3.97 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x (d+e x)} \, dx\)

Optimal. Leaf size=46 \[ -\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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Rubi [A]  time = 0.190585, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)),x]

[Out]

-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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Rubi in Sympy [A]  time = 24.1138, size = 36, normalized size = 0.78 \[ - \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(e*x/sqrt(d**2 - e**2*x**2)) - atanh(sqrt(d**2 - e**2*x**2)/d)

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Mathematica [A]  time = 0.0242314, size = 46, normalized size = 1. \[ -\log \left (\sqrt{d^2-e^2 x^2}+d\right )-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)),x]

[Out]

-ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + Log[x] - Log[d + Sqrt[d^2 - e^2*x^2]]

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Maple [B]  time = 0.016, size = 137, normalized size = 3. \[{\frac{1}{d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{d\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{d}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{e\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289655, size = 73, normalized size = 1.59 \[ 2 \, \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + log(-(d - sqrt(-e^2*x^2 + d^2))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.28776, size = 65, normalized size = 1.41 \[ -\arcsin \left (\frac{x e}{d}\right ){\rm sign}\left (d\right ) -{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(x*e/d)*sign(d) - ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/ab
s(x))