Integrand size = 13, antiderivative size = 10 \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\text {arctanh}(c+d x)}{d} \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {253, 212} \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\text {arctanh}(c+d x)}{d} \]
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Rule 212
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\tanh ^{-1}(c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(10)=20\).
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 3.20 \[ \int \frac {1}{1-(c+d x)^2} \, dx=-\frac {\log (1-c-d x)}{2 d}+\frac {\log (1+c+d x)}{2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(22\) vs. \(2(10)=20\).
Time = 0.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.30
method | result | size |
parallelrisch | \(-\frac {\ln \left (d x +c -1\right )-\ln \left (d x +c +1\right )}{2 d}\) | \(23\) |
default | \(-\frac {\ln \left (d x +c -1\right )}{2 d}+\frac {\ln \left (d x +c +1\right )}{2 d}\) | \(26\) |
norman | \(-\frac {\ln \left (d x +c -1\right )}{2 d}+\frac {\ln \left (d x +c +1\right )}{2 d}\) | \(26\) |
risch | \(-\frac {\ln \left (d x +c -1\right )}{2 d}+\frac {\ln \left (-d x -c -1\right )}{2 d}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.20 \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\log \left (d x + c + 1\right ) - \log \left (d x + c - 1\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (7) = 14\).
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.20 \[ \int \frac {1}{1-(c+d x)^2} \, dx=- \frac {\frac {\log {\left (x + \frac {c - 1}{d} \right )}}{2} - \frac {\log {\left (x + \frac {c + 1}{d} \right )}}{2}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.50 \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\log \left (d x + c + 1\right )}{2 \, d} - \frac {\log \left (d x + c - 1\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.70 \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\log \left ({\left | d x + c + 1 \right |}\right )}{2 \, d} - \frac {\log \left ({\left | d x + c - 1 \right |}\right )}{2 \, d} \]
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Time = 10.47 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-(c+d x)^2} \, dx=\frac {\mathrm {atanh}\left (c+d\,x\right )}{d} \]
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