3.21.82 \(\int \frac {1}{\sqrt {-d+e x} \sqrt {d^2-e^2 x^2}} \, dx\) [2082]

Optimal. Leaf size=53 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {-d+e x}}\right )}{\sqrt {d} e} \]

[Out]

arctan(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 = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {675, 211} \begin {gather*} \frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {e x-d}}\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]*ArcTan[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[-d + e*x])])/(Sqrt[d]*e)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[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} \tan ^{-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 = 0.09, size = 54, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {2} \tan ^{-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]*ArcTan[(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 = 0.62, size = 63, normalized size = 1.19

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-e x -d}\, \sqrt {2}}{2 \sqrt {d}}\right )}{\sqrt {e x -d}\, \sqrt {-e x -d}\, e \sqrt {d}}\)	\(63\)

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*x-d)^(1/2)/e/d^(1/2)*(-e^2*x^2+d^2)^(1/2)*2^(1/2)*arctan(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 = 2.82, size = 144, normalized size = 2.72 \begin {gather*} \left [\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{d}} e^{\left (-1\right )} \log \left (\frac {2 \, \sqrt {2} \sqrt {-x^{2} e^{2} + d^{2}} \sqrt {x e - d} d \sqrt {-\frac {1}{d}} - x^{2} e^{2} - 2 \, d x e + 3 \, d^{2}}{x^{2} e^{2} - 2 \, d x e + d^{2}}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} e^{2} + d^{2}} \sqrt {x e - d} \sqrt {d}}{x^{2} e^{2} - d^{2}}\right ) e^{\left (-1\right )}}{\sqrt {d}}\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)*sqrt(-1/d)*e^(-1)*log((2*sqrt(2)*sqrt(-x^2*e^2 + d^2)*sqrt(x*e - d)*d*sqrt(-1/d) - x^2*e^2 - 2*d*
x*e + 3*d^2)/(x^2*e^2 - 2*d*x*e + d^2)), sqrt(2)*arctan(sqrt(2)*sqrt(-x^2*e^2 + d^2)*sqrt(x*e - d)*sqrt(d)/(x^
2*e^2 - d^2))*e^(-1)/sqrt(d)]

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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 = 1.00, size = 49, normalized size = 0.92 \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 {e\,x-d}} \,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)*(e*x - d)^(1/2)),x)

[Out]

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

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