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

Optimal. Leaf size=37 \[ -\frac{\tanh ^{-1}\left (\frac{x \sqrt{c-d}}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}} \]

[Out]

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

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Rubi [A]  time = 0.0688485, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{\tanh ^{-1}\left (\frac{x \sqrt{c-d}}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}} \]

Antiderivative was successfully verified.

[In]  Int[(-c - d + (c - d)*x^2)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.58743, size = 31, normalized size = 0.84 \[ - \frac{\operatorname{atanh}{\left (\frac{x \sqrt{c - d}}{\sqrt{c + d}} \right )}}{\sqrt{c - d} \sqrt{c + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-c-d+(c-d)*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.0289556, size = 44, normalized size = 1.19 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{c-d}}{\sqrt{-c-d}}\right )}{\sqrt{-c-d} \sqrt{c-d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-c - d + (c - d)*x^2)^(-1),x]

[Out]

ArcTan[(Sqrt[c - d]*x)/Sqrt[-c - d]]/(Sqrt[-c - d]*Sqrt[c - d])

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Maple [A]  time = 0.008, size = 33, normalized size = 0.9 \[ -{1{\it Artanh} \left ({ \left ( c-d \right ) x{\frac{1}{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-c-d+(c-d)*x^2),x)

[Out]

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

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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(1/((c - d)*x^2 - c - d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233139, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \,{\left (c^{2} - d^{2}\right )} x -{\left ({\left (c - d\right )} x^{2} + c + d\right )} \sqrt{c^{2} - d^{2}}}{{\left (c - d\right )} x^{2} - c - d}\right )}{2 \, \sqrt{c^{2} - d^{2}}}, -\frac{\arctan \left (\frac{\sqrt{-c^{2} + d^{2}} x}{c + d}\right )}{\sqrt{-c^{2} + d^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c - d)*x^2 - c - d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.600821, size = 87, normalized size = 2.35 \[ \frac{\sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} \log{\left (- c \sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} - d \sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} + x \right )}}{2} - \frac{\sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} \log{\left (c \sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} + d \sqrt{\frac{1}{\left (c - d\right ) \left (c + d\right )}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-c-d+(c-d)*x**2),x)

[Out]

sqrt(1/((c - d)*(c + d)))*log(-c*sqrt(1/((c - d)*(c + d))) - d*sqrt(1/((c - d)*(
c + d))) + x)/2 - sqrt(1/((c - d)*(c + d)))*log(c*sqrt(1/((c - d)*(c + d))) + d*
sqrt(1/((c - d)*(c + d))) + x)/2

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GIAC/XCAS [A]  time = 0.209535, size = 45, normalized size = 1.22 \[ \frac{\arctan \left (\frac{c x - d x}{\sqrt{-c^{2} + d^{2}}}\right )}{\sqrt{-c^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c - d)*x^2 - c - d),x, algorithm="giac")

[Out]

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