Optimal. Leaf size=10 \[ \frac {\tanh ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {247, 206} \[ \frac {\tanh ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{1-(c+d x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\tanh ^{-1}(c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 32, normalized size = 3.20 \[ \frac {\log (c+d x+1)}{2 d}-\frac {\log (-c-d x+1)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 22, normalized size = 2.20 \[ \frac {\log \left (d x + c + 1\right ) - \log \left (d x + c - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 27, normalized size = 2.70 \[ \frac {\log \left ({\left | d x + c + 1 \right |}\right )}{2 \, d} - \frac {\log \left ({\left | d x + c - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 26, normalized size = 2.60 \[ -\frac {\ln \left (d x +c -1\right )}{2 d}+\frac {\ln \left (d x +c +1\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 25, normalized size = 2.50 \[ \frac {\log \left (d x + c + 1\right )}{2 \, d} - \frac {\log \left (d x + c - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.05, size = 10, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 22, normalized size = 2.20 \[ - \frac {\frac {\log {\left (x + \frac {c - 1}{d} \right )}}{2} - \frac {\log {\left (x + \frac {c + 1}{d} \right )}}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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