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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]