3.66 \(\int \frac{1}{\sqrt{-1+a^{2 x}}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]

[Out]

ArcTan[Sqrt[-1 + a^(2*x)]]/Log[a]

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Rubi [A]  time = 0.0268088, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-1 + a^(2*x)],x]

[Out]

ArcTan[Sqrt[-1 + a^(2*x)]]/Log[a]

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Rubi in Sympy [A]  time = 1.78313, size = 14, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\sqrt{a^{2 x} - 1} \right )}}{\log{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(a**(2*x) - 1))/log(a)

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Mathematica [A]  time = 0.0120483, size = 17, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[-1 + a^(2*x)]]/Log[a]

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Maple [A]  time = 0.01, size = 16, normalized size = 0.9 \[{\frac{1}{\ln \left ( a \right ) }\arctan \left ( \sqrt{-1+{a}^{2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+a^(2*x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.50184, size = 20, normalized size = 1.18 \[ \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{\log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(a^(2*x) - 1))/log(a)

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Fricas [A]  time = 0.251136, size = 20, normalized size = 1.18 \[ \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{\log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(a^(2*x) - 1))/log(a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a^{2 x} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a**(2*x) - 1), x)

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GIAC/XCAS [A]  time = 0.221577, size = 20, normalized size = 1.18 \[ \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{{\rm ln}\left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(a^(2*x) - 1))/ln(a)