Optimal. Leaf size=39 \[ \frac {c+d x}{2 d \left (1-(c+d x)^2\right )}+\frac {\tanh ^{-1}(c+d x)}{2 d} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 199, 206} \[ \frac {c+d x}{2 d \left (1-(c+d x)^2\right )}+\frac {\tanh ^{-1}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (1-(c+d x)^2\right )^2} \, dx &=\frac {\operatorname {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 {\operatorname {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*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 1.15 \[ \frac {-\frac {2 (c+d x)}{(c+d x)^2-1}-\log (-c-d x+1)+\log (c+d x+1)}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 85, normalized size = 2.18 \[ -\frac {2 \, d x - {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c + 1\right ) + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c - 1\right ) + 2 \, c}{4 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} - 1\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 56, normalized size = 1.44 \[ \frac {\log \left ({\left | d x + c + 1 \right |}\right )}{4 \, d} - \frac {\log \left ({\left | d x + c - 1 \right |}\right )}{4 \, d} - \frac {d x + c}{2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 1.33 \[ -\frac {\ln \left (d x +c -1\right )}{4 d}+\frac {\ln \left (d x +c +1\right )}{4 d}-\frac {1}{4 \left (d x +c -1\right ) d}-\frac {1}{4 \left (d x +c +1\right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 56, normalized size = 1.44 \[ -\frac {d x + c}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + {\left (c^{2} - 1\right )} d\right )}} + \frac {\log \left (d x + c + 1\right )}{4 \, d} - \frac {\log \left (d x + c - 1\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.06, size = 43, normalized size = 1.10 \[ \frac {\mathrm {atanh}\left (c+d\,x\right )}{2\,d}-\frac {\frac {x}{2}+\frac {c}{2\,d}}{c^2+2\,c\,d\,x+d^2\,x^2-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 54, normalized size = 1.38 \[ \frac {- c - d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} - 2 d} + \frac {- \frac {\log {\left (x + \frac {c - 1}{d} \right )}}{4} + \frac {\log {\left (x + \frac {c + 1}{d} \right )}}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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