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

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

[Out]

Sqrt[-1 + c*x]*Sqrt[1 + c*x] + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]

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Rubi [A]  time = 0.198525, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \sqrt{c x-1} \sqrt{c x+1}+\tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + c*x]*Sqrt[1 + c*x] + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]

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Rubi in Sympy [A]  time = 9.2897, size = 34, normalized size = 0.85 \[ \sqrt{c x - 1} \sqrt{c x + 1} + \operatorname{atan}{\left (\sqrt{c x - 1} \sqrt{c x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x - 1)*sqrt(c*x + 1) + atan(sqrt(c*x - 1)*sqrt(c*x + 1))

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Mathematica [A]  time = 0.0540141, size = 42, normalized size = 1.05 \[ \sqrt{c x-1} \sqrt{c x+1}-\tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]

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Maple [A]  time = 0.024, size = 53, normalized size = 1.3 \[{1\sqrt{cx-1}\sqrt{cx+1} \left ( \sqrt{{c}^{2}{x}^{2}-1}-\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*((c^2*x^2-1)^(1/2)-arctan(1/(c^2*x
^2-1)^(1/2)))

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Maxima [A]  time = 1.52933, size = 34, normalized size = 0.85 \[ \sqrt{c^{2} x^{2} - 1} - \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241092, size = 127, normalized size = 3.18 \[ -\frac{c^{2} x^{2} - \sqrt{c x + 1} \sqrt{c x - 1} c x - 2 \,{\left (c x - \sqrt{c x + 1} \sqrt{c x - 1}\right )} \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 1}{c x - \sqrt{c x + 1} \sqrt{c x - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 31.06, size = 148, normalized size = 3.7 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), 1/(c*
*2*x**2))/(4*pi**(3/2)) - meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5
/4, 3/2), (0,)), 1/(c**2*x**2))/(4*pi**(3/2)) + I*meijerg(((-1, -3/4, -1/2, -1/4
, 0, 1), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), exp_polar(2*I*pi)/(c**2*x**2)
)/(4*pi**(3/2)) + I*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4), (0, 1/2
, 1/2, 0)), exp_polar(2*I*pi)/(c**2*x**2))/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.219661, size = 54, normalized size = 1.35 \[ \sqrt{c x + 1} \sqrt{c x - 1} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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