3.640 \(\int \frac{\sqrt{c+d x}}{1-x^2} \, dx\)

Optimal. Leaf size=58 \[ \sqrt{c+d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+d}}\right )-\sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-d}}\right ) \]

[Out]

-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c +
d*x]/Sqrt[c + d]]

_______________________________________________________________________________________

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]

-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c +
d*x]/Sqrt[c + d]]

_______________________________________________________________________________________

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]

-sqrt(c - d)*atanh(sqrt(c + d*x)/sqrt(c - d)) + sqrt(c + d)*atanh(sqrt(c + d*x)/
sqrt(c + d))

_______________________________________________________________________________________

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]

-(Sqrt[c - d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - d]]) + Sqrt[c + d]*ArcTanh[Sqrt[c +
d*x]/Sqrt[c + d]]

_______________________________________________________________________________________

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]

-(-c+d)^(1/2)*arctan((d*x+c)^(1/2)/(-c+d)^(1/2))+arctanh((d*x+c)^(1/2)/(c+d)^(1/
2))*(c+d)^(1/2)

_______________________________________________________________________________________

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]

Exception raised: ValueError

_______________________________________________________________________________________

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]

[1/2*sqrt(c - d)*log((d*x - 2*sqrt(d*x + c)*sqrt(c - d) + 2*c - d)/(x + 1)) + 1/
2*sqrt(c + d)*log((d*x + 2*sqrt(d*x + c)*sqrt(c + d) + 2*c + d)/(x - 1)), -sqrt(
-c + d)*arctan(sqrt(d*x + c)/sqrt(-c + d)) + 1/2*sqrt(c + d)*log((d*x + 2*sqrt(d
*x + c)*sqrt(c + d) + 2*c + d)/(x - 1)), sqrt(-c - d)*arctan(sqrt(d*x + c)/sqrt(
-c - d)) + 1/2*sqrt(c - d)*log((d*x - 2*sqrt(d*x + c)*sqrt(c - d) + 2*c - d)/(x
+ 1)), -sqrt(-c + d)*arctan(sqrt(d*x + c)/sqrt(-c + d)) + sqrt(-c - d)*arctan(sq
rt(d*x + c)/sqrt(-c - d))]

_______________________________________________________________________________________

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]

2*(-d*(c - d)*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c + d))/sqrt(-c + d), -c + d
> 0), (acoth(sqrt(c + d*x)/sqrt(c - d))/sqrt(c - d), (-c + d < 0) & (c - d < c +
d*x)), (atanh(sqrt(c + d*x)/sqrt(c - d))/sqrt(c - d), (-c + d < 0) & (c - d > c
+ d*x)))/2 + d*(c + d)*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c - d))/sqrt(-c - d
), -c - d > 0), (acoth(sqrt(c + d*x)/sqrt(c + d))/sqrt(c + d), (-c - d < 0) & (c
+ d < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c + d))/sqrt(c + d), (c + d > c + d*
x) & (-c - d < 0)))/2)/d

_______________________________________________________________________________________

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]

(c - d)*arctan(sqrt(d*x + c)/sqrt(-c + d))/sqrt(-c + d) - (c + d)*arctan(sqrt(d*
x + c)/sqrt(-c - d))/sqrt(-c - d)