3.1017 \(\int \frac{1}{x \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

-(ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/Sqrt[a])

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Rubi [A]  time = 0.0475632, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

-(ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/Sqrt[a])

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Rubi in Sympy [A]  time = 5.42323, size = 22, normalized size = 0.88 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**2+a)**(1/2),x)

[Out]

-atanh(sqrt(a + b*x**2)/sqrt(a))/sqrt(a)

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Mathematica [A]  time = 0.0273563, size = 31, normalized size = 1.24 \[ \frac{\log (x)-\log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

(Log[x] - Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/Sqrt[a]

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Maple [A]  time = 0.006, size = 29, normalized size = 1.2 \[ -{1\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a)^(1/2),x)

[Out]

-1/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

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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/(sqrt(b*x^2 + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279876, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{a}}, -\frac{\arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.47171, size = 19, normalized size = 0.76 \[ - \frac{\operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2+a)**(1/2),x)

[Out]

-asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a)

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GIAC/XCAS [A]  time = 0.265499, size = 30, normalized size = 1.2 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*x),x, algorithm="giac")

[Out]

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