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 = 0.136803, 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 \[ \sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(1 - x^2),x]
[Out]
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Rubi in Sympy [A] time = 22.4584, size = 46, normalized size = 0.79 \[ - \sqrt{c - d} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c - d}} \right )} + \sqrt{c + d} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c + d}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(-x**2+1),x)
[Out]
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Mathematica [A] time = 0.0488128, 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.
[In] Integrate[Sqrt[c + d*x]/(1 - x^2),x]
[Out]
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Maple [A] time = 0.023, size = 47, normalized size = 0.8 \[ -\sqrt{-c+d}\arctan \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{-c+d}}}} \right ) +{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c+d}}}} \right ) \sqrt{c+d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(-x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x + c)/(x^2 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231954, size = 1, normalized size = 0.02 \[ \left [\frac{1}{2} \, \sqrt{c - d} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c - d} + 2 \, c - d}{x + 1}\right ) + \frac{1}{2} \, \sqrt{c + d} \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt{-c + d} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c + d}}\right ) + \frac{1}{2} \, \sqrt{c + d} \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c + d} + 2 \, c + d}{x - 1}\right ), \sqrt{-c - d} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c - d}}\right ) + \frac{1}{2} \, \sqrt{c - d} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c - d} + 2 \, c - d}{x + 1}\right ), -\sqrt{-c + d} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c + d}}\right ) + \sqrt{-c - d} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c - d}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x + c)/(x^2 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.27425, size = 199, normalized size = 3.43 \[ \frac{2 \left (- \frac{d \left (c - d\right ) \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c + d}} \right )}}{\sqrt{- c + d}} & \text{for}\: - c + d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c - d}} \right )}}{\sqrt{c - d}} & \text{for}\: - c + d < 0 \wedge c - d < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c - d}} \right )}}{\sqrt{c - d}} & \text{for}\: c - d > c + d x \wedge - c + d < 0 \end{cases}\right )}{2} + \frac{d \left (c + d\right ) \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c - d}} \right )}}{\sqrt{- c - d}} & \text{for}\: - c - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c + d}} \right )}}{\sqrt{c + d}} & \text{for}\: - c - d < 0 \wedge c + d < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c + d}} \right )}}{\sqrt{c + d}} & \text{for}\: c + d > c + d x \wedge - c - d < 0 \end{cases}\right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(-x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.211769, size = 84, normalized size = 1.45 \[ \frac{{\left (c - d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c + d}}\right )}{\sqrt{-c + d}} - \frac{{\left (c + d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c - d}}\right )}{\sqrt{-c - d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(d*x + c)/(x^2 - 1),x, algorithm="giac")
[Out]