Optimal. Leaf size=58 \[ \sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right ) \]
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Rubi [A] time = 0.0580653, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {700, 1130, 206} \[ \sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right ) \]
Antiderivative was successfully verified.
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Rule 700
Rule 1130
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{1-x^2} \, dx &=(2 d) \operatorname{Subst}\left (\int \frac{x^2}{-c^2+d^2+2 c x^2-x^4} \, dx,x,\sqrt{c+d x}\right )\\ &=-\left ((c-d) \operatorname{Subst}\left (\int \frac{1}{c-d-x^2} \, dx,x,\sqrt{c+d x}\right )\right )+(c+d) \operatorname{Subst}\left (\int \frac{1}{c+d-x^2} \, dx,x,\sqrt{c+d x}\right )\\ &=-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right )+\sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )\\ \end{align*}
Mathematica [A] time = 0.0346304, size = 58, normalized size = 1. \[ \sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 47, normalized size = 0.8 \begin{align*}{\it Artanh} \left ({\sqrt{dx+c}{\frac{1}{\sqrt{c+d}}}} \right ) \sqrt{c+d}-\sqrt{-c+d}\arctan \left ({\sqrt{dx+c}{\frac{1}{\sqrt{-c+d}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40727, size = 741, normalized size = 12.78 \begin{align*} \left [\frac{1}{2} \, \sqrt{c - d} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c - d} + 2 \, c - d}{x + 1}\right ) + \frac{1}{2} \, \sqrt{c + d} \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt{-c + d} \arctan \left (-\frac{\sqrt{d x + c} \sqrt{-c + d}}{c - d}\right ) + \frac{1}{2} \, \sqrt{c + d} \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt{-c - d} \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c - d}}{c + d}\right ) + \frac{1}{2} \, \sqrt{c - d} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c - d} + 2 \, c - d}{x + 1}\right ), -\sqrt{-c + d} \arctan \left (-\frac{\sqrt{d x + c} \sqrt{-c + d}}{c - d}\right ) - \sqrt{-c - d} \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c - d}}{c + d}\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.65656, size = 61, normalized size = 1.05 \begin{align*} \frac{2 \left (\frac{d \left (c - d\right ) \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d}} \right )}}{2 \sqrt{- c + d}} - \frac{d \left (c + d\right ) \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c - d}} \right )}}{2 \sqrt{- c - d}}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36741, size = 84, normalized size = 1.45 \begin{align*} \frac{{\left (c - d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c + d}}\right )}{\sqrt{-c + d}} - \frac{{\left (c + d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c - d}}\right )}{\sqrt{-c - d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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