,\left (2\,d^2-6\,c^2\,d^2\right )-x\,\left (4\,c\,d-4\,c^3\,d\right )+d^4\,x^4+4\,c\,d^3\,x^3+1} \]
[In]
[Out]