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

Optimal. Leaf size=40 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{1-c^4 x^4}}{c x \sqrt{\frac{1}{c^2 x^2}+1}}\right )}{c} \]

[Out]

-(ArcTanh[Sqrt[1 - c^4*x^4]/(c*Sqrt[1 + 1/(c^2*x^2)]*x)]/c)

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Rubi [A]  time = 0.275145, antiderivative size = 44, normalized size of antiderivative = 1.1, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{x \sqrt{\frac{1}{c^2 x^2}+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 + 1/(c^2*x^2)]*x*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 + c^2*x^2])

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Rubi in Sympy [A]  time = 20.0079, size = 56, normalized size = 1.4 \[ - \frac{x \sqrt{1 + \frac{1}{c^{2} x^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{- c^{4} x^{4} + 1}}{c \sqrt{x^{2} + \frac{1}{c^{2}}}} \right )}}{c \sqrt{x^{2} + \frac{1}{c^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+1/c**2/x**2)**(1/2)/(-c**4*x**4+1)**(1/2),x)

[Out]

-x*sqrt(1 + 1/(c**2*x**2))*atanh(sqrt(-c**4*x**4 + 1)/(c*sqrt(x**2 + c**(-2))))/
(c*sqrt(x**2 + c**(-2)))

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Mathematica [A]  time = 0.0767345, size = 79, normalized size = 1.98 \[ \frac{x \sqrt{\frac{1}{c^2 x^2}+1} \left (\log \left (c^2 x^3+x\right )-\log \left (c^2 x^2+\sqrt{c^2 x^2+1} \sqrt{1-c^4 x^4}+1\right )\right )}{\sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + 1/(c^2*x^2)]*x*(Log[x + c^2*x^3] - Log[1 + c^2*x^2 + Sqrt[1 + c^2*x^2]
*Sqrt[1 - c^4*x^4]]))/Sqrt[1 + c^2*x^2]

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Maple [C]  time = 0.083, size = 101, normalized size = 2.5 \[ -{\frac{x{\it csgn} \left ({c}^{-1} \right ) }{ \left ({c}^{2}{x}^{2}+1 \right ) c}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}\sqrt{-{c}^{4}{x}^{4}+1}\ln \left ( 2\,{\frac{1}{x{c}^{2}} \left ({\it csgn} \left ({c}^{-1} \right ) c\sqrt{-{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}}}}+1 \right ) } \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+1/c^2/x^2)^(1/2)/(-c^4*x^4+1)^(1/2),x)

[Out]

-((c^2*x^2+1)/c^2/x^2)^(1/2)*x*(-c^4*x^4+1)^(1/2)*csgn(1/c)*ln(2*(csgn(1/c)*c*(-
1/c^2*(c^2*x^2-1))^(1/2)+1)/x/c^2)/(c^2*x^2+1)/(-1/c^2*(c^2*x^2-1))^(1/2)/c

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Maxima [A]  time = 0.762961, size = 42, normalized size = 1.05 \[ -\frac{\log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/(c^2*x^2) + 1)/sqrt(-c^4*x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.292123, size = 162, normalized size = 4.05 \[ -\frac{\log \left (\frac{c^{2} x^{2} + \sqrt{-c^{4} x^{4} + 1} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) - \log \left (-\frac{c^{2} x^{2} - \sqrt{-c^{4} x^{4} + 1} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/(c^2*x^2) + 1)/sqrt(-c^4*x^4 + 1),x, algorithm="fricas")

[Out]

-1/2*(log((c^2*x^2 + sqrt(-c^4*x^4 + 1)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(
c^2*x^2 + 1)) - log(-(c^2*x^2 - sqrt(-c^4*x^4 + 1)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x
^2)) + 1)/(c^2*x^2 + 1)))/c

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{1 + \frac{1}{c^{2} x^{2}}}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+1/c**2/x**2)**(1/2)/(-c**4*x**4+1)**(1/2),x)

[Out]

Integral(sqrt(1 + 1/(c**2*x**2))/sqrt(-(c*x - 1)*(c*x + 1)*(c**2*x**2 + 1)), x)

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GIAC/XCAS [A]  time = 0.299391, size = 78, normalized size = 1.95 \[ \frac{{\left ({\rm ln}\left (\sqrt{2} + 1\right ) -{\rm ln}\left (\sqrt{2} - 1\right ) -{\rm ln}\left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) +{\rm ln}\left (-\sqrt{-c^{2} x^{2} + 1} + 1\right )\right )}{\left | c \right |}{\rm sign}\left (x\right )}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/(c^2*x^2) + 1)/sqrt(-c^4*x^4 + 1),x, algorithm="giac")

[Out]

1/2*(ln(sqrt(2) + 1) - ln(sqrt(2) - 1) - ln(sqrt(-c^2*x^2 + 1) + 1) + ln(-sqrt(-
c^2*x^2 + 1) + 1))*abs(c)*sign(x)/c^2