3.263 \(\int \frac{1}{c (a-d)-(b-c) x^2} \, dx\)

Optimal. Leaf size=50 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]

[Out]

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

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Rubi [A]  time = 0.104129, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.6212, size = 39, normalized size = 0.78 \[ \frac{\operatorname{atanh}{\left (\frac{x \sqrt{b - c}}{\sqrt{c} \sqrt{a - d}} \right )}}{\sqrt{c} \sqrt{a - d} \sqrt{b - c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(x*sqrt(b - c)/(sqrt(c)*sqrt(a - d)))/(sqrt(c)*sqrt(a - d)*sqrt(b - c))

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Mathematica [A]  time = 0.0372223, size = 50, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{c-b}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{c-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[{1{\it Artanh} \left ({ \left ( b-c \right ) x{\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \right ){\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(a-d)-(b-c)*x^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232153, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \,{\left (a b c - a c^{2} -{\left (b c - c^{2}\right )} d\right )} x + \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}{\left ({\left (b - c\right )} x^{2} + a c - c d\right )}}{{\left (b - c\right )} x^{2} - a c + c d}\right )}{2 \, \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}}, -\frac{\arctan \left (-\frac{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d} x}{a c - c d}\right )}{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.03231, size = 104, normalized size = 2.08 \[ - \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (- a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} - c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210557, size = 78, normalized size = 1.56 \[ -\frac{\arctan \left (\frac{b x - c x}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}}\right )}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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