3.2070 \(\int \frac{1}{\sqrt{d+e x} \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]

[Out]

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

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Rubi [A]  time = 0.0736239, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.8134, size = 48, normalized size = 0.92 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{d^{2} - e^{2} x^{2}}}{2 \sqrt{d} \sqrt{d + e x}} \right )}}{\sqrt{d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0419587, size = 52, normalized size = 1. \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 58, normalized size = 1.1 \[ -{\frac{\sqrt{2}}{e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(e*x+d)^(1/2)*(-e^2*x^2+d^2)^(1/2)/(-e*x+d)^(1/2)/e*2^(1/2)/d^(1/2)*arctanh(1
/2*(-e*x+d)^(1/2)*2^(1/2)/d^(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(1/(sqrt(-e^2*x^2 + d^2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221973, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (-\frac{e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt{2} \sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d} \sqrt{d} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt{d} e}, \frac{\sqrt{2} \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{1}{d}}}\right )}{e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*sqrt(2)*log(-(e^2*x^2 - 2*d*e*x + 2*sqrt(2)*sqrt(-e^2*x^2 + d^2)*sqrt(e*x +
d)*sqrt(d) - 3*d^2)/(e^2*x^2 + 2*d*e*x + d^2))/(sqrt(d)*e), sqrt(2)*sqrt(-1/d)*
arctan(sqrt(2)*sqrt(-e^2*x^2 + d^2)*sqrt(e*x + d)/((e^2*x^2 - d^2)*sqrt(-1/d)))/
e]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*sqrt(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*sqrt(e*x + d)), x)