3.21.81 \(\int \frac {1}{\sqrt {d+e x} \sqrt {d^2-e^2 x^2}} \, dx\) [2081]

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]

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

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {675, 214} \begin {gather*} -\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} \end {gather*}

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))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 675

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {d^2-e^2 x^2}} \, dx &=(2 e) \text {Subst}\left (\int \frac {1}{-2 d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\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}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 52, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.55, size = 58, normalized size = 1.12

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {-e x +d}\, \sqrt {2}}{2 \sqrt {d}}\right )}{\sqrt {e x +d}\, \sqrt {-e x +d}\, e \sqrt {d}}\)	\(58\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.09, size = 141, normalized size = 2.71 \begin {gather*} \left [\frac {\sqrt {2} e^{\left (-1\right )} \log \left (-\frac {x^{2} e^{2} - 2 \, d x e + 2 \, \sqrt {2} \sqrt {-x^{2} e^{2} + d^{2}} \sqrt {x e + d} \sqrt {d} - 3 \, d^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )}{2 \, \sqrt {d}}, -\sqrt {2} \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} e^{2} + d^{2}} \sqrt {x e + d} d \sqrt {-\frac {1}{d}}}{x^{2} e^{2} - d^{2}}\right ) e^{\left (-1\right )}\right ] \end {gather*}

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, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \sqrt {d + e x}}\, dx \end {gather*}

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 [A]
time = 0.86, size = 53, normalized size = 1.02 \begin {gather*} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x e + d}}{2 \, \sqrt {-d}}\right )}{\sqrt {-d}} - \frac {\sqrt {2} \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right )}{\sqrt {-d}}\right )} e^{\left (-1\right )} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {d^2-e^2\,x^2}\,\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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